Physical quantities — Paper 5 · A Level Physics 9702

1 It is useful to know how the speed of an object is affected by its size when it moves through liquid in a confined space. In a laboratory this can be modelled by dropping small steel balls through oil. It is suggested that the terminal velocity v is related to the radius r of a steel ball by the equation v = kr 2 where k is a constant. Design a laboratory experiment to investigate whether v is related to r as indicated in the above equation. You should draw a diagram showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) how the radius of the steel ball would be measured, (c) how the terminal velocity of the steel ball in oil would be measured, (d) the control of variables, (e) how the data would be analysed, (f) any safety precautions that you would take. [15] Examiner’s Use Diagram … … … … … … … … … … … … … … … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … … …

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks)

This question in 9702/51 May/June 2007

1 A student wishes to measure the resistivity of glass. A teacher suggests that its resistivity is For of the order of 106 Ω m which is very large. Examiner’s Use Resistivity ρ is defined by the equation RA ρ = l where R is resistance, A is cross-sectional area and l is the length of the material. The student is given a number of sheets of glass of the same thickness and of different areas. Design a laboratory experiment to determine the resistivity of glass. You should draw a diagram showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) how the glass would be connected to the circuit, (c) the measurements that would be taken, (d) the control of variables, (e) how the data would be analysed, (f) any safety precautions that you would take. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … … … …

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 A is the independent variable or vary A. [1] P2 R is the dependent variable or determine R for different A. [1] P3 Keep the temperature (of glass) constant. Do not allow “controlled variable”. [1] Methods of data collection (5 marks) M1 Basic circuit diagram. [1] Ammeter and voltmeter with power supply, or ohmmeter without power supply, or bridge methods. M2 Correct orientation of glass between electrodes – largest cross-sectional area. [1] M3 A distance (thickness) measured using a micrometer/vernier scale/vernier callipers. [1] M4 Method of determining area perpendicular to current flow. [1] Distances measured and multiplied together. This mark may only be scored if it is clear that the correct dimensions are being used. M5 Method of determining resistance. [1] Ohmmeter. R = V/I justified. Description of balancing bridge with correct equation. Method of analysis (2 marks) A1 A2 R against 1/A ρ = gradient/l R against l /A ρ = gradient 1/A against R or 1/R against A ρ = 1/(gradient × l) l /A against R or l /R against A ρ = 1/gradient lg R against lg A ρ = 10l × y-intercept [2] Safety considerations (1 mark) S1 Relevant safety precaution related to: [1] EHT power supply (>100 V) – switch off before changing circuit/use of rubber gloves; or handling glass – wear (thick) gloves. Additional detail (4 marks) D1/2/3/4 Relevant points might include [4] Calculation of typical resistance of glass using value of resistivity given. Range of ammeter or ohmmeter with reasoning. Use of EHT or power supply >1000 V or microammeter/galvanometer. Take many readings of thickness and average. Good contact between circuit and glass e.g. metal plates, foil, conducting putty. Metal plates/foil/conducting putty to cover all of the cross-sectional area in use. Method of securing good contact between circuit and glass, e.g. g clamps, weights. Clean/dry the glass. [Total: 15] GCE A/AS LEVEL – May/June 2008 9702 05

This question in 9702/51 May/June 2008

1 A student wishes to determine the Young modulus E of wood from the period of oscillation of For a loaded wooden rule, as shown in Fig. 1.1. Examiner’s Use fixed end load l Fig. 1.1 An equation relating the period of oscillation T to the overhanging length l of the rule is kl 3 T 2 = . E The constant k is given by 16π 2M k = wd 3 where M is the mass of the load, w is the width of the rule and d is the thickness of the rule. Design a laboratory experiment to determine the Young modulus of wood. You should draw a diagram showing the arrangement of your equipment. In your account, you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) how to analyse the data, (e) how to determine E, (f) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … … …

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 Vary l or l is the independent variable [Allow M] [1] P2 Determine the period T (for each l [M]) or T is the dependent variable [1] P3 M is kept constant [l is kept constant] [1] Methods of data collection (5 marks) M1 Diagram showing the cantilever is fixed e.g. g-clamp & bench, retort stand & clamp [1] M2 Many oscillations repeated to determine average T (n [ 10 or t [ 10 s for stopwatch) [1] M3 Weigh M using balance [1] M4 Measure w and d and measure/record l [1] M5 Use of vernier caliper/micrometer to measure d and/or w [1] Method of analysis (2 marks) A1 Appropriate graph plotted i.e. T 2 against l 3; [T 2 against M] or lg T against lg l or lg M [1] 16π 2 M k 16 π 23l A2 E = 3 = E = 3 wd × gradient gradient wd × gradient k k Allow logarithmic solutions e.g. E = 2 xy - intercept = y - intercept [1] 10 100 Safety considerations (1 mark) S1 Relevant safety precaution related to the use of loads [1] e.g. cushion/sand in case load falls, keep feet away, keep distance from experiment. Additional detail (4 marks) D Relevant points might include [4] 1. Use same rule or keep w and/or d constant. 2. Repeat measurements of d and/or w along rule and average. 3. Discussion of use of motion sensor e.g. orientation or light gates with detail. 4. Use small amplitude or small angle oscillations (to ensure equation is valid). 5. Method of securing load to rule e.g. with tape/glue. 6. Discussion of magnitude of load: large enough to make T large enough. 7. Use of fiducial marker to help to time. 8. Start timing after oscillations have settled. Do not allow vague use of computers/light gates, video cameras, dataloggers. [Total: 15] GCE A/AS LEVEL – May/June 2009 9702 05

This question in 9702/51 May/June 2009

1 A hammer is often used to force a nail into wood. The faster the hammer moves, the deeper For the nail moves into the wood. Examiner’s Use This can be represented in a laboratory by a mass falling vertically onto a nail. It is suggested that the depth d of the nail in the wood (see Fig. 1.1) is related to the velocity v of the mass at the instant it hits the nail by the equation d = kv n where k and n are constants. nail d wood Fig. 1.1 Design a laboratory experiment to investigate the relationship between v and d so as to determine a value for n. You should draw a diagram showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 Vary v and measure d, or v is the independent variable and d is the dependent variable [1] P2 Keep mass constant [1] P3 Keep the wood constant/keep same type of nails [1] Methods of data collection (5 marks) M1 Diagram of apparatus showing mass falling onto centre of nail [1] M2 Change height of falling mass (to change v) [1] M3 Measurement(s) from which v can be determined, e.g. measure height fallen; light gate(s) connected to timer/data-logger measuring time, and ticker tape/motion sensor. Do not award stopwatch methods. [1] M4 Appropriate equation to determine v (the velocity of the mass at the instant it hits the nail) [1] M5 Detail on measuring d ; subtract, needle, mark nail, depth gauge [1] Method of analysis (2 marks) A1 Plot a graph of log d against log v [1] A2 n = gradient [1] Safety considerations (1 mark) S1 Precaution linked to falling masses, e.g. keep well away/sand trays [1] Additional detail (4 marks) D 1/2/3/4 Relevant points might include [4] 1. Method to create a large d, e.g. large mass, thin nails, soft wood 2. Use of a guide for falling mass/guide for nail 3. Use of vernier scale to measure d 4. Repeat experiment and determine an average 5. Use different part of wood for each test 6. Method to make nail vertical e.g. set square 7. Discussion / preliminary experiment about thin nails going totally into wood 8. lg d = n lg v + lg k [Total: 15] GCE AS/A LEVEL – May/June 2010 9702 51

This question in 9702/51 May/June 2010

1 A student wishes to determine the resistivity of aluminium. For Examiner’s The resistivity ρ of a conductor is defined as Use RA ρ = l for a conductor of resistance R, cross-sectional area A and length l. Fig. 1.1 shows the typical dimensions of a strip of aluminium of lengths c, d and t. The resistivity of aluminium is about 10–8 Ωm. t = 1 mm d 1 cm c 1 m Fig. 1.1 (not to scale) Design a laboratory experiment to determine the resistivity of aluminium using this strip. The usual apparatus of a school laboratory is available, including a metal cutter. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 c, d or A is the independent variable and R is the dependent variable or vary c, d or A and measure R. [1] P2 If c varied then (t and) d or A kept constant, if d varied then (t and) c or A kept constant, if A varied then c or d kept constant. [1] P3 Keep temperature constant. [1] Methods of data collection (5 marks) M1 Circuit diagram to measure resistance. [1] M2 Use micrometer screw gauge to measure d or t. (Allow digital or vernier callipers) [1] M3 Measure c with a ruler/metre rule. [1] M4 Method of making contact with the strip e.g. use electrodes of at least same dimension as c or d or t or conducting paint methods. Do not allow crocodile clips, unless it is clear that the whole area of the end of the strip is covered. [1] M5 Method to determine resistance. [1] Method of analysis (2 marks) A1 Plot a graph of R against c, 1/d or 1/A depending on orientation. Other alternatives possible, e.g. R against 1/c depending on orientation [1] A2 Must be consistent with A1: ρ = A × gradient or t × gradient/c [1] Other alternatives possible, e.g. ρ = d × gradient/t Safety considerations (1 mark) S1 Reference sharp edges or cutting metals, e.g. wear gloves. [1] Additional detail (4 marks) D1/2/3/4 Relevant points might include [4] 1. Insulate aluminium strip 2. Take many readings of t or d and average 3. Use a protective resistor/circuit designed to reduce current 4. Rearrange equation to determine graph using c, d and t or A 5. Determine typical resistance of aluminium strip 6. Likely meter range of ammeter/voltmeter/ohmmeter 7. Detail on cutting strip e.g. mark using set square Do not allow vague computer methods. [Total: 15] GCE A/AS LEVEL – October/November 2010 9702 53

This question in 9702/53 Oct/Nov 2010

1 When light is incident on the front of a photocell, an e.m.f. is generated in the photocell. For Examiner’s A student wishes to investigate the effect of adding various thicknesses of glass in front of Use a photocell. This may be carried out in the laboratory by varying the number of identical thin glass sheets between a light source and the front of the photocell. It is suggested that the e.m.f. V is related to the number n of glass sheets by the equation V = V0e –αnt where t is the thickness of one sheet, α is the absorption coefficient of glass and V0 is the e.m.f. for n = 0. Design a laboratory experiment to determine the absorption coefficient of glass. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 n is the independent variable and V is the dependent variable or vary n and measure V [1] P2 Keep distance from light to photocell constant [1] P3 Keep intensity of light constant. Allow constant voltage across lamp/current through lamp/brightness. Do not allow ‘same lamp/output’. [1] Methods of data collection (5 marks) M1 Labelled diagram of apparatus: lamp, glass sheet and photocell in line. [1] M2 Voltmeter connected to photocell. Penalise unworkable photocell circuit. [1] M3 Use micrometer (screw gauge) to measure thickness of glass sheet. [1] M4 Take many readings of thickness and average. [1] M5 Perform experiment in a dark room or shield apparatus. [1] Method of analysis (2 marks) A1 Plot a graph of ln V against n. Allow ln V against nt [1] A2 α = (–)gradient / t. (ln V against nt then α = (–)gradient) [1] Safety considerations (1 mark) S Reasoned method to prevent burns from hot source, e.g. use gloves Reasoned method to prevent eye damage from bright/intense source, e.g. shield lamp/ dark glasses/do not look at source directly Reasoned method to prevent cuts from glass e.g. use gloves. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use small distance/high intensity to gain large reading. 2 Method to check output of lamp is constant e.g. measure current through/p.d. across lamp/regularly check V0 with no glass. 3 Reasoned method to ensure output of lamp is constant e.g. workable circuit diagram with variable resistor or variable power supply. 4 Clean sheets of glass before use. 5 Direction of light is perpendicular to glass sheets/constant orientation. 6 ln V = –αnt + ln V0. 7 Further safety consideration. Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2011 9702 51

This question in 9702/51 May/June 2011

1 A student wishes to investigate projectile motion. For Examiner’s A small ball is rolled with velocity v along a horizontal surface. When the ball reaches the Use end of the horizontal surface, it falls and lands on a lower horizontal surface. The vertical displacement of the ball is p and the horizontal displacement of the ball is q, as shown in Fig 1.1. v p q Fig. 1.1 It is suggested that gq 2 = 2pv 2 where g is the acceleration of free fall. Design a laboratory experiment to investigate how q is related to p and how v may be determined from the results. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 p is the independent variable or vary p [1] P2 q is the dependent variable or measure q [1] P3 Keep (horizontal) velocity (v) constant [1] Methods of data collection (5 marks) M1 Labelled diagram of apparatus including method to vary p. [1] M2 Method to determine position of ball on surface e.g. carbon paper/dye/video/sand. [1] M3 Use ruler/caliper to measure p and/or q. [1] M4 Method to ensure velocity is constant e.g. releasing ball from same height on a track/ spring loaded device or impulse device set to a constant value. [1] M5 Method to ensure that the moved surface remains horizontal, e.g. spirit level/check height at different places. [1] Method of analysis (2 marks) A1 q2 against p q against p p against q2 p against q [1] g × gradient g g 1 g A2 v = v = gradient × v = v = × [1] 2 2 2× gradient gradient 2 Allow valid logarithmic graph e.g. lg q against lg p, lg 2p, lg 2p/g and valid calculation of v from y-intercept. Safety considerations (1 mark) S Reasoned method to prevent ball rolling on floor e.g. box below/storage box for balls/ sand box. Reasoned method to prevent ball causing injury e.g. goggles/safety screen [1] Additional detail (4 marks) D Relevant points might include [4] 1 Method to ensure that velocity of ball is horizontal only when it reaches table, e.g. curved track. 2 Ensure that the ball leaves the table at 90°, e.g. set square/protractor on upper surface. 3 Detail on measuring q – location of landing position e.g. centre of crater/start of track. 4 Detail on determining location of zero position for p and q e.g. set square, plumb line. 5 Detail on method of determining position of ball e.g. slow motion playback including scale. 6 Take many readings of q for each p and average. 7 Straight line through the origin shows that p is proportional to q2/relationship is valid – this mark may only be awarded when A1 is given. 8 Use of high density ball to minimise the effects of air resistance. Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2011 9702 52

This question in 9702/52 May/June 2011

1 A student wishes to investigate projectile motion. For Examiner’s A small ball is rolled with velocity v along a horizontal surface. When the ball reaches the Use end of the horizontal surface, it falls and lands on a lower horizontal surface. The vertical displacement of the ball is p and the horizontal displacement of the ball is q, as shown in Fig 1.1. v p q Fig. 1.1 It is suggested that gq 2 = 2pv 2 where g is the acceleration of free fall. Design a laboratory experiment to investigate how q is related to p and how v may be determined from the results. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P1 p is the independent variable or vary p [1] P2 q is the dependent variable or measure q [1] P3 Keep (horizontal) velocity (v) constant [1] Methods of data collection (5 marks) M1 Labelled diagram of apparatus including method to vary p. [1] M2 Method to determine position of ball on surface e.g. carbon paper/dye/video/sand. [1] M3 Use ruler/caliper to measure p and/or q. [1] M4 Method to ensure velocity is constant e.g. releasing ball from same height on a track/ spring loaded device or impulse device set to a constant value. [1] M5 Method to ensure that the moved surface remains horizontal, e.g. spirit level/check height at different places. [1] Method of analysis (2 marks) A1 q2 against p q against p p against q2 p against q [1] g × gradient g g 1 g A2 v = v = gradient × v = v = × [1] 2 2 2× gradient gradient 2 Allow valid logarithmic graph e.g. lg q against lg p, lg 2p, lg 2p/g and valid calculation of v from y-intercept. Safety considerations (1 mark) S Reasoned method to prevent ball rolling on floor e.g. box below/storage box for balls/ sand box. Reasoned method to prevent ball causing injury e.g. goggles/safety screen [1] Additional detail (4 marks) D Relevant points might include [4] 1 Method to ensure that velocity of ball is horizontal only when it reaches table, e.g. curved track. 2 Ensure that the ball leaves the table at 90°, e.g. set square/protractor on upper surface. 3 Detail on measuring q – location of landing position e.g. centre of crater/start of track. 4 Detail on determining location of zero position for p and q e.g. set square, plumb line. 5 Detail on method of determining position of ball e.g. slow motion playback including scale. 6 Take many readings of q for each p and average. 7 Straight line through the origin shows that p is proportional to q2/relationship is valid – this mark may only be awarded when A1 is given. 8 Use of high density ball to minimise the effects of air resistance. Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2011 9702 53

This question in 9702/53 May/June 2011

2 A scientist is observing some of the moons orbiting the planet Jupiter. For Examiner’s Use Fig. 2.1 For six different moons, the scientist records the distance r from the centre of Jupiter and the period T of the orbit. Question 2 continues on the next page. It is suggested that T and r are related by the equation For Examiner’s T 2 = kr 3 Use where k is a constant. (a) A graph is plotted of lg T on the y-axis against lg r on the x-axis. Determine the value of the gradient and express the y-intercept in terms of k. gradient = … y-intercept = … [1] (b) Values of r and T are given in Fig. 2.2. r / 106 m T / 103 s lg (r / m) lg (T / s) 129 24 ± 4 181 42 ± 4 422 154 ± 8 671 304 ± 8 1070 590 ± 15 1880 1420 ± 15 Fig. 2.2 Calculate and record values of lg (r / m) and lg (T / s) in Fig. 2.2. Include the absolute uncertainties in lg (T / s). [3] (c) (i) Plot a graph of lg (T / s) against lg (r / m). Include error bars for lg (T / s). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] For 6.4 Examiner’s Use 6.2 lg (T / s) 6.0 5.8 5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 lg (r / m)

10 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Part Mark Expected Answer Additional Guidance (a) A1 1.5 or 3/2 Ignore y-intercept (incorrect y-intercept will be penalised in (d)(i)). (b) T1 Allow a mixture of decimal places. 8.111 or 8.1106 4.38 T2 T1 must be table values. 8.258 or 8 2577 4.62 T2 must be a minimum of 2 d.p. 8.625 or 8.6253 5.188 (5.19) Ignore rounding errors. 8.827 or 8.8267 5.483 (5.48) 9.029 or 9.0294 5.771 (5.77) 9.274 or 9.2742 6.152 (6.15) U1 From ± 0.07 or ± 0.08, to ± 0.005 Allow more than one significant figure. (c) (i) G1 Six points plotted correctly Must be within half a small square. Do not allow ‘blobs’ (more than half a small square). Ecf allowed from table. U2 Error bars in lg T plotted correctly All error bars to be plotted. Must be accurate to less than half a small square. (c) (ii) G2 Line of best fit If points are plotted correctly then lower end of line should pass between (8.0, 4.20) and (8.0, 4.28) and upper end of line should pass between (9.4, 6.32) and (9.4, 6.38). Allow ecf from points plotted incorrectly – examiner judgement. G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Steepest or shallowest possible line Examiner judgement on worst acceptable that passes through all the error bars. line. Lines must cross. Mark scored only if error bars are plotted. (c) (iii) C1 Gradient of best fit line The triangle used should be at least half the length of the drawn line. Check the read offs. Work to half a small square. Do not penalise POT. U3 Uncertainty in gradient Method of determining absolute uncertainty Difference in worst gradient and gradient. (c) (iv) C2 Negative y-intercept Must be negative. FOX does not score. Check substitution into y = mx + c. Allow ecf from (c)(iii). U4 Uncertainty in y-intercept Uses worst gradient and point on WAL. Do not check calculation. FOX does not score. (d) (i) C3 Method to determine k k= 102 × y-intercept [k is about 10–16, if FOX 108] U5 Uncertainty in k Best k – worst k using y-intercept. Allow ecf for method from (c)(iv). (d) (ii) C4 M between 2.36 × 1026 and 2.36 × 1028 Must be in range. Allow between 2.4 × 1026 given to 2 or 3 s.f. and 2.4 × 1028 for 2 s.f. [Total: 15] GCE AS/A LEVEL – October/November 2011 9702 51 Uncertainties in Question 2 (c) (iii) Gradient [E3] Uncertainty = gradient of line of best fit – gradient of worst acceptable line Uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (iv) [E4] Uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line Uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (d) (i) [E5] Uncertainty = best k –worst k

This question in 9702/51 Oct/Nov 2011

2 Different light-emitting diodes (LEDs) can emit electromagnetic radiation of different For wavelengths. Examiner’s Use A student uses the circuit of Fig. 2.1 to investigate the minimum potential difference required to cause an LED to emit its characteristic wavelength. V Fig. 2.1 For different LEDs emitting radiation of wavelength λ, the minimum potential difference V is recorded. Question 2 continues on the next page. It is suggested that V and λ are related by the equation For Examiner’s hc Use = B + eV λ where c is the speed of light in a vacuum, e is the elementary charge, h is the Planck constant and B is a constant. (a) A graph is plotted of V on the y-axis against 1/λ on the x-axis. Determine expressions for the gradient and y-intercept in terms of B, c, e and h. gradient = … y-intercept = … [1] (b) Values of λ and V are given in Fig. 2.2. λ/ 10–9 m V / V 950 0.60 ± 0.05 875 0.70 ± 0.05 655 1.20 ± 0.05 560 1.55 ± 0.05 505 1.80 ± 0.05 430 2.25 ± 0.05 Fig. 2.2 Calculate and record values of (1 / λ) / 106 m–1 in Fig. 2.2. [2] (c) (i) Plot a graph of V / V against (1 / λ) / 106 m–1. Include error bars for V. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] 2.4 For Examiner’s Use V / V 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (1 / h) / 106 m–1

9 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Part Mark Expected Answer Additional Guidance (a) A1 Gradient = hc/e Note y-intercept must be negative y-intercept = – B/e (b) T1 1/λ / 106 m–1 Appropriate column heading T2 Must be values in table. 1.05 or 1.053 A mixture of 3 s.f. and 4 s.f. is allowed. 1.14 or 1.143 1.53 or 1.527 1.79 or 1.786 1.98 or 1.980 2.33 or 2.326 (c) (i) G1 Six points plotted correctly Must be within half a small square. Penalise ‘blobs’. Ecf allowed from table. U1 All error bars in V / V plotted Do not allow near misses correctly. (c) (ii) G2 Line of best fit If points are plotted correctly then lower end of line should pass between (1.12, 0.7) and (1.16,0.7) and upper end of line should pass between (2.32, 2.25) and (2.34, 2.25). Allow ecf from points plotted incorrectly – examiner judgement. G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Steepest or shallowest possible line Should pass from top of top error bar to that passes through all the error bottom of bottom error bar or bottom of bars. top error bar to top of bottom error bar. Mark scored only if error bars are plotted. (c) (iii) C1 Gradient of best fit line The triangle used should be at least half the length of the drawn line. Check the read offs. Work to half a small square. Do not penalise POT. Should be about 1.3 × 10 –6. U2 Uncertainty in gradient Method of determining absolute uncertainty Difference in worst gradient and gradient. [± 0.08] (c) (iv) C2 y-intercept Must be negative Expect to see point substituted into y = mx + c FOX does not score. Do not penalise POT. Should be between –0.72 and –0.86 U3 Method of determining uncertainty in Difference in worst y-intercept and y-intercept y-intercept. [Should be about ± 0.14]. FOX does not score. Allow ecf from (c)(iv). GCE AS/A LEVEL – October/November 2012 9702 51 (d) (i) C3 h in the range 6.77 × 10–34 to Gradient must be used. Penalise 1 s.f. or >3 7.14 × 10–34 and given to 2 or 3 s.f. significant figures h = gradient × e/c = gradient × 5.33 × 10–28 Allow 6.8 × 10–34 to 7.1 × 10–34 to 2 s.f. (d) (ii) U4 Percentage uncertainty in h ∆m ∆h × 100 or × 100 m h [should be about 6%] (e) C4 B = –e × y-intercept Ignore ‘–’ signs. y-intercept must be used and J or C V or V C but allow ecf from FOX. Should be between 1.16 × 10–19 J and 1.37 × 10–19 J. If FOX 8.3 × 10–20 J U5 Absolute uncertainty in B Uncertainty = best B – worst B = ∆y-intercept × e [Total: 15] Uncertainties in Question 2 (c) (iii) Gradient [U2] Uncertainty = gradient of line of best fit – gradient of worst acceptable line Uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (iv) [U3] Uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line Uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) (d) (ii) [U4] ∆m Percentage uncertainty = × 100 m ∆h 1 (max h − min h ) Percentage uncertainty = × 100 = 2 × 100 h h (e) [U5] Absolute uncertainty = best B – worst B Absolute uncertainty = ∆y-intercept × e ∆c Absolute uncertainty = × B c

This question in 9702/51 Oct/Nov 2012

1 As a bar magnet is dropped through a coil, an e.m.f. is induced in the coil. The maximum For e.m.f. E is induced as the magnet leaves the coil with speed v. Examiner’s Use It is suggested that E is directly proportional to v. Design a laboratory experiment to test the relationship between E and v. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P v is the independent variable or vary v. [1] P E is the dependent variable or measure E. [1] P Keep the number of turns on the coil constant. [1] Methods of data collection (5 marks) M1 Labelled diagram showing magnet falling vertically through coil. [1] M2 Voltmeter or c.r.o. connected to the coil. Allow voltage sensor connected to datalogger. [1] M3 Method to change speed e.g. change height. [1] M4 Measurements to determine v. Use metre rule to measure distance magnet falls to the bottom of the coil or metre rule/ruler to measure length of coil or ruler to measure length of the magnet. [Allow timing instrument to measure the time of the fall from the start to the bottom of the coil.] [1] M5 Method of determining v corresponding to appropriate distance e.g. v = √2gh or v=2h/t (for height method) or v = L/t for length of magnet or coil and by stopwatch, timer or lightgate(s) connected to datalogger. [Allow v = gt for timing fall to bottom of coil.] [1] Method of analysis (2 marks) A Plot a graph of E against v. [Allow lg E against lg v] [1] A Relationship valid if straight line through origin. [1] [If lg-lg then straight line with gradient = (+)1 (ignore reference to y-intercept)] Safety considerations (1 mark) S Keep away from falling magnet/use sand tray/cushion to catch magnet. [1] Additional detail (4 marks) D1/2/3/4 Relevant points might include [4] Use coil with large number of turns/drop magnet from large heights/strong magnet 1 Detailed use of datalogger/storage oscilloscope to determine maximum E; allow video camera including slow motion play back 2 Use same magnet or magnet of same strength. 3 Use of short magnet so that v is (nearly) constant 4 Use short/thin coil so that v is (nearly) constant 5 Use a non-metallic vertical guide/tube 6 Method to support vertical coil or guide/tube 7 Repeat experiment for each v and average Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – October/November 2012 9702 52

This question in 9702/52 Oct/Nov 2012

2 Different light-emitting diodes (LEDs) can emit electromagnetic radiation of different For wavelengths. Examiner’s Use A student uses the circuit of Fig. 2.1 to investigate the minimum potential difference required to cause an LED to emit its characteristic wavelength. V Fig. 2.1 For different LEDs emitting radiation of wavelength λ, the minimum potential difference V is recorded. Question 2 continues on the next page. It is suggested that V and λ are related by the equation For Examiner’s hc Use = B + eV λ where c is the speed of light in a vacuum, e is the elementary charge, h is the Planck constant and B is a constant. (a) A graph is plotted of V on the y-axis against 1/λ on the x-axis. Determine expressions for the gradient and y-intercept in terms of B, c, e and h. gradient = … y-intercept = … [1] (b) Values of λ and V are given in Fig. 2.2. λ/ 10–9 m V / V 950 0.60 ± 0.05 875 0.70 ± 0.05 655 1.20 ± 0.05 560 1.55 ± 0.05 505 1.80 ± 0.05 430 2.25 ± 0.05 Fig. 2.2 Calculate and record values of (1 / λ) / 106 m–1 in Fig. 2.2. [2] (c) (i) Plot a graph of V / V against (1 / λ) / 106 m–1. Include error bars for V. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] 2.4 For Examiner’s Use V / V 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 (1 / h) / 106 m–1

9 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Part Mark Expected Answer Additional Guidance (a) A1 Gradient = hc/e Note y-intercept must be negative y-intercept = – B/e (b) T1 1/λ / 106 m–1 Appropriate column heading T2 Must be values in table. 1.05 or 1.053 A mixture of 3 s.f. and 4 s.f. is allowed. 1.14 or 1.143 1.53 or 1.527 1.79 or 1.786 1.98 or 1.980 2.33 or 2.326 (c) (i) G1 Six points plotted correctly Must be within half a small square. Penalise ‘blobs’. Ecf allowed from table. U1 All error bars in V / V plotted Do not allow near misses correctly. (c) (ii) G2 Line of best fit If points are plotted correctly then lower end of line should pass between (1.12, 0.7) and (1.16,0.7) and upper end of line should pass between (2.32, 2.25) and (2.34, 2.25). Allow ecf from points plotted incorrectly – examiner judgement. G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Steepest or shallowest possible line Should pass from top of top error bar to that passes through all the error bottom of bottom error bar or bottom of bars. top error bar to top of bottom error bar. Mark scored only if error bars are plotted. (c) (iii) C1 Gradient of best fit line The triangle used should be at least half the length of the drawn line. Check the read offs. Work to half a small square. Do not penalise POT. Should be about 1.3 × 10 –6. U2 Uncertainty in gradient Method of determining absolute uncertainty Difference in worst gradient and gradient. [± 0.08] (c) (iv) C2 y-intercept Must be negative Expect to see point substituted into y = mx + c FOX does not score. Do not penalise POT. Should be between –0.72 and –0.86 U3 Method of determining uncertainty in Difference in worst y-intercept and y-intercept y-intercept. [Should be about ± 0.14]. FOX does not score. Allow ecf from (c)(iv). GCE AS/A LEVEL – October/November 2012 9702 52 (d) (i) C3 h in the range 6.77 × 10–34 to Gradient must be used. Penalise 1 s.f. or >3 7.14 × 10–34 and given to 2 or 3 s.f. significant figures h = gradient × e/c = gradient × 5.33 × 10–28 Allow 6.8 × 10–34 to 7.1 × 10–34 to 2 s.f. (d) (ii) U4 Percentage uncertainty in h ∆m ∆h × 100 or × 100 m h [should be about 6%] (e) C4 B = –e × y-intercept Ignore ‘–’ signs. y-intercept must be used and J or C V or V C but allow ecf from FOX. Should be between 1.16 × 10–19 J and 1.37 × 10–19 J. If FOX 8.3 × 10–20 J U5 Absolute uncertainty in B Uncertainty = best B – worst B = ∆y-intercept × e [Total: 15] Uncertainties in Question 2 (c) (iii) Gradient [U2] Uncertainty = gradient of line of best fit – gradient of worst acceptable line Uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (iv) [U3] Uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line Uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) (d) (ii) [U4] ∆m Percentage uncertainty = × 100 m ∆h 1 (max h − min h ) Percentage uncertainty = × 100 = 2 × 100 h h (e) [U5] Absolute uncertainty = best B – worst B Absolute uncertainty = ∆y-intercept × e ∆c Absolute uncertainty = × B c

This question in 9702/52 Oct/Nov 2012

2 A trolley is attached to springs, as shown in Fig 2.1. When the trolley is displaced and then For released, the trolley oscillates. Examiner’s Use stand stand card light gate trolley springs springs Fig. 2.1 A student investigates how the maximum speed v of a trolley varies with the total mass M of the trolley. The maximum speed is determined using the time t taken for the card to pass through a light gate connected to an electronic timer. The length of the card is 5.0 ± 0.1 cm. Question 2 continues on the next page. It is suggested that v and M are related by the equation For Examiner’s k Use v = A M where A is the initial displacement and k is the spring constant of the springs. (a) A graph is plotted of v 2 on the y-axis against 1/M on the x-axis. Determine an expression for the gradient in terms of A and k. gradient = … [1] (b) Values of M and t are given in Fig. 2.2. M / kg t / s (1/M ) / kg–1 v 2 / m2 s–2 0.75 0.046 ± 0.002 1.25 0.058 ± 0.002 1.75 0.068 ± 0.002 2.25 0.078 ± 0.002 2.75 0.086 ± 0.002 3.25 0.092 ± 0.002 Fig. 2.2 Calculate and record values of (1/M ) / kg–1 and v 2 / m2 s–2 in Fig. 2.2. Include the absolute uncertainties in v 2. [3] (c) (i) Plot a graph of v 2 / m2 s–2 against (1/M ) / kg–1. Include error bars for v 2. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] 1.4 For Examiner’s Use 1.3 1.2 v 2 / m2 s–2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (1 / M ) / kg–1

10 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Part Mark Expected Answer Additional Guidance (a) A1 Gradient = kA2 (b) T1 T1 must be values in 1/M. Ignore row 2. T2 1.3 or 1.33 1.2 0.8(0)(0)(0) 0.74 T2 must be to 2 s.f. or 3 s.f. 0.571 or 0.54 or 0.55 0.5714 0.444 or 0.41 or 0.411 0.4444 or 0.410 0.364 or 0.34 0.3636 0.308 or 0.29 or 0.30 0.3077 U1 From ± 0.2 or ± 0.15 to ± 0.02 or Allow more than one significant figure. ± 0.03 Do not allow ± 0.1 for row 1. (c) (i) G1 Six points plotted correctly Must be within half a small square. Penalise ‘blobs’. Ecf allowed from table. U2 All error bars in v2 plotted Must be accurate within half a small square. correctly (c) (ii) G2 Line of best fit There must be a balance of points about the line of best fit – examiner judgement. Allow ecf from points plotted incorrectly. G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Should Steepest or shallowest possible pass from top of top error bar to bottom of bottom line that passes through all the error bar or bottom of top error bar to top of bottom error bars. error bar. Mark scored only if error bars are plotted. (c) (iii) C1 Gradient of best fit line The triangle used should be at least half the length of the drawn line. Check the read offs. Work to half a small square. Do not penalise POT. Should be about 0.9. U3 Uncertainty in gradient Method of determining absolute uncertainty. Difference in worst gradient and gradient. (d) (i) C2 k = gradient / A2 Should be about 22. = gradient / 0.04 C3 N m–1 Allow kg s–2 (d) (ii) U4 Percentage uncertainty in k ∆m ∆A ∆m × 100 + 2 × × 100 = × 100 + 5% m A m GCE AS/A LEVEL – October/November 2012 9702 53 (e) C4 v in the range 0.534 to 0.559 For 2 s.f. 0.53 to 0.56 and given to 2 or 3 s.f. U5 Uncertainty in v [Total: 15] Uncertainties in Question 2 (c) (iii) Gradient [U3] Uncertainty = gradient of line of best fit – gradient of worst acceptable line Uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (d) (ii) [U4] ∆m ∆A ∆m Percentage uncertainty = × 100 + 2 × × 100 = × 100 + 5% m A m max m Maximum k = (min A ) 2 min m Minimum k = (max A ) 2 ∆k 1 (max k − min k ) Percentage uncertainty = × 100 = 2 × 100 k k (e) [U5] ∆A 1 ∆k Percentage uncertainty = × 100 + 2 × × 100 A k Absolute uncertainty = v × percentage uncertainty/100 max k Maximum v = max A× 0.75 min k Minimum v = min A × 0.75 Absolute uncertainty = max v – v or v – min v or 12 (max v − min v )

This question in 9702/53 Oct/Nov 2012

1 A student is investigating the flow of water through a horizontal tube. For Examiner’s The rate Q (volume per unit time) at which water flows through a tube depends on the Use pressure difference per unit length across the tube. The student has the use of a metal can with two holes. A narrow horizontal tube goes through the hole in the side of the can. The can is continuously supplied with water from a tap. The level of water in the can is kept constant by the position of a wide vertical tube which passes through the hole in the bottom of the can as shown in Fig. 1.1. Both tubes may be moved along the holes. to tap metal can h water flow rate Q l wide tube narrow tube overflow Fig. 1.1 It is suggested that the relationship between the flow rate Q of water through the narrow horizontal tube and the vertical height h is 2 π ρ g h d 4 Q = l η where ρ is the density of water, g is the acceleration of free fall, d is the internal diameter of the tube, l is the length of the tube and η is a constant. Design a laboratory experiment to test the relationship between Q and h and determine a value for η . You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P h is the independent variable or vary h. [1] P Q is the dependent variable or measure Q (allow t). [1] P Keep l constant. [1] Methods of data collection (5 marks) M Labelled diagram of apparatus: including labelled measuring cylinder/calibrated beaker to receive water. (Measurement may be credited in the text.) [1] M Vary position of vertical/larger tube. [1] M Measure h and l with a rule/caliper. [1] M Measure d with a travelling microscope or vernier calipers. [1] M Measure t with stopwatch. [1] Method of analysis (2 marks) A Plot a graph of Q against h. [Allow lg Q against lg h] [1] 2πρ gd 4 A η = l × gradient Must include gradient and η must be subject of formula. [1] Safety considerations (1 mark) S Reasoned method to prevent spills, e.g. use tray/sink/cloths on floor. Reasoned method to prevent injury when adjusting metal/glass tubes by wearing protective gloves. Mark may not be scored from the diagram. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Repeat experiment for same h and average 2 Method to determine the density of water including method to measure mass and volume and equation 3 Take many readings of d and average 4 Relationship is valid if straight line passing through origin [if lg-lg graph allow straight line with gradient = 1] 5 Method to check that tube is horizontal 6 Detail on measuring h to the centre of the horizontal tube e.g. add radius of tube 7 Keep temperature of water constant Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2013 9702 51

This question in 9702/51 May/June 2013

1 An aluminium ring is placed on a coil with the rod of a metal retort stand passing through For their centres, as shown in Fig. 1.1. Examiner’s Use aluminium ring when alternating current applied to coil h metal retort stand aluminium ring with zero current applied to coil coil Fig. 1.1 When an alternating current of frequency f is applied to the coil, the ring rises until it is in equilibrium at a height h above the coil. It is suggested that the relationship between h and f is h = kf n where k and n are constants. Design a laboratory experiment to test the relationship between h and f and determine values for k and n. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P f is the independent variable or vary f. [1] P h is the dependent variable or measure h. [1] P Keep current in coil constant. [1] Methods of data collection (5 marks) M Labelled diagram of apparatus: including working circuit with a.c. supply, coil, stand and ring. [1] M Signal generator (or variable frequency power generator). [1] M Measure h with a rule/calipers. [1] M Measure f using oscilloscope/read off signal generator. [1] M Measure h from opposite sides of the ring (and determine mean)/wait for ring to stabilise. [1] Method of analysis (2 marks) A Plot a graph of lg h against lg f. (Allow ln – ln graphs.) [1] A k = 10y-intercept and n = gradient. [1] Safety considerations (1 mark) S Reasoned method to prevent the coil overheating – switch off when not in use. Or reasoned method to prevent injury from hot coil – do not touch hot coil, use gloves. Do not allow “small currents”. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use iron/steel retort stand 2 Method to keep current constant 3 Method to determine period from oscilloscope using time-base 4 When using oscilloscope, f = 1/T 5 Use coil of many turns/large current to have measurable heights 6 Logarithmic equation e.g. lg h = n lg f + lg k 7 Relationship is valid if a straight line is produced Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – October/November 2013 9702 51

This question in 9702/51 Oct/Nov 2013

1 An aluminium ring is placed on a coil with the rod of a metal retort stand passing through For their centres, as shown in Fig. 1.1. Examiner’s Use aluminium ring when alternating current applied to coil h metal retort stand aluminium ring with zero current applied to coil coil Fig. 1.1 When an alternating current of frequency f is applied to the coil, the ring rises until it is in equilibrium at a height h above the coil. It is suggested that the relationship between h and f is h = kf n where k and n are constants. Design a laboratory experiment to test the relationship between h and f and determine values for k and n. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram For Examiner’s Use … … … … … … … … … … … … … … For … Examiner’s Use … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional For problem data collection analysis considerations detail Examiner’s Use

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P f is the independent variable or vary f. [1] P h is the dependent variable or measure h. [1] P Keep current in coil constant. [1] Methods of data collection (5 marks) M Labelled diagram of apparatus: including working circuit with a.c. supply, coil, stand and ring. [1] M Signal generator (or variable frequency power generator). [1] M Measure h with a rule/calipers. [1] M Measure f using oscilloscope/read off signal generator. [1] M Measure h from opposite sides of the ring (and determine mean)/wait for ring to stabilise. [1] Method of analysis (2 marks) A Plot a graph of lg h against lg f. (Allow ln – ln graphs.) [1] A k = 10y-intercept and n = gradient. [1] Safety considerations (1 mark) S Reasoned method to prevent the coil overheating – switch off when not in use. Or reasoned method to prevent injury from hot coil – do not touch hot coil, use gloves. Do not allow “small currents”. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use iron/steel retort stand 2 Method to keep current constant 3 Method to determine period from oscilloscope using time-base 4 When using oscilloscope, f = 1/T 5 Use coil of many turns/large current to have measurable heights 6 Logarithmic equation e.g. lg h = n lg f + lg k 7 Relationship is valid if a straight line is produced Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – October/November 2013 9702 52

This question in 9702/52 Oct/Nov 2013

1 A ball rolls forwards and backwards on a curved track as shown in Fig. 1.1. flexible track ball Fig. 1.1 It is suggested that the period T of the oscillations is related to the radius r of the ball and the radius of curvature C of the track by the relationship 2 28p2 T = (C – r ) 5g where g is the acceleration of free fall. You are provided with a flexible track. Design a laboratory experiment to test the relationship between T and r. Explain how your results could be used to determine a value for C. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P r is the independent variable or vary r. [1] P T (or t) is the dependent variable or measure T (or t). [1] P Keep the radius of curvature (of the track) or C constant (or radius of track constant). Do not allow “use same track”. [1] Methods of data collection (5 marks) M Diagram showing ball in a (curved) track with supports for track, e.g. retort stands. Minimum of two labels (from ball, track, supports; not stopwatch, bench, micrometer). Supports making contact with track higher than ball / at least half way up. [1] M Measure time using stopwatch or light gates and timer or datalogger with motion sensor. Detail needed for video camera. [1] M Use many oscillations (at least 10 or at least 10 s of timing) and determine T = t / n. [1] M Measure diameter (radius) of ball with a micrometer / vernier calipers. Do not allow travelling microscope. [1] M radius = diameter / 2. [1] Method of analysis (2 marks) A Plot a graph of T 2 against r (or r against T 2) Do not allow log graphs. [1] 5 g y − intercept A C = y-intercept × = (or for r against T 2, C = y-intercept) [1] 2 gradient 28 π Safety considerations (1 mark) S Precaution linked to ball escaping on to floor, e.g. use barrier / safety screen / sand tray to prevent balls rolling on to floor. [1] GCE AS/A LEVEL – May/June 2014 9702 51 Additional detail (4 marks) D Relevant points might include [4] 1 Add weights to / G-clamp retort stands 2 Keep the material / density of the ball constant 3 Use of fiducial marker near centre of track / mark on the track 4 Clean track / balls. Do not allow oil the track. 5 Repeat measurements of t (for each ball) and average 6 Repeat measurement for d (or r) and average 7 Relationship is valid if straight line, provided plotted graph is correct 8 Relationship is valid if straight line not passing through origin or has an intercept, provided plotted graph is correct (any quoted expression must be correct, e.g. 28 π 2C y-intercept = ) 5 g Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2014 9702 51

This question in 9702/51 May/June 2014

1 A ball rolls forwards and backwards on a curved track as shown in Fig. 1.1. flexible track ball Fig. 1.1 It is suggested that the period T of the oscillations is related to the radius r of the ball and the radius of curvature C of the track by the relationship 2 28π2 T = (C – r ) 5g where g is the acceleration of free fall. You are provided with a flexible track. Design a laboratory experiment to test the relationship between T and r. Explain how your results could be used to determine a value for C. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P r is the independent variable or vary r. [1] P T (or t) is the dependent variable or measure T (or t). [1] P Keep the radius of curvature (of the track) or C constant (or radius of track constant). Do not allow “use same track”. [1] Methods of data collection (5 marks) M Diagram showing ball in a (curved) track with supports for track, e.g. retort stands. Minimum of two labels (from ball, track, supports; not stopwatch, bench, micrometer). Supports making contact with track higher than ball / at least half way up. [1] M Measure time using stopwatch or light gates and timer or datalogger with motion sensor. Detail needed for video camera. [1] M Use many oscillations (at least 10 or at least 10 s of timing) and determine T = t / n. [1] M Measure diameter (radius) of ball with a micrometer / vernier calipers. Do not allow travelling microscope. [1] M radius = diameter / 2. [1] Method of analysis (2 marks) A Plot a graph of T 2 against r (or r against T 2) Do not allow log graphs. [1] 5 g y − intercept A C = y-intercept × = (or for r against T 2, C = y-intercept) [1] 2 gradient 28 π Safety considerations (1 mark) S Precaution linked to ball escaping on to floor, e.g. use barrier / safety screen / sand tray to prevent balls rolling on to floor. [1] GCE AS/A LEVEL – May/June 2014 9702 53 Additional detail (4 marks) D Relevant points might include [4] 1 Add weights to / G-clamp retort stands 2 Keep the material / density of the ball constant 3 Use of fiducial marker near centre of track / mark on the track 4 Clean track / balls. Do not allow oil the track. 5 Repeat measurements of t (for each ball) and average 6 Repeat measurement for d (or r) and average 7 Relationship is valid if straight line, provided plotted graph is correct 8 Relationship is valid if straight line not passing through origin or has an intercept, provided plotted graph is correct (any quoted expression must be correct, e.g. 28 π 2C y-intercept = ) 5 g Do not allow vague computer methods. [Total: 15] GCE AS/A LEVEL – May/June 2014 9702 53

This question in 9702/53 May/June 2014

1 A student investigates the power dissipated by a lamp connected to a model wind turbine as shown in Fig. 1.1. wind turbine lamp Fig. 1.1 The power P dissipated in the lamp depends on the angle θ between the axis of the turbine and the direction of the wind, as shown by the top view in Fig. 1.2. turbine wind direction e Fig. 1.2 It is suggested that P = k cosθ where k is a constant. Design a laboratory experiment to test the relationship between P and θ and determine a value for k. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P (cos) θ is the independent variable, or vary (cos) θ. [1] P P is the dependent variable, or measure P. [1] P Keep the speed of the air constant. Allow keep power to the fan/hairdryer constant. [1] Methods of data collection (5 marks) M Labelled diagram showing method to produce air flow in line with turbine. Method of producing “wind” must be labelled. [1] M Circuit connecting turbine to lamp with ammeter and voltmeter connected correctly. No additional power supplies in the lamp circuit. [1] M P = IV. Do not allow I2R or V2/R unless it is clear that R is determined from V / I. Allow wattmeter or joule meter and stopwatch. [1] M Measure angle with protractor or use rule to measure appropriate distances. [1] M Ensure that there are no other draughts or airflows. [1] Method of analysis (2 marks) A Plot a graph of P against cos θ. [1] A k = gradient. [1] Safety considerations (1 mark) S Precaution linked to avoiding air flow entering eyes or avoid moving blades. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use of large wind speed to gain measurable readings. 2 Use of low wattage/low resistance lamp or turbine with low friction. 3 Additional detail on measuring (cos) θ – correct angle must be determined. 4 Wait until airflow/turbine/meter readings constant. 5 Avoid turbulence or reflection of air flow. 6 Ensure distance from fan to turbine is constant. 7 Relationship is valid if the graph is a straight line passing through the origin. 8 Method to check that wind speed is constant. Do not allow vague computer methods. [Total: 15]

This question in 9702/51 Oct/Nov 2014

1 A student investigates the power dissipated by a lamp connected to a model wind turbine as shown in Fig. 1.1. wind turbine lamp Fig. 1.1 The power P dissipated in the lamp depends on the angle θ between the axis of the turbine and the direction of the wind, as shown by the top view in Fig. 1.2. turbine wind direction e Fig. 1.2 It is suggested that P = k cosθ where k is a constant. Design a laboratory experiment to test the relationship between P and θ and determine a value for k. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P (cos) θ is the independent variable, or vary (cos) θ. [1] P P is the dependent variable, or measure P. [1] P Keep the speed of the air constant. Allow keep power to the fan/hairdryer constant. [1] Methods of data collection (5 marks) M Labelled diagram showing method to produce air flow in line with turbine. Method of producing “wind” must be labelled. [1] M Circuit connecting turbine to lamp with ammeter and voltmeter connected correctly. No additional power supplies in the lamp circuit. [1] M P = IV. Do not allow I2R or V2/R unless it is clear that R is determined from V / I. Allow wattmeter or joule meter and stopwatch. [1] M Measure angle with protractor or use rule to measure appropriate distances. [1] M Ensure that there are no other draughts or airflows. [1] Method of analysis (2 marks) A Plot a graph of P against cos θ. [1] A k = gradient. [1] Safety considerations (1 mark) S Precaution linked to avoiding air flow entering eyes or avoid moving blades. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use of large wind speed to gain measurable readings. 2 Use of low wattage/low resistance lamp or turbine with low friction. 3 Additional detail on measuring (cos) θ – correct angle must be determined. 4 Wait until airflow/turbine/meter readings constant. 5 Avoid turbulence or reflection of air flow. 6 Ensure distance from fan to turbine is constant. 7 Relationship is valid if the graph is a straight line passing through the origin. 8 Method to check that wind speed is constant. Do not allow vague computer methods. [Total: 15]

This question in 9702/52 Oct/Nov 2014

1 A thin card is inserted between two separate iron cores. A coil is wound around one core as shown in Fig. 1.1. thin card iron cores Fig. 1.1 A current in the coil may induce an e.m.f. in another coil wound on the other core. The induced e.m.f. V depends on the thickness t of the card. A student suggests that V = V0e–σt where V0 is the induced e.m.f. without card between the cores and σ is a constant. Design a laboratory experiment to test the relationship between V and t and determine the value of σ. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P t is the independent variable or vary t. [1] P V is the dependent variable or measure V. [1] P Keep the current (in the primary coil) constant. [1] Methods of data collection (5 marks) M Diagram showing two independent labelled coils wound on iron cores. [1] M AC power supply / signal generator connected to one coil. [1] M Voltmeter / oscilloscope connected to other coil in a workable circuit. [1] M Measure thickness of card using micrometer / vernier calipers / digital calipers. [1] M Method to keep current constant – rheostat (or variable power supply) and ammeter correctly positioned in primary circuit and explained. Diagram and text required. [1] Method of analysis (2 marks) M Plot a graph of ln V against t (allow lg V against t) or ln V / V0 against t [1] M σ = – gradient [1] Safety considerations (1 mark) S Precaution linked to hot coil(s) e.g. switch off when not in use / do not touch / wear gloves. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Use large current (in primary coil)/large number of turns on the secondary to achieve measurable V (allow more turns on secondary than primary). 2 Keep frequency of power supply constant or keep the number of turns on each coil constant. 3 Use laminated cores or use insulated wire for turns. 4 Repeat measurements of t and average. 5 Measurement of V0 stating that no card is present. 6 Logarithmic equation: ln V = ln V0 – σt 7 Relationship is valid if the graph is a straight line with y–intercept = ln V0 8 Discussion of compression of card / measure t when secured. Do not allow vague computer methods. [Total: 15]

This question in 9702/53 Oct/Nov 2014

2 A student is investigating the performance of a motor vehicle. The vehicle is driven at a constant speed v on a test track, as shown in Fig. 2.1. Fig. 2.1 The performance P of the vehicle is the distance travelled per unit volume of fuel, measured in kilometres per litre (km l –1). This is obtained from the vehicle’s computer system. The experiment is repeated for different speeds. It is suggested that P and v are related by the equation P = kv m where k and m are constants. (a) A graph is plotted of lg P on the y-axis against lg v on the x-axis. Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of v and P are given in Fig. 2.2. v / km h–1 P / km l –1 lg (v / km h–1) lg (P / km l –1) 50 20.5 ± 0.5 61 16.0 ± 0.5 71 13.0 ± 0.5 80 11.0 ± 0.5 90 9.5 ± 0.5 99 8.0 ± 0.5 Fig. 2.2 Calculate and record values of lg (v / km h–1) and lg (P / km l –1) in Fig. 2.2. Include the absolute uncertainties in lg (P / km l –1). [3] (c) (i) Plot a graph of lg (P / km l –1) against lg (v / km h–1). Include error bars for lg (P / km l –1). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] 1.35 1.30 1.25 lg (P / km l –1) 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.85 0.80 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 lg (v / km h–1)

10 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Mark Expected Answer Additional Guidance (a) A1 gradient = m y-intercept = lg k (b) T1 Allow a mixture of significant figures. T2 1.70 or 1.699 1.312 or 1.3118 T1 (first column) and T2 (second column) must be values in table. 1.79 or 1.785 1.204 or 1.2041 1.85 or 1.851 1.114 or 1.1139 1.90 or 1.903 1.041 or 1.0414 1.95 or 1.954 0.98 or 0.978 2.00 or 1.996 0.90 or 0.903 U1 From ±0.01 to ±0.03 Allow more than one significant figure. (c) (i) G1 Six points plotted correctly Must be within half a small square. Do not allow “blobs”. Ecf allowed from table. U2 Error bars in lg P plotted correctly All error bars to be plotted. Must be accurate to less than half a small square. (ii) G2 Line of best fit Upper end of line must pass between (1.75, 1.24) and (1.75, 1.255) and lower end of line must pass between (2.00, 0.900) and (2.00, 0.915). G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Steepest or shallowest possible Examiner judgement on worst acceptable line that passes through all the line. Lines must cross. Mark scored only if error bars. error bars are plotted. (iii) C1 Gradient of line of best fit Must be negative. The triangle used should be at least half the length of the drawn line. Check the read-offs. Work to half a small square. Do not penalise POT. (Should be about –1.35.) U3 Uncertainty in gradient Method of determining absolute uncertainty: difference in worst gradient and gradient. (iv) C2 y-intercept Check substitution into y = mx + c. Allow ecf from (c)(iii). (Should be about 4.) Do not allow read-off of false origin. U4 Uncertainty in y-intercept Uses worst gradient and point on worst acceptable line. Do not check calculation. Do not allow if false origin used. (d) (i) C3 k = 10y-intercept C4 m = gradient and given to 2 or 3 s.f. Must be negative. and in the range –1.30 to –1.44 Allow –1.3 or –1.4 (2 s.f.) (ii) U5 Percentage uncertainty in k Uncertainties in Question 2 (c) (iii) Gradient [U3] uncertainty = gradient of line of best fit – gradient of worst acceptable line uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (iv) [U4] uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) (d) (ii) [U5] max k = 10max y-intercept and min k = 10min y-intercept 1 (max k − min k ) max k − k k − min k 2 percentage uncertainty = × 100 = × 100 = × 100 k k k

This question in 9702/51 May/June 2015

2 A student is investigating the performance of a motor vehicle. The vehicle is driven at a constant speed v on a test track, as shown in Fig. 2.1. Fig. 2.1 The performance P of the vehicle is the distance travelled per unit volume of fuel, measured in kilometres per litre (km l –1). This is obtained from the vehicle’s computer system. The experiment is repeated for different speeds. It is suggested that P and v are related by the equation P = kv m where k and m are constants. (a) A graph is plotted of lg P on the y-axis against lg v on the x-axis. Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of v and P are given in Fig. 2.2. v / km h–1 P / km l –1 lg (v / km h–1) lg (P / km l –1) 50 20.5 ± 0.5 61 16.0 ± 0.5 71 13.0 ± 0.5 80 11.0 ± 0.5 90 9.5 ± 0.5 99 8.0 ± 0.5 Fig. 2.2 Calculate and record values of lg (v / km h–1) and lg (P / km l –1) in Fig. 2.2. Include the absolute uncertainties in lg (P / km l –1). [3] (c) (i) Plot a graph of lg (P / km l –1) against lg (v / km h–1). Include error bars for lg (P / km l –1). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer. gradient = … [2] 1.35 1.30 1.25 lg (P / km l –1) 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0.85 0.80 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 lg (v / km h–1)

10 marks

Mark scheme: 2 Analysis, conclusions and evaluation (15 marks) Mark Expected Answer Additional Guidance (a) A1 gradient = m y-intercept = lg k (b) T1 Allow a mixture of significant figures. T2 1.70 or 1.699 1.312 or 1.3118 T1 (first column) and T2 (second column) must be values in table. 1.79 or 1.785 1.204 or 1.2041 1.85 or 1.851 1.114 or 1.1139 1.90 or 1.903 1.041 or 1.0414 1.95 or 1.954 0.98 or 0.978 2.00 or 1.996 0.90 or 0.903 U1 From ±0.01 to ±0.03 Allow more than one significant figure. (c) (i) G1 Six points plotted correctly Must be within half a small square. Do not allow “blobs”. Ecf allowed from table. U2 Error bars in lg P plotted correctly All error bars to be plotted. Must be accurate to less than half a small square. (ii) G2 Line of best fit Upper end of line must pass between (1.75, 1.24) and (1.75, 1.255) and lower end of line must pass between (2.00, 0.900) and (2.00, 0.915). G3 Worst acceptable straight line. Line should be clearly labelled or dashed. Steepest or shallowest possible Examiner judgement on worst acceptable line that passes through all the line. Lines must cross. Mark scored only if error bars. error bars are plotted. (iii) C1 Gradient of line of best fit Must be negative. The triangle used should be at least half the length of the drawn line. Check the read-offs. Work to half a small square. Do not penalise POT. (Should be about –1.35.) U3 Uncertainty in gradient Method of determining absolute uncertainty: difference in worst gradient and gradient. (iv) C2 y-intercept Check substitution into y = mx + c. Allow ecf from (c)(iii). (Should be about 4.) Do not allow read-off of false origin. U4 Uncertainty in y-intercept Uses worst gradient and point on worst acceptable line. Do not check calculation. Do not allow if false origin used. (d) (i) C3 k = 10y-intercept C4 m = gradient and given to 2 or 3 s.f. Must be negative. and in the range –1.30 to –1.44 Allow –1.3 or –1.4 (2 s.f.) (ii) U5 Percentage uncertainty in k Uncertainties in Question 2 (c) (iii) Gradient [U3] uncertainty = gradient of line of best fit – gradient of worst acceptable line uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) (iv) [U4] uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) (d) (ii) [U5] max k = 10max y-intercept and min k = 10min y-intercept 1 (max k − min k ) max k − k k − min k 2 percentage uncertainty = × 100 = × 100 = × 100 k k k

This question in 9702/53 May/June 2015

1 A student is investigating the angle at which a glass cylinder containing oil topples, as shown in Fig. 1.1. q glass cylinder oil bench Fig. 1.1 A cylinder containing a mass m of oil can be tilted through a maximum angle φ from the vertical before it topples. It is suggested that the relationship between m and φ is 1 am = + b tan φ ρd 3 where d is the diameter of the cylinder, ρ is the density of the oil and a and b are constants. Design a laboratory experiment to test the relationship between φ and m. Explain how your results could be used to determine values for a and b. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P m is the independent variable, or vary m. [1] P (tan)φ is the dependent variable, or measure (tan) φ. [1] P Keep the temperature of the oil constant. [1] Methods of data collection (5 marks) M Labelled diagram showing labelled protractor positioned to determine φ for tilted cylinder. Allow distances marked to determine φ and use of a rule. [1] M Use of balance/scales to measure the mass of the oil/cylinder. [1] M Mass of oil = mass of (oil + cylinder) – mass of cylinder. [1] M Use of vernier calipers/micrometer/rule to measure d. [1] M Repeat each experiment for the same value of m and average φ. [1] Method of analysis (2 marks) 1 A Plot a graph of against m. tan φ m m m or (Allow d 3 ρ d 3 or ρ . Do not allow log-log graphs.) [1] A a = gradient ×ρd3 and b = y-intercept; must be consistent with suggested graph. [1] Safety considerations (1 mark) S Precaution linked to preventing spilling oil, e.g. use a tray/lid/cloth to absorb oil (do not allow just wiping or mopping) or precaution linked to preventing glass cylinder breaking, e.g. padding/cushion or use of gloves to prevent skin irritation (do not allow “because oil is slippery”). [1] Additional detail (4 marks) D Relevant points might include [4] 1 Repeat measurements of d in different directions and average 2 Use of video with slow motion/frame by frame playback to determine φ 3 Use large protractor to reduce percentage uncertainty or trigonometry relationship related to measurements to be taken 4 Use the same (diameter) cylinder (not “same size” but allow “same size and shape”) 5 Slowly/gently/gradually tilt cylinder of oil/use of rough surface (to prevent sliding) 6 Experimental method to determine density of oil and ρ = m / V 7 Relationship is valid if the graph is a straight line that does NOT pass through the origin / has an intercept; must be consistent with suggested graph Do not allow vague computer methods.

This question in 9702/51 Oct/Nov 2015

1 A beaker contains water and some metal blocks as shown in Fig. 1.1. water heater metal block metal block Fig. 1.1 A student uses an electrical heater to produce a particular temperature increase in the water. It is suggested that the electrical energy E supplied to the heater is related to the mass m of metal blocks by the relationship E = am + b where a and b are constants. Design a laboratory experiment to test the relationship between E and m. Explain how your results could be used to determine values for a and b. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to (a) the procedure to be followed, (b) the measurements to be taken, (c) the control of variables, (d) the analysis of the data, (e) the safety precautions to be taken. [15] Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … Defining the Methods of Method of Safety Additional problem data collection analysis considerations detail

15 marks

Mark scheme: 1 Planning (15 marks) Defining the problem (3 marks) P m is the independent variable and E is the dependent variable or vary m and measure E. Do not allow time. [1] P Keep the temperature change of water constant. Allow two specified temperatures. Do not allow “keep temperature constant”. [1] P Keep the mass or volume of water constant. [1] Methods of data collection (5 marks) M Labelled diagram including labelled thermometer with bulb in water and at least one other label. [1] M Workable circuit diagram to determine E: power supply, heater and ammeter and voltmeter, or joulemeter or wattmeter. [1] M Method to determine change in temperature: measure initial temperature, measure final temperature and subtract, or measure initial temperature and specific temperature change. [1] M Use balance/scales to measure mass of blocks. [1] M Stir water (so that metal is in thermal equilibrium). [1] Method of analysis (2 marks) A Plot a graph of E against m. Do not allow log–log graphs. [1] A a = gradient and b = y-intercept; must be consistent with suggested graph. [1] Safety considerations (1 mark) S Precaution linked to hot heater/water, e.g. use gloves or use tongs for hot blocks. Do not allow goggles. [1] Additional detail (4 marks) D Relevant points might include [4] 1 Method to ensure that e.m.f. of the power supply is constant/current in heater is constant, e.g. adjust variable power supply/variable resistor to ensure p.d./current is constant 2 Keep the starting temperature of water/metal constant 3 Wait for water and metal temperatures to equalise 4 Add insulation to sides of beaker/lid (to prevent energy losses) 5 Use of timer and equation, e.g. E = Pt = ItV for candidate’s method 6 Use large temperature change to reduce percentage uncertainty 7 Relationship is valid if the graph is a straight line that does not pass through the origin Do not allow vague computer methods.

This question in 9702/53 Oct/Nov 2015

2 A student is investigating the motion of a small steel ball in cooking oil. A measure of the oil’s resistance to the ball’s motion is called viscosity. Viscosity has the units pascal second (Pa s). The student drops a ball into a cylinder of oil as shown in Fig. 2.1. ball cooking oil heat Fig. 2.1 The velocity of the ball is measured when it becomes constant and then the viscosity of the oil is determined. The experiment is repeated for different temperatures of oil. It is suggested that the viscosity η and the Celsius temperature θ are related by the equation η = p θ q where p and q are constants. (a) A graph is plotted of lg η on the y-axis against lg θ on the x-axis. Determine expressions for the gradient and the y-intercept. gradient = … y-intercept = … [1] (b) Values of θ and η are given in Fig. 2.2. θ/ °C η/ 10–3 Pa s lg (θ/ °C) lg (η/ 10–3 Pa s) 38 41 ± 1 46 32 ± 1 55 25 ± 1 64 20 ± 1 72 17 ± 1 79 14 ± 1 Fig. 2.2 Calculate and record values of lg (θ/ °C) and lg (η/ 10–3 Pa s) in Fig. 2.2. Include the absolute uncertainties in lg (η/ 10–3 Pa s). [2] (c) (i) Plot a graph of lg (η/ 10–3 Pa s) against lg (θ/ °C). Include error bars for lg (η/ 10–3 Pa s). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = q y-intercept = lg p 1 2(b) 1.58 or 1.580 1.61 or 1.613 1.66 or 1.663 1.51 or 1.505 1.74 or 1.740 1.40 or 1.398 1.81 or 1.806 1.30 or 1.301 1.86 or 1.857 1.23 or 1.230 1.90 or 1.898 1.15 or 1.146 1 absolute uncertainties in lg η: ± 0.01 to ± 0.03 1 2(c)(i) six points plotted correctly must be accurate to the nearest half small square diameter of points must be less than half a small square 1 error bars in lg η plotted correctly all error bars to be plotted total length of bar must be accurate to less than half a small square and symmetrical 1 2(c)(ii) line of best fit drawn points must be balanced do not allow line from top plot to bottom plot if points are plotted correctly then lower end of line should pass between (1.820, 1.275) and (1.835, 1.275) and upper end of line should pass between (1.640, 1.525) and (1.650, 1.525) 1 worst acceptable line drawn steepest or shallowest possible line mark scored only if all error bars are plotted 1 Question Answer Marks 2(c)(iii) gradient determined with clear substitution of data points into ∆y / ∆x; distance between data points must be at least half the length of the drawn line must be negative 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c 1 y-intercept of worst acceptable line determined by substitution into y = mx + c uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) no ECF from false origin method 1 2(d) -intercept 10y p = and given to 2 or 3 sf 1 q = gradient and q and p have correct power of ten from (c)(iii) and (c)(iv) 1 absolute uncertainty in p = intercept of WAL 10y p − − absolute uncertainty in q = uncertainty in gradient correct substitution of numbers must be seen 1 2(e) 100 p q θ = or ( ) lg 100 lg 2 intercept lg gradient p y q θ − − − = = correct substitution of numbers must be seen 1

This question in 9702/52 Feb/March 2019

1 When a light plastic ball is placed in a vertical column of moving air, the ball becomes stationary at a height h, as shown in Fig. 1.1. ball h moving air air blower Fig. 1.1 A student is using an air blower to create the vertical column of moving air. The student connects the motor of the air blower to a d.c. power supply. It is suggested that the relationship between the radius r of the ball and h is 4 r r 3 gh = PK 3 where g is the acceleration of free fall, P is the power of the motor and K is a constant. Design a laboratory experiment to test the relationship between r and h. Explain how your results could be used to determine a value for K. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: 1 Defining the problem r is the independent variable and h is the dependent variable or vary r and measure/determine h 1 keep P constant 1 Methods of data collection labelled diagram of workable experiment including: • labelled air blower • labelled ball vertically (by eye) above the blower • vertical (by eye) rule at least from top of blower to ball 1 circuit diagram to determine P, e.g. voltmeter and ammeter connected to motor and power supply or wattmeter connected to motor and power supply 1 use of micrometer/calipers/rule to measure diameter of ball 1 measure the distance between the top/middle/bottom of the ball and the top of the blower 1 Method of Analysis plot a graph of h against 1 / r3 (or lg h against lg r) 1 relationship valid if a straight line through (0,0) (for lg h against lg r straight line with gradient = –3) 1 4 gradient 3 g K P π × = (for lg h against lg r, intercept 4 10 3 y g K P − π × = ) 1 Question Answer Marks 1 Additional detail including safety considerations D1 use large box/tray to collect ball (to prevent ball rolling on floor/bouncing) or reasoned method to avoid draughts, e.g. switch off fans, close windows, use a screen 6 D2 method to determine P from correct circuit, e.g. P = I × V or use wattmeter to measure P D3 stand on bench with clamped rule vertically to measure vertical distance D4 method to ensure clamped rule to measure h is vertical, e.g. correctly positioned set square indicated at right angles between the rule and the horizontal surface or plumb line shown in appropriate position D5 r = d / 2 when diameter measured D6 repeat diameter measurement in different directions and find average D7 repeat experiment for each value of r and determine average h D8 method to determine h, e.g. h = reading top of ball – r – top of blower or h = reading bottom of ball + r – top of blower or h = (distance from top of ball to top of blower + distance from bottom of ball to top of blower) / 2 D9 wait for the ball to become stationary (vertically) D10 video (camera) shown level with elevated ball and description of playback frame-by-frame or slow motion

This question in 9702/51 Oct/Nov 2019

2 A student is investigating the oscillations of a mass attached to two springs connected in series, as shown in Fig. 2.1. springs mass Fig. 2.1 A stopwatch is used to measure the time t for 10 oscillations. The measurement of t is repeated and the average period T is determined. The experiment is repeated for different masses. It is suggested that T and mass M are related by the equation 2 r M q T = k where k is the spring constant of the two springs in series and q is a constant. (a) A graph is plotted of lg T on the y-axis against lg M on the x-axis. Determine expressions for the gradient and the y-intercept. gradient = … y-intercept = … [1] (b) Values of M, lg (M / g) and measurements of t are given in Fig. 2.2. M / g t / s t / s T / s lg (M / g) lg (T / s) 155 15.2 16.0 2.190 205 18.3 17.5 2.312 250 19.3 20.1 2.398 305 21.0 21.8 2.484 355 23.5 22.7 2.550 410 24.1 24.9 2.613 Fig. 2.2 Calculate and record values of T / s and lg (T / s) in Fig. 2.2. Include the absolute uncertainties in T / s and lg (T / s). [4] (c) (i) Plot a graph of lg (T / s) against lg (M / g). Include error bars for lg (T / s). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

11 marks

Mark scheme: 2(a) gradient = q and y-intercept = π  = π −     2 1 lg lg 2 lg 2 k k 1 2(b) T / s lg (T / s) 1.56 0.193 or 0.1931 1.79 0.253 or 0.2529 1.97 0.294 or 0.2945 2.14 0.330 or 0.3304 2.31 0.364 or 0.3636 2.45 0.389 or 0.3892 Values of T as above. 1 Values of lg T as above. 1 Uncertainties in T all ±0.04. 1 Uncertainties in lg (T / s) consistent with uncertainties in T e.g. from ±0.011 to ±0.007. 1 Question Answer Marks 2(c)(i) Six points plotted correctly. Must be accurate to the nearest half a small square. Diameter of points must be less than half a small square. 1 Error bars in lg T plotted correctly. All error bars must be plotted. Length of bar must be accurate to less than half a small square and symmetrical. 1 2(c)(ii) Line of best fit drawn. Lower end of line should pass between (2.22, 0.22) and (2.25, 0.22) and upper end of line should pass between (2.49, 0.34) and (2.51, 0.34). Do not accept line from first to last plot. 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted. 1 2(c)(iii) Gradient determined with clear substitution of data points from the line of best fit into ∆y / ∆x. Distance between data points must be greater than half the length of the drawn line. 1 uncertainty = gradient of line of best fit – gradient of worst acceptable line or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined by substitution of correct point from the line of best fit into y = mx + c. 1 2(d) k determined from y-intercept. 2 2 intercept 2 2 10 10 y k −   π π     = =               (c)(iv) 1 q = answer to (c)(iii) and given to 2 or 3 significant figures. 1 Question Answer Marks 2(e) M determined from (d) or (c)(iii) and (c)(iv) with correct substitution shown. 2 2 2 2 4 q q q T k k k M = = = π π π or π   −     = 2 (lg1) lg lg k M q = − = − -intercept lg gradient y M (c)(iv) (c)(iii) 10 M   −     = (c)(iv) (c)(iii) 1

This question in 9702/51 Oct/Nov 2019

1 When a light plastic ball is placed in a vertical column of moving air, the ball becomes stationary at a height h, as shown in Fig. 1.1. ball h moving air air blower Fig. 1.1 A student is using an air blower to create the vertical column of moving air. The student connects the motor of the air blower to a d.c. power supply. It is suggested that the relationship between the radius r of the ball and h is 4 r r 3 gh = PK 3 where g is the acceleration of free fall, P is the power of the motor and K is a constant. Design a laboratory experiment to test the relationship between r and h. Explain how your results could be used to determine a value for K. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: 1 Defining the problem r is the independent variable and h is the dependent variable or vary r and measure/determine h 1 keep P constant 1 Methods of data collection labelled diagram of workable experiment including: • labelled air blower • labelled ball vertically (by eye) above the blower • vertical (by eye) rule at least from top of blower to ball 1 circuit diagram to determine P, e.g. voltmeter and ammeter connected to motor and power supply or wattmeter connected to motor and power supply 1 use of micrometer/calipers/rule to measure diameter of ball 1 measure the distance between the top/middle/bottom of the ball and the top of the blower 1 Method of Analysis plot a graph of h against 1 / r3 (or lg h against lg r) 1 relationship valid if a straight line through (0,0) (for lg h against lg r straight line with gradient = –3) 1 4 gradient 3 g K P π × = (for lg h against lg r, intercept 4 10 3 y g K P − π × = ) 1 Question Answer Marks 1 Additional detail including safety considerations D1 use large box/tray to collect ball (to prevent ball rolling on floor/bouncing) or reasoned method to avoid draughts, e.g. switch off fans, close windows, use a screen 6 D2 method to determine P from correct circuit, e.g. P = I × V or use wattmeter to measure P D3 stand on bench with clamped rule vertically to measure vertical distance D4 method to ensure clamped rule to measure h is vertical, e.g. correctly positioned set square indicated at right angles between the rule and the horizontal surface or plumb line shown in appropriate position D5 r = d / 2 when diameter measured D6 repeat diameter measurement in different directions and find average D7 repeat experiment for each value of r and determine average h D8 method to determine h, e.g. h = reading top of ball – r – top of blower or h = reading bottom of ball + r – top of blower or h = (distance from top of ball to top of blower + distance from bottom of ball to top of blower) / 2 D9 wait for the ball to become stationary (vertically) D10 video (camera) shown level with elevated ball and description of playback frame-by-frame or slow motion

This question in 9702/53 Oct/Nov 2019

2 A student is investigating the oscillations of a mass attached to two springs connected in series, as shown in Fig. 2.1. springs mass Fig. 2.1 A stopwatch is used to measure the time t for 10 oscillations. The measurement of t is repeated and the average period T is determined. The experiment is repeated for different masses. It is suggested that T and mass M are related by the equation 2 r M q T = k where k is the spring constant of the two springs in series and q is a constant. (a) A graph is plotted of lg T on the y-axis against lg M on the x-axis. Determine expressions for the gradient and the y-intercept. gradient = … y-intercept = … [1] (b) Values of M, lg (M / g) and measurements of t are given in Fig. 2.2. M / g t / s t / s T / s lg (M / g) lg (T / s) 155 15.2 16.0 2.190 205 18.3 17.5 2.312 250 19.3 20.1 2.398 305 21.0 21.8 2.484 355 23.5 22.7 2.550 410 24.1 24.9 2.613 Fig. 2.2 Calculate and record values of T / s and lg (T / s) in Fig. 2.2. Include the absolute uncertainties in T / s and lg (T / s). [4] (c) (i) Plot a graph of lg (T / s) against lg (M / g). Include error bars for lg (T / s). [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

11 marks

Mark scheme: 2(a) gradient = q and y-intercept = π  = π −     2 1 lg lg 2 lg 2 k k 1 2(b) T / s lg (T / s) 1.56 0.193 or 0.1931 1.79 0.253 or 0.2529 1.97 0.294 or 0.2945 2.14 0.330 or 0.3304 2.31 0.364 or 0.3636 2.45 0.389 or 0.3892 Values of T as above. 1 Values of lg T as above. 1 Uncertainties in T all ±0.04. 1 Uncertainties in lg (T / s) consistent with uncertainties in T e.g. from ±0.011 to ±0.007. 1 Question Answer Marks 2(c)(i) Six points plotted correctly. Must be accurate to the nearest half a small square. Diameter of points must be less than half a small square. 1 Error bars in lg T plotted correctly. All error bars must be plotted. Length of bar must be accurate to less than half a small square and symmetrical. 1 2(c)(ii) Line of best fit drawn. Lower end of line should pass between (2.22, 0.22) and (2.25, 0.22) and upper end of line should pass between (2.49, 0.34) and (2.51, 0.34). Do not accept line from first to last plot. 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted. 1 2(c)(iii) Gradient determined with clear substitution of data points from the line of best fit into ∆y / ∆x. Distance between data points must be greater than half the length of the drawn line. 1 uncertainty = gradient of line of best fit – gradient of worst acceptable line or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined by substitution of correct point from the line of best fit into y = mx + c. 1 2(d) k determined from y-intercept. 2 2 intercept 2 2 10 10 y k −   π π     = =               (c)(iv) 1 q = answer to (c)(iii) and given to 2 or 3 significant figures. 1 Question Answer Marks 2(e) M determined from (d) or (c)(iii) and (c)(iv) with correct substitution shown. 2 2 2 2 4 q q q T k k k M = = = π π π or π   −     = 2 (lg1) lg lg k M q = − = − -intercept lg gradient y M (c)(iv) (c)(iii) 10 M   −     = (c)(iv) (c)(iii) 1

This question in 9702/53 Oct/Nov 2019

1 A student investigates springs made of metal wire, as shown in Fig. 1.1. cross-sectional area A metal wire Fig. 1.1 The student constructs several springs from wire of thickness t. Each spring has a different cross-sectional area A. The student investigates how the spring constant k varies with A. It is suggested that the relationship between k and A is βρt 4 k = 3 A 2N where ρ is the density of the metal, N is the number of turns of wire on the spring and β is a constant. Design a laboratory experiment to test the relationship between k and A. Explain how your results could be used to determine a value for β. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to: ● the procedure to be followed ● the measurements to be taken ● the control of variables ● the analysis of the data ● any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: 1 Defining the problem A is the independent variable and k is the dependent variable or vary A and measure k 1 keep N constant 1 Methods of data collection labelled diagram of workable experiment including: • spring fixed at one end to a support • load attached to the other end of the spring • labelled load 1 method to measure mass or weight of load: use top-pan balance to measure mass or newton meter to measure weight 1 use of a micrometer/calipers to determine t and rule/calipers to measure the diameter of the spring 1 method to measure extension, e.g. labelled ruler drawn parallel to spring, equilibrium position and displaced position indicated and x indicated or description of use of ruler to measure equilibrium position and displaced position and difference determined 1 Method of analysis plot a graph of k against 1 / A3/2 (or A–3/2) or equivalent e.g. lg k against lg A 1 relationship valid if a straight line passing through the origin is produced (for lg k against lg A, relationship valid if a straight line with gradient –3/2) 1 4 gradient N t β ρ × = [for lg k against lg A, β = 10y-intercept × N / (ρt4) ] 1 Question Answer Marks 1 Additional detail including safety considerations 6 D1 use safety goggles/safety screen to prevent injury to eyes from (moving) spring/load or use cushion/sand box in case load falls D2 keep t constant D3 or mg F k x x = D4 use of set square when taking measurements to determine extension of spring D5 repeat measurement of t along wire/spring and average D6 repeat measurement of diameter D of spring (to determine A) in different directions and average D7 use of 2 4 D A π = D8 method to ensure clamped rule to measure extension is vertical, e.g. correctly positioned set square indicated at right angles between the rule and the horizontal surface or plumb line shown in appropriate position D9 method to determine the density of the wire or additional detail on construction of coil D10 method to determine the mean diameter of the spring, e.g. subtract t from external diameter of spring Question Answer Marks

This question in 9702/51 May/June 2020

1 A student investigates springs made of metal wire, as shown in Fig. 1.1. cross-sectional area A metal wire Fig. 1.1 The student constructs several springs from wire of thickness t. Each spring has a different cross-sectional area A. The student investigates how the spring constant k varies with A. It is suggested that the relationship between k and A is βρt 4 k = 3 A 2N where ρ is the density of the metal, N is the number of turns of wire on the spring and β is a constant. Design a laboratory experiment to test the relationship between k and A. Explain how your results could be used to determine a value for β. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to: ● the procedure to be followed ● the measurements to be taken ● the control of variables ● the analysis of the data ● any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: 1 Defining the problem A is the independent variable and k is the dependent variable or vary A and measure k 1 keep N constant 1 Methods of data collection labelled diagram of workable experiment including: • spring fixed at one end to a support • load attached to the other end of the spring • labelled load 1 method to measure mass or weight of load: use top-pan balance to measure mass or newton meter to measure weight 1 use of a micrometer/calipers to determine t and rule/calipers to measure the diameter of the spring 1 method to measure extension, e.g. labelled ruler drawn parallel to spring, equilibrium position and displaced position indicated and x indicated or description of use of ruler to measure equilibrium position and displaced position and difference determined 1 Method of analysis plot a graph of k against 1 / A3/2 (or A–3/2) or equivalent e.g. lg k against lg A 1 relationship valid if a straight line passing through the origin is produced (for lg k against lg A, relationship valid if a straight line with gradient –3/2) 1 4 gradient N t β ρ × = [for lg k against lg A, β = 10y-intercept × N / (ρt4) ] 1 Question Answer Marks 1 Additional detail including safety considerations 6 D1 use safety goggles/safety screen to prevent injury to eyes from (moving) spring/load or use cushion/sand box in case load falls D2 keep t constant D3 or mg F k x x = D4 use of set square when taking measurements to determine extension of spring D5 repeat measurement of t along wire/spring and average D6 repeat measurement of diameter D of spring (to determine A) in different directions and average D7 use of 2 4 D A π = D8 method to ensure clamped rule to measure extension is vertical, e.g. correctly positioned set square indicated at right angles between the rule and the horizontal surface or plumb line shown in appropriate position D9 method to determine the density of the wire or additional detail on construction of coil D10 method to determine the mean diameter of the spring, e.g. subtract t from external diameter of spring Question Answer Marks

This question in 9702/53 May/June 2020

2 A student investigates the image of an object formed on a screen by a converging lens, as shown in Fig. 2.1. d ho object screen lens Fig. 2.1 The student measures the height ho of the object and the distance d from the lens to the screen. The height hi of the image is measured as shown in Fig. 2.2. screen hi Fig. 2.2 The experiment is repeated for different values of d. It is suggested that hi and d are related by the equation 1 t h i d + = + 1 f c 2 m h o where f is a property of the lens called the focal length and t is the thickness of the lens. hi (a) A graph is plotted of on the y-axis against d on the x-axis. ho Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) The value of ho is (2.4 ± 0.1) cm. Values of d and hi are given in Table 2.1. Table 2.1 hi d / cm hi / cm ho 54.0 1.7 ± 0.1 57.5 1.9 ± 0.1 61.5 2.2 ± 0.1 67.0 2.6 ± 0.1 74.0 3.1 ± 0.1 80.5 3.6 ± 0.1 hi Calculate and record values of in Table 2.1. ho hi Include the absolute uncertainties in . [2] ho hi(c) (i) Plot a graph of against d / cm. ho hi Include error bars for . [2] ho (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = 1 f y-intercept = 1 2 t f − 1 2(b) i o h h 0.71 or 0.708 0.79 or 0.792 0.92 or 0.917 1.1 or 1.08 1.3 or 1.29 1.5 or 1.50 1 Absolute uncertainties in i o h h from ± 0.07 to ± 0.1 (or ± 0.10 or ± 0.11). 1 2(c)(i) Six points plotted correctly. Must be accurate to nearest half a small square. Diameter of points must be less than half a small square. 1 Error bars in i o h h plotted correctly. All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 1 Question Answer Marks 2(c)(ii) Line of best fit drawn. Points must be balanced. Do not accept line from top to bottom point. Line must pass between (55, 0.75) and (56, 0.75) and between (77, 1.40) and (78, 1.40). 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted. 1 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx. Distance between data points must be at least half the length of the drawn line. 1 Gradient of worst acceptable line determined. uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined from substitution into y = mx + c. 1 y-intercept determined using gradient from worst acceptable line. uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) No ECF from false origin method. 1 Question Answer Marks 2(d)(i) f determined using gradient with correct substitution shown. = = 1 1 gradient f (c)(iii) 1 f determined using gradient and given to two or three significant figures and correct SI unit shown with correct power of ten e.g. 33 cm or 0.33 m or 33.1 cm or 0.331 m 1 Absolute uncertainty in f determined. Δ = × gradient absolute uncertainty in gradient f f 1 2(d)(ii) t determined using y-intercept and given to two or three significant figures. Correct substitution of numbers required. ( ) 2 -intercept 1 t f y = × + or ( ) × + = 2 -intercept 1 gradient y t 1

This question in 9702/51 Oct/Nov 2020

2 A student investigates the image of an object formed on a screen by a converging lens, as shown in Fig. 2.1. d ho object screen lens Fig. 2.1 The student measures the height ho of the object and the distance d from the lens to the screen. The height hi of the image is measured as shown in Fig. 2.2. screen hi Fig. 2.2 The experiment is repeated for different values of d. It is suggested that hi and d are related by the equation 1 t h i d + = + 1 f c 2 m h o where f is a property of the lens called the focal length and t is the thickness of the lens. hi (a) A graph is plotted of on the y-axis against d on the x-axis. ho Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) The value of ho is (2.4 ± 0.1) cm. Values of d and hi are given in Table 2.1. Table 2.1 hi d / cm hi / cm ho 54.0 1.7 ± 0.1 57.5 1.9 ± 0.1 61.5 2.2 ± 0.1 67.0 2.6 ± 0.1 74.0 3.1 ± 0.1 80.5 3.6 ± 0.1 hi Calculate and record values of in Table 2.1. ho hi Include the absolute uncertainties in . [2] ho hi(c) (i) Plot a graph of against d / cm. ho hi Include error bars for . [2] ho (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = 1 f y-intercept = 1 2 t f − 1 2(b) i o h h 0.71 or 0.708 0.79 or 0.792 0.92 or 0.917 1.1 or 1.08 1.3 or 1.29 1.5 or 1.50 1 Absolute uncertainties in i o h h from ± 0.07 to ± 0.1 (or ± 0.10 or ± 0.11). 1 2(c)(i) Six points plotted correctly. Must be accurate to nearest half a small square. Diameter of points must be less than half a small square. 1 Error bars in i o h h plotted correctly. All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 1 Question Answer Marks 2(c)(ii) Line of best fit drawn. Points must be balanced. Do not accept line from top to bottom point. Line must pass between (55, 0.75) and (56, 0.75) and between (77, 1.40) and (78, 1.40). 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted. 1 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx. Distance between data points must be at least half the length of the drawn line. 1 Gradient of worst acceptable line determined. uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined from substitution into y = mx + c. 1 y-intercept determined using gradient from worst acceptable line. uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) No ECF from false origin method. 1 Question Answer Marks 2(d)(i) f determined using gradient with correct substitution shown. = = 1 1 gradient f (c)(iii) 1 f determined using gradient and given to two or three significant figures and correct SI unit shown with correct power of ten e.g. 33 cm or 0.33 m or 33.1 cm or 0.331 m 1 Absolute uncertainty in f determined. Δ = × gradient absolute uncertainty in gradient f f 1 2(d)(ii) t determined using y-intercept and given to two or three significant figures. Correct substitution of numbers required. ( ) 2 -intercept 1 t f y = × + or ( ) × + = 2 -intercept 1 gradient y t 1

This question in 9702/53 Oct/Nov 2020

1 An electric pump is placed in a container of liquid. A model wind turbine is connected to the pump by a cable, as shown in Fig. 1.1. blades moving air pipe h cable pump wind turbine bench liquid Fig. 1.1 (not to scale) The turbine is placed in moving air. As the turbine blades turn, electricity is generated and the pump pushes liquid through a vertical pipe. The frequency of rotation of the turbine blades is f. The height the liquid moves is h. The mass per unit time of the liquid leaving the top of the pipe is Q. It is suggested that Q is related to f by the relationship Qgh = C + Df 3 where g is the acceleration of free fall, and C and D are constants. Plan a laboratory experiment to test the relationship between Q and f. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for C and D. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem f is the independent variable and Q is the dependent variable, or vary f and measure Q. 1 Keep h constant 1 Methods of data collection Labelled diagram of workable experiment including: 1 • fan is positioned in line with the turbine so that blades of both fan and turbine overlap • base of fan on same bench as turbine • fan labelled and one other label from bench, (wind) turbine, cable, pump, pipe, liquid Labelled apparatus showing workable method to collect the liquid from the top of the pipe, e.g. hose / pipe / tube connected 1 to top of pipe with the other end over a beaker / measuring cylinder below top of pipe. At least one label related to collection of liquid. Use of stop-watch / timer to measure time to collect liquid or to measure time for blades to rotate. 1 Use of (top pan) balance to measure mass of liquid leaving the pipe. 1 Method of Analysis Plots a graph of Q against f 3 or equivalent (e.g. f 3 against Q). 1 Do not accept logarithmic graphs. C = gh -intercepty (for f 3 against Q: C = −D -intercepty ) 1 D = gh gradient 1 3 gh (for f against Q: D = ) gradient 1 Additional detail including safety considerations 6 Any six from: D1 Precaution with reason linked to prevent liquid spilling (on bench / floor) e.g. use of large bucket / bowl / tray to contain any spilled liquid or Precaution with reason linked to prevent air / dust particles in eye, e.g. use of goggles or Precaution with reason linked to turbine falling, e.g. clamp turbine to bench. D2 Use rule to measure h. D3 Method to determine mass of liquid, e.g. mass of beaker + liquid – mass of empty beaker or mass of container / pipe before – mass of container / pipe after. D4 Method to determine f, e.g. measure time t for many rotations / revolutions N and period T = t /N and f = 1/T or measure time t for many rotations / revolutions N and f = N/t or video rotating blades, playback frame by frame and use a time stamp to determine period T and f = 1/T. D5 Mark one of the blades to assist in counting number of rotations. D6 Method to vary f, e.g. change speed of fan / change distance between fan and blades / vary current in fan. D7 Wait for steady air flow before starting timing and / or collecting liquid. mass (of liquid) D8 Q = time (to collect liquid) or method and explanation to reduce uncertainty in Q, e.g. use large value of time or mass of liquid collected. D9 Repeat measurements of Q for the same value of f and average Q. 1 D10 Relationship valid if a straight line is produced (not passing through the origin). Do not accept passing through the origin.

This question in 9702/52 Feb/March 2023

2 A block of modelling clay of mass M is attached to a string as shown in Fig. 2.1. string block pellet h pellet Fig. 2.1 A pellet travelling at speed u enters the block and causes the block to move through a vertical height h. The experiment is repeated for different values of M. It is suggested that h and M are related by the equation 1 M + Z 2 = 2 g h c uZ m where g is the acceleration of free fall and Z is a constant. 1 (a) A graph is plotted of on the y-axis against M on the x-axis. h Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of M and h are given in Table 2.1. Table 2.1 1 1 - M / g h / cm / cm 2 h 565 21.0 ± 0.2 637 17.8 ± 0.2 675 16.2 ± 0.2 723 14.6 ± 0.2 790 12.6 ± 0.2 892 10.2 ± 0.2 1 1 - Calculate and record values of / cm 2 in Table 2.1. h 1 Include the absolute uncertainties in . [2] h 1 1 - 2 against M / g.(c) (i) Plot a graph of / cm h 1 Include error bars for . [2] h (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) 2g 1 gradient = uZ 2g y-intercept = u 2(b) 1 −1 1 2 / cm h 0.218 or 0.2182 0.237 or 0.2370 0.248 or 0.2485 0.262 or 0.2617 0.282 or 0.2817 0.313 or 0.3131 Values correct as shown above. 1 1 Uncertainties in from ± 0.001 to ± 0.003. h 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. h All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (605, 0.230) and (615, 0.230) and between (845, 0.300) and (855, 0.300) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) u determined using y-intercept and u and Z given to 2, 3 or 4 significant figures. 1 2  981 44.29 u = = y -intercept (c)(iv) Z determined using gradient with method shown and u and Z given with SI units with appropriate powers of ten. 1 2  981 44.29 y -intercept (c)(iv) Z = = or Z = = u  gradient u (c)(iii) gradient (c)(iii) 2(d)(ii) Percentage uncertainty in Z with method shown. 1  y -intercept gradient  percentage uncertainty in Z =  +   100  y -intercept gradient  or Correct substitution for u and  u gradient  percentage uncertainty in Z =  +   100  u gradient  or Correct substitution for max/min methods. 2(e) M determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1  1  − y -intercept    25  M = gradient or uZ uZ M = − Z = − Z 2gh 221.5

This question in 9702/51 Oct/Nov 2023

2 A block of modelling clay of mass M is attached to a string as shown in Fig. 2.1. string block pellet h pellet Fig. 2.1 A pellet travelling at speed u enters the block and causes the block to move through a vertical height h. The experiment is repeated for different values of M. It is suggested that h and M are related by the equation 1 M + Z 2 = 2 g h c uZ m where g is the acceleration of free fall and Z is a constant. 1 (a) A graph is plotted of on the y-axis against M on the x-axis. h Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of M and h are given in Table 2.1. Table 2.1 1 1 - M / g h / cm / cm 2 h 565 21.0 ± 0.2 637 17.8 ± 0.2 675 16.2 ± 0.2 723 14.6 ± 0.2 790 12.6 ± 0.2 892 10.2 ± 0.2 1 1 - Calculate and record values of / cm 2 in Table 2.1. h 1 Include the absolute uncertainties in . [2] h 1 1 - 2 against M / g.(c) (i) Plot a graph of / cm h 1 Include error bars for . [2] h (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) 2g 1 gradient = uZ 2g y-intercept = u 2(b) 1 −1 1 2 / cm h 0.218 or 0.2182 0.237 or 0.2370 0.248 or 0.2485 0.262 or 0.2617 0.282 or 0.2817 0.313 or 0.3131 Values correct as shown above. 1 1 Uncertainties in from ± 0.001 to ± 0.003. h 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. h All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (605, 0.230) and (615, 0.230) and between (845, 0.300) and (855, 0.300) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) u determined using y-intercept and u and Z given to 2, 3 or 4 significant figures. 1 2  981 44.29 u = = y -intercept (c)(iv) Z determined using gradient with method shown and u and Z given with SI units with appropriate powers of ten. 1 2  981 44.29 y -intercept (c)(iv) Z = = or Z = = u  gradient u (c)(iii) gradient (c)(iii) 2(d)(ii) Percentage uncertainty in Z with method shown. 1  y -intercept gradient  percentage uncertainty in Z =  +   100  y -intercept gradient  or Correct substitution for u and  u gradient  percentage uncertainty in Z =  +   100  u gradient  or Correct substitution for max/min methods. 2(e) M determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1  1  − y -intercept    25  M = gradient or uZ uZ M = − Z = − Z 2gh 221.5

This question in 9702/53 Oct/Nov 2023

2 A student investigates the resonant frequency of a metal rod. The metal rod of length L is suspended from two rubber loops. A sensitive microphone with a cone is positioned at one end of the rod. The microphone is attached to an oscilloscope, as shown in Fig. 2.1. cone rubber loop microphone to oscilloscope hammer rod stand bench Fig. 2.1 (not to scale) The rod is hit gently with a hammer. The period T of the trace produced on the oscilloscope is determined. The experiment is repeated for different values of L. It is suggested that T and L are related by the equation 2Ln T = C where C and n are constants. (a) A graph is plotted of lg T on the y-axis against lg L on the x-axis. Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of L and T are given in Table 2.1. Table 2.1 L / cm T / 10–5 s lg (L / cm) lg (T / 10–5 s) 54 24 ± 1 70 32 ± 1 86 39 ± 1 108 49 ± 2 140 64 ± 2 167 74 ± 2 Calculate and record values of lg (L / cm) and lg (T / 10–5 s) in Table 2.1. Include the absolute uncertainties in lg T. [2] (c) (i) Plot a graph of lg (T / 10–5 s) against lg (L / cm). Include error bars for lg T. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = n y-intercept = lg 2 C 1 2(b) lg (L / cm) lg (T / 10–5 s) 1.73 or 1.732 1.38 or 1.380  0.02 1.85 or 1.845 1.51 or 1.505  0.01 1.93 or 1.934 1.59 or 1.591  0.01 2.033 or 2.0334 1.69 or 1.690  0.02 2.146 or 2.1461 1.81 or 1.806  0.01 2.223 or 2.2227 1.87 or 1.869  0.01 Values of lg (L / cm) and lg (T/ 10–5 s) correct as shown above. 1 Uncertainties in lg (T/ 10–5 s) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. Must be within half a small square. Diameter of points must be less than half a small square. 1 Error bars in lg T plotted correctly. All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 1 Question Answer Marks 2(c)(ii) Straight line of best fit drawn. Do not accept line from top point to bottom point. Line must pass between (1.780, 1.45) and (1.800, 1.45) and between (2.085, 1.75) and (2.100, 1.75) 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted. 1 2(c)(iii) Gradient determined with clear substitution of data points into y / x. Distance between data points must be greater than half the length of the drawn line. 1 Gradient determined of worst acceptable line with clear substitution of data points into y / x. uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 1 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 1 Question Answer Marks 2(d) Value of n determined using gradient (n = gradient) and C given to 2 or 3 significant figures. 1 Value of C determined using y-intercept with method shown. -intercept 2 10y C  1 Absolute uncertainties in n and C. uncertainty in n = uncertainty in gradient and   worst y-intercept 2 = 10 C C or   min worst -intercept max worst -intercept 2 2 10 10 = 2 y y C Clear method must be shown with C correctly evaluated. 1 2(e) Value of L determined (non-zero) to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d) with correct substitution and correct power of ten. Units of T and either C or y-intercept must be consistent. 2 log log log10 -intercept log T y C L n n     log10 -intercept 10 y n L   or 2 n TC L  1

This question in 9702/52 May/June 2024

2 A student investigates the refraction of white light entering a transparent rectangular block. A narrow beam of light enters the block at the midpoint of one of the shorter sides. The angle of incidence θ is measured, as shown in Fig. 2.1. block beam of light θ d Fig. 2.1 (not to scale) The distance d between the corner of the block and the point where the beam of light touches the boundary of the block is measured. The experiment is repeated for different values of θ. It is suggested that d and θ are related by the equation B sin2 θ = B + d 2 n2 where B and n are constants. 1 (a) A graph is plotted of d 2 on the y-axis against on the x-axis. sin2 θ Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] 1 (b) Values of θ, and d are given in Table 2.1. sin2 θ Table 2.1 1 θ / ° d / cm d 2 / cm2 sin2 θ 28.5 4.39 24.8 ± 0.2 33.5 3.28 21.4 ± 0.2 42.5 2.19 17.1 ± 0.2 50.0 1.70 14.7 ± 0.2 57.5 1.41 12.9 ± 0.2 63.5 1.25 11.8 ± 0.2 Calculate and record values of d 2 / cm2 in Table 2.1. Include the absolute uncertainties in d 2. [2] 1(c) (i) Plot a graph of d 2 / cm2 against sin2 θ. Include error bars for d 2. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = Bn2 1 y-intercept = − B 2(b) 1 d2 / cm2 615 or 615.0 458 or 458.0 292 or 292.4 216 or 216.1 166 or 166.4 139 or 139.2 Values correct as shown above. Uncertainties in d2 decreasing from 10 to 4 or 5. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in d2 plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.90, 250) and (2.00, 250) and between (3.55, 500) and (3.65, 500). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) B determined using y-intercept (B = – y-intercept) and B and n given to 2 or 3 or 4 significant figures. 1 n determined using gradient 1 and B and n given with SI units with correct powers of ten. gradient gradient n = or n = B − y -intercept Unit for B: cm2 No unit for n. 2(d)(ii) Percentage uncertainty in n determined with method shown. 1 1  y -intercept gradient  percentage uncertainty =  +   100 2  y -intercept gradient  or correct substitution for max/min methods. 2(e) determined to a minimum of two significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and correct 1 power of ten. −1 gradient  = sin − y -intercept + 900 or −1 n 2 B  = sin B + 900

This question in 9702/52 Oct/Nov 2024

1 A thin solid disc of radius r and thickness z is attached to a thin axle. String is wrapped around the axle, as shown in Fig. 1.1. z disc axle r string block Fig. 1.1 A block of mass m is attached to the string. The block is released from rest and falls downwards. The block has speed v when it has fallen through a distance h from the point of release. The value of v is determined using one light gate connected to a timer. It is suggested that v is related to m by the relationship h πr 2z 1 = + v 2 2PQm P where P and Q are constants. Plan a laboratory experiment to test the relationship between v and m. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for P and Q. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem vary m and measure v or m is the independent variable and v is the dependent variable 1 keep h constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • axle resting on support(s) (on stands) • supports placed on bench • light gate (connected to timer) positioned at a distance h • light gate labelled and h indicated vertical metre rule clamped to a stand in a position close to block to measure h 1 method to determine v using an interrupt length, e.g. v = length of block / time recorded by the timer 1 method to measure m, e.g. use a (top-pan) balance 1 Method of analysis 1 1 1 plot a graph of against or equivalent v 2 m Do not accept logarithms. 1 1 1 1 1 1 against against v 2 m m v 2 1 gradient P = P = − h  y -intercept y -intercept  h or r 2 z  gradient P = 2Qh 1 1 1 1 1 against against v 2 m m v 2 r 2 z r 2 z  y -intercept Q = Q = − 2Ph  gradient 2 or r 2 z  y -intercept Q = 2  gradient Additional detail including safety considerations 6 D1 precaution linked to falling block resulting in damage to block / bench, e.g. use a cushion / sand tray to prevent damage to bench or precaution linked to stands falling, e.g. clamp stand(s) to the bench prevent stand falling D2 keep r and z constant D3 method to determine r e.g. use calipers / ruler to measure diameter d and r = d / 2 D4 measure z with a micrometer / calipers 1 D5 set square correctly positioned between rule and bench to ensure that rule to measure h is vertical D6 method to keep h constant by identifying constant initial position of the bottom of the block, e.g. clamped pin / rod to indicate the starting point each time or (fiducial) marker on rule D7 description of method to ensure that axle can rotate, e.g. axle is lubricated at the supports to enable axle to rotate, axle is not fixed at the supports so axle can rotate D8 use a large length of block to reduce (percentage) uncertainty in interrupt time or increase the time light gate is interrupted D9 repeat experiment for the same value of m and determine the average v 1 D10 relationship valid if a straight line is produced (with y-intercept = ). hP Do not accept line passing through the origin.

This question in 9702/52 May/June 2025

2 A student investigates the relationship between the luminosity of a star and its mass. The student obtains data of relative luminosity λ and relative mass µ for six stars, where luminosity of star λ = luminosity of Sun and mass of star µ = mass of Sun. It is suggested that λ and µ are related by the equation λ = kµn where k and n are constants. (a) A graph is plotted of lg λ on the y-axis against lg µ on the x-axis. Determine expressions for the gradient and y-intercept. gradient = … y-intercept = … [1] (b) Values of µ and λ are given in Table 2.1. Table 2.1 µ λ lg µ lg λ 4.6 ± 0.4 500 5.4 ± 0.4 800 8.4 ± 0.4 3200 11 ± 1 7000 16 ± 1 25 000 18 ± 1 38 000 Calculate and record values of lg µ and lg λ in Table 2.1. Include the absolute uncertainties in lg µ. [2] (c) (i) Plot a graph of lg λ against lg µ. Include error bars for lg µ. [2] (ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines. [2] (iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer. gradient = … [2]

9 marks

Mark scheme: 2(a) gradient = n 1 y-intercept = lg k 2(b) 1 lg  lg  0.66 or 0.663  0.04 2.70 or 2.699 0.73 or 0.732  0.03 2.90 or 2.903 0.92 or 0.924  0.02 3.51 or 3.505 1.04 or 1.041  0.04 3.85 or 3.845 1.20 or 1.204  0.03 4.40 or 4.398 1.26 or 1.255  0.02 or  0.03 4.58 or 4.580 Values of lg and lg correct as shown above. Uncertainties in lg correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (0.82, 3.20) and (0.84, 3.20) and between (1.13, 4.20) and (1.16, 4.20). Worst acceptable straight line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow methods using a false origin. 2(d) Value of k determined using y-intercept. 1 k = 10y −intercept n = gradient and n and k given to 2 or 3 significant figures. 1 absolute uncertainty in n = absolute uncertainty in gradient 1 and absolute uncertainty in k = (10y-intercept – 10y-intercept of WAL) Correct method must be seen. 2(e) M determined to a minimum of 2 significant figures from (d) or (c)(iii) and (c)(iv) with correct substitution and correct 1 power(s) of ten. Do not accept incorrect POT for n or k. Correct substitution must be seen.  0.46 lg0.46 − y -intercept lg0.46 − lg k = n = gradient or lg = or lg = k (d) gradient gradient and M =  2.0  1030

This question in 9702/54 May/June 2025

1 On a bench, a steel ball of radius r is used to compress a spring by a distance x. The ball is held at rest in this position, as shown in Fig. 1.1. compressed spring ball bench P Fig. 1.1 The ball is released and rolls along the bench. At a fixed point P, the ball has speed v. The speed of the ball at P is determined using one light gate connected to a timer. Several steel balls of different radii are available. It is suggested that v is related to r by the relationship Ykx2 v 2 = r nρ where k is the spring constant of the spring, ρ is the density of the steel, and Y and n are constants. Plan a laboratory experiment to test the relationship between v and r. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for Y and n. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem vary r and measure v or r is the independent variable and v is the dependent variable 1 keep x constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • one end of spring resting against block clamped to bench using G-clamp • light gate positioned at P • light gate connected to timer • apparatus shown on bench • labels for light gate and P and at least one other label from bench, block, stand, spring, ball, timer method to determine r, e.g. use calipers or micrometer to measure diameter d and r = d / 2 1 description of method to determine v, use diameter of ball (to interrupt beam) ÷ measured time at light gate positioned at P 1 instrument to determine x, e.g. rule(r) or calipers 1 Method of Analysis plot a graph of (2 lg v) or (lg v2) against lg r 1 or plot a graph of (lg v) against (lg r) or equivalent, e.g. (ln v) against (ln r) n = −gradient for (2 lg v) or (lg v2) against lg r 1 or n = −2 gradient for (lg v) against lg r 1  10y -intercept 1 Y = for (2 lg v) or (lg v2) against lg r kx 2 or  10 2 y -intercept Y = for (lg v) against lg r kx 2 Additional detail including safety considerations 6 D1 precaution to prevent ball leaving bench, e.g. screens around apparatus / cushions on bench (to stop the ball) D2 keep k and constant D3 description of method to determine k, e.g. add mass to spring and k = mg / extension or use newton meter to measure force applied to spring and k = force / extension or take several readings of force and extension, plot a force–extension graph and k = gradient m D4 description of experimental method to determine , e.g. measure mass of ball using a balance and = 4 3 r 3 D5 repeat measurements of diameter or d in different directions and determine the average value of d D6 method to keep x constant, e.g. use a pin / ruler / card to indicate the starting point each time to keep x constant D7 x = original length of spring – compressed length of spring D8 adjust (vertical) position of light gate so that the diameter of (each) ball cuts the beam D9 repeat experiment for the same value of r and determine the average v 1  Ykx 2  1  Ykx 2  D10 relationship valid if a straight line is produced (with y-intercept = lg   or lg   ).  2      Do not accept line through the origin.

This question in 9702/51 Oct/Nov 2025

1 Fig. 1.1 shows a model wind turbine with blades, each of length L, placed in moving air. L moving air blade terminals turbine bench Fig. 1.1 The area of the circle swept by the blades of the turbine is A. The output of the turbine has two terminals. The turbine is connected to a resistor of resistance R. At a speed v of the moving air, the current in the resistor is I. The atmospheric pressure is P and the thermodynamic temperature of the air is T. It is suggested that I is related to v by the relationship I2R APv3 = Q 2T where Q is a constant. Plan a laboratory experiment to test the relationship between I and v. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine a value for Q. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem vary v and measure I or v is the independent variable and I is the dependent variable 1 keep A and R constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • fan positioned in line with the turbine so that blades of both fan and turbine overlap • fan on bench • fan labelled and one other label from bench, turbine, blade(s) (of turbine), terminals, L workable circuit diagram showing resistor connected to an ammeter in series with the terminals of the turbine using correct 1 circuit symbols method to vary v, e.g. change speed of fan / change distance between fan and blades / vary current in or p.d. across fan 1 method to determine temperature T, e.g. use a thermometer 1 Method of Analysis plot a graph of I2 against v3 or equivalent (e.g. lg I against lg v or 2 ln I against ln v) 1 relationship valid if a straight line is produced passing through the origin 1 (For lg I against lg v: relationship valid if a straight line with gradient = 1.5 is produced) 2TR  gradient 1 Q = AP 2TR  10 2 y -intercept (For lg I against lg v: Q = ) AP 1 Additional detail including safety considerations 6 D1 precaution with reason linked to (moving) fan blades / turbine blades, e.g. keep away from the fan to avoid (moving) blades or use a screen around the fan / turbine to avoid (moving) blades or precaution with reason linked to prevent air / dust particles in eye, e.g. use goggles to avoid air stream (into eye) D2 clamp turbine / fan to bench D3 keep P and T constant D4 T = t + 273 D5 method to determine A: use a rule(r) / calipers to measure L and A = L2 D6 repeat measurements of L in different positions / different blades and average D7 method to measure v, e.g. use an anemometer or air speed meter or method to measure P, e.g. use a manometer or barometer or pressure gauge D8 wait for steady / constant air flow / movement of blades / current D9 method to determine R, e.g.: separate circuit showing ohmmeter connected to R only or terminals of turbine connected correctly to resistor and ammeter and voltmeter across resistor and R = V / I or separate workable circuit with power supply resistor, ammeter and voltmeter across R and R = V / I D10 method to check temperature / pressure is constant, e.g. measure temperature / pressure several times / before and after

This question in 9702/52 Oct/Nov 2025

1 On a bench, a steel ball of radius r is used to compress a spring by a distance x. The ball is held at rest in this position, as shown in Fig. 1.1. compressed spring ball bench P Fig. 1.1 The ball is released and rolls along the bench. At a fixed point P, the ball has speed v. The speed of the ball at P is determined using one light gate connected to a timer. Several steel balls of different radii are available. It is suggested that v is related to r by the relationship Ykx2 v 2 = r nρ where k is the spring constant of the spring, ρ is the density of the steel, and Y and n are constants. Plan a laboratory experiment to test the relationship between v and r. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for Y and n. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem vary r and measure v or r is the independent variable and v is the dependent variable 1 keep x constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • one end of spring resting against block clamped to bench using G-clamp • light gate positioned at P • light gate connected to timer • apparatus shown on bench • labels for light gate and P and at least one other label from bench, block, stand, spring, ball, timer method to determine r, e.g. use calipers or micrometer to measure diameter d and r = d / 2 1 description of method to determine v, use diameter of ball (to interrupt beam) ÷ measured time at light gate positioned at P 1 instrument to determine x, e.g. rule(r) or calipers 1 Method of Analysis plot a graph of (2 lg v) or (lg v2) against lg r 1 or plot a graph of (lg v) against (lg r) or equivalent, e.g. (ln v) against (ln r) n = −gradient for (2 lg v) or (lg v2) against lg r 1 or n = −2 gradient for (lg v) against lg r 1  10y -intercept 1 Y = for (2 lg v) or (lg v2) against lg r kx 2 or  10 2 y -intercept Y = for (lg v) against lg r kx 2 Additional detail including safety considerations 6 D1 precaution to prevent ball leaving bench, e.g. screens around apparatus / cushions on bench (to stop the ball) D2 keep k and constant D3 description of method to determine k, e.g. add mass to spring and k = mg / extension or use newton meter to measure force applied to spring and k = force / extension or take several readings of force and extension, plot a force–extension graph and k = gradient m D4 description of experimental method to determine , e.g. measure mass of ball using a balance and = 4 3 r 3 D5 repeat measurements of diameter or d in different directions and determine the average value of d D6 method to keep x constant, e.g. use a pin / ruler / card to indicate the starting point each time to keep x constant D7 x = original length of spring – compressed length of spring D8 adjust (vertical) position of light gate so that the diameter of (each) ball cuts the beam D9 repeat experiment for the same value of r and determine the average v 1  Ykx 2  1  Ykx 2  D10 relationship valid if a straight line is produced (with y-intercept = lg   or lg   ).  2      Do not accept line through the origin.

This question in 9702/53 Oct/Nov 2025

1 Fig. 1.1 shows a horizontal turntable. turntable P C r Fig. 1.1 Point C is at the centre of the turntable. Point P is a distance r from the centre. Fig. 1.2 shows a side view of a d.c. motor attached to the turntable with a belt. terminals turntable C belt motor turntable base Fig. 1.2 The motor is used to rotate the turntable at frequency f0. The motor is switched off and the turntable continues to rotate at frequency f0. A sphere of adhesive putty of mass m is dropped onto the turntable at point P. The frequency of the turntable is now f. It is suggested that f is related to m by the relationship Kf0 = βK + mr 2 f where β and K are constants. Plan a laboratory experiment to test the relationship between f and m. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for β and K. In your plan you should include: • the procedure to be followed • the measurements to be taken • the control of variables • the analysis of the data • any safety precautions to be taken. Diagram … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … [15]

15 marks

Mark scheme: Question Answer Marks 1 Defining the problem vary m and measure f or m is the independent variable and f is the dependent variable 1 keep f0 constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • turntable base and motor on bench • two labels from turntable, motor, belt, C, putty, bench, release mechanism workable circuit showing terminals of motor connected to (d.c.) supply 1 method to measure time to determine f and f0, e.g. use a stopwatch to measure (n)t 1 and method to measure m, e.g. use a (top-pan) balance measure the time t for n revolutions 1 and T = t / n and f = 1 / T or f = n / t Method of Analysis 1 1 plots a graph of against m or equivalent f Do not accept logarithms. 1 1 1 1 against m m against f f r 2 r 2 × gradient K = K = f0 × gradient f0 1 1 1 against m m against f f = f0  y -intercept f0  y -intercept  = − gradient or r 2×y -intercept = − K 1 Additional detail including safety considerations 6 D1 precaution linked to putty leaving turntable, e.g. screens around apparatus / cushions to prevent putty from leaving bench D2 keep r constant D3 use rule(r) to measure r D4 stand and fixed point located above P (to ensure r is constant) D5 method to ensure putty is dropped at (constant) distance r, e.g.; fixed location above the turntable to drop putty to keep r constant or draw a circle of radius r on the turntable D6 method to keep f0 constant, e.g. adjust the rheostat to keep the current in the motor constant / check ammeter D7 clamp motor and/or turntable (base) to bench D8 lubricate the turntable to reduce friction D9 repeat experiment for the same value of m and determine the average f D10 relationship valid if a straight line is produced (with y-intercept = ) f0 Do not accept line through the origin.

This question in 9702/54 Oct/Nov 2025