Cambridge A Level Physics 9702 — 2021 Oct/Nov Paper 5 · Variant 3

1 A student investigates stationary sound waves in cylindrical tubes. Fig. 1.1 shows a stationary wave pattern in a tube which is open at both ends. L d cylindrical tube Fig. 1.1 The tube has length L and diameter d. The frequency of the sound for the stationary wave pattern shown is f. There are a number of different tubes available. It is suggested that the relationship between f and d is v = 2L + kd f where v is the speed of sound in air and k is a constant. Design a laboratory experiment to test the relationship between f and d. Explain how your results could be used to determine values for k 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: ● 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. 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Mark scheme: 1 Defining the problem diameter/d is the independent variable and frequency/f is the dependent variable or vary d and measure f 1 keep L constant or length (of tube) constant 1 Methods of data collection labelled diagram of workable experiment including: • tube supported • (loud)speaker positioned in line with the tube • (loud)speaker labelled 1 labelled microphone, positioned outside tube in line with tube, connected to labelled oscilloscope or correct circuit symbol 1 adjust/change frequency until maximum amplitude detected 1 use calipers to measure d 1 Question Answer Marks 1 Method of analysis plot a graph of 1 f against d or d against 1 f (Do not accept logarithmic graphs.) 1 for 1 f against d or for d against 1 f 2 -intercept L v y = v = gradient × k or gradient 2 -intercept L v y × =− 1 for 1 f against d or for d against 1 f k = gradient × v or gradient 2 -intercept L k y × = 2 -intercept L k y = − 1 Question Answer Marks 1 Additional detail including safety considerations 6 D1 wear ear defenders (to prevent damage to hearing/to avoid loud sounds) or use a low volume to prevent damage to hearing/to avoid loud sounds D2 use a rule to measure L D3 increase frequency from a low frequency to the first maximum amplitude D4 method to determine f at maximum amplitude, e.g. increase frequency to f, then continue increasing frequency, and then decrease frequency until value of f determined D5 method to determine period from oscilloscope, e.g. no. of divisions × time-base D6 for frequency/time period determined by oscilloscope, f = 1 / T D7 repeat measurements of d and average in different directions/positions or along the tube D8 perform experiment in a quiet room D9 signal generator connected to (loud)speaker in diagram D10 relationship valid if a straight line produced (Do not accept through the origin.)

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2 A student investigates the discharge of a capacitor in the circuit shown in Fig. 2.1. E C V R1 R2 P Q Fig. 2.1 The student closes the switch and charges the capacitor. The switch is opened and a stop-watch is started. The capacitor discharges through the two resistors of resistance R1 and R2 connected between P and Q. At a fixed time t the potential difference V across the capacitor is measured. The experiment is repeated for different values of R1 and R2. It is suggested that V, R1 and R2 are related by the equation ⎛⎞V t ln = – ⎝⎠E C(R1 + R2) where E is the electromotive force (e.m.f.) of the battery and C is the capacitance of the capacitor. 1 (a) A graph is plotted of ln V on the y-axis against on the x-axis. R1 + R2 Determine expressions for the gradient and y-intercept. gradient = ............................................................... y-intercept = ............................................................... [1] (b) Values of R1, R2, V and ln V are given in Table 2.1. Each resistance value has a percentage uncertainty of ± 5%. Table 2.1 1 R1 / kΩ R2 / kΩ (R1 + R2) / kΩ / 10−6 Ω−1 V / V ln (V / V) R1 + R2 22 33 1.28 0.247 22 47 1.98 0.683 22 68 2.87 1.054 33 47 2.39 0.871 33 68 3.28 1.188 47 68 3.55 1.267 1 Calculate and record values of (R1 + R2) / kΩ and / 10−6 Ω−1 in Table 2.1. R1 + R2 1 Include the absolute uncertainties in (R1 + R2) and . [2] R1 + R2 1(c) (i) Plot a graph of ln (V / V) against / 10−6 Ω−1. R1 + R2 1 Include error bars for . [2] R1 + R2 (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]

Mark scheme: 2(a) gradient = – t C y-intercept = ln E 2(b) (R1 + R2) / kΩ 1 2 1 R R + / 10–6 Ω 55 (± 3) 18 or 18.2 ± 0.9 69 (± 3 or 4) 14 or 14.5 ± 0.7 90 (± 4 or 5) 11 or 11.1 ± 0.6 80 (± 4) 13 or 12.5 ± 0.6 101 (± 5) 9.9 or 9.90 or 9.901 ± 0.5 115 (± 6) 8.7 or 8.70 or 8.696 ± 0.4 Values of (R1 + R2) and 1 2 1 R R + correct as shown above. 1 Absolute uncertainties in 1 2 1 R R + from ± 0.9 or ± 1 to ± 0.4 or ± 0.5. 1 2(c)(i) Six points plotted correctly. Must be accurate to half a small square. Diameter of points must be less than half a small square. 1 Error bars in 1 2 1 R R + 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 point to bottom point. Line must pass between (10.2, 1.10) and (10.8, 1.10) and between (16.7, 0.40) and (17.2, 0.40). 1 Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). All error bars must be plotted. 1 2(c)(iii) Negative 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 by substitution of point on line into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution of point on line 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 2(d)(i) C determined using gradient and C and E both given to two or three significant figures. 60 gradient t C = − = −(c)(iii) 1 E determined using y-intercept and C and E both given with correct SI unit. = -intercept ey E unit of C: F or C V–1 or s Ω–1 unit of E: V 1 Question Answer Marks 2(d)(ii) Percentage uncertainty determined with method shown.   Δ = + ×     1 gradient percentage uncertainty 100 60 gradient Clear substitution must be shown for maximum/minimum methods. 1 2(e) (R1 + R2) determined to at least two significant figures from (d)(i) or (c)(iii) and (c)(iv) with correct substitution including signs and correct power of ten(s). Do not accept ECF for POT from (c)(iii), (c)(iv) or (d). ( ) 1 2 1 60 1 ln ln ln t R R V C C V E E + = − × = − × − or ( ) + = = − − 1 2 gradient ln5.0 -intercept 1.61 R R y (c)(iii) (c)(iv) 1

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