Cambridge A Level Physics 9702 — 2024 Feb/March Paper 5 · Variant 2

1 Fig. 1.1 shows a thin cylindrical metal rod of length L. L Fig. 1.1 One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency f. A microphone placed at the other end of the rod detects the sound wave emitted from the rod. The frequency of the detected sound is also f. A number of rods of different length are available. It is suggested that f is related to L by the relationship E 2f Ln = ρ where ρ is the density of the metal, and E and n are constants. Plan a laboratory experiment to test the relationship between f and L. Draw a diagram showing the arrangement of your equipment. Explain how the results could be used to determine values for E 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. 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Mark scheme: Question Answer Marks 1 Defining the problem L is the independent variable and f is the dependent variable, or vary L and measure f. 1 Keep constant 1 Methods of data collection Labelled diagram of workable experiment including: 1 • rod supported by string / elastic bands from a clamp • clamp attached to stand, with stand on bench • two labels from stand, clamp, hammer, microphone, rod, string. Diagram showing labelled microphone connected to labelled oscilloscope. 1 Method to measure L, e.g. use a metre rule 1 Method to measure mass (m) (of metal rod), e.g. use a (top-pan) balance 1 Method of Analysis 1 1 Plots a graph of log f against log L or equivalent e.g. log f against log L n = − gradient 1 1 (for log f against log : n = gradient) L E = 4 10 2 y -intercept 1 1 2 y -intercept (for lg f vs lg : E = 4 10 ) L (for ln f against ln L; E = 4 e 2 y -intercept ) 1 Additional detail including safety considerations 6 Any six from: D1 Precaution linked to falling rod, e.g. sand tray / cushion (in case rod falls) OR gently hit rod prevent rod falling D2 Method to determine area of rod (A) e.g. measure diameter (d) of rod using a micrometer / calipers D3 Repeat measurements of diameter along the length of rod / around the rod and average diameter m d 2 D4 Method to determine ρ from experimental method, e.g. = and A = AL 4 4m or = d 2 L d m or r = and = 2 r 2 L D5 Perform experiment in a quiet room D6 Reasoned method to prevent rod hitting microphone, e.g. have a gap between rod and microphone / gently hit rod or method to obtain measurable signal from the microphone, e.g. use a cone to increase the sound detected by the microphone D7 Method to determine frequency from oscilloscope, e.g. T = time-base  (horizontal) length (of one wave) and f = 1/T D8 Method to reduce uncertainties e.g. use large values of L to reduce (percentage) uncertainty in L or adjust time-base to display as few waves as possible or Z waves on oscilloscope and divide time by Z or wait for the wave(form) / frequency to stabilise (and reach resonance) D9 Repeat measurements of f for each value of L and average f 1  1 E   E  D10 Relationship is valid if a straight line is produced (passing through log   OR 1log   ).  2   2  4     Do not accept passing through the origin.

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2 A student investigates an electrical circuit. A power supply with negligible internal resistance is connected to six resistors, each of resistance Z, and a resistor of resistance R, as shown in Fig. 2.1. power supply Z Z Z R A Z Z Z Fig. 2.1 The current measured by the ammeter is I. The experiment is repeated for different values of R. It is suggested that I and R are related by the equation E = 3IR + 4IZ where E is the electromotive force (e.m.f.) of the power supply. 1 (a) A graph is plotted of on the y-axis against R on the x-axis. I Determine expressions for the gradient and y-intercept. gradient = .............................................................. y-intercept = ............................................................... [1] (b) Values of R and I are given in Table 2.1. Table 2.1 1 R / kΩ I / μA / A–1 I 1.25 225 ± 5 2.55 185 ± 5 3.90 160 ± 5 5.25 140 ± 5 6.55 125 ± 5 7.80 115 ± 5 1 Calculate and record values of / A–1 in Table 2.1. I 1 Include the absolute uncertainties in . [2] I 1 1(c) (i) Plot a graph of / A–1 against R / kΩ. Include error bars for [2] I I. (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]

Mark scheme: 2(a) 3 1 Gradient = E 4Z y-intercept = E 2(b) 1 1 / A−1 I 4440 or 4444 5410 or 5405 6250 or 6250 7140 or 7143 8000 or 8000 8700 or 8696 1 1 Uncertainties in I From  90–110 to  360–400 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. I 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 plot to bottom plot. Points must be balanced. Line must pass between (1.8, 5000) and (2.1, 5000) and between (7.2, 8500) and (7.5, 8500) Worst acceptable line drawn. 1 Steepest or shallowest possible line that passes through all the error bars. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x; distance between data points must be greater than 1 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) 2(d)(i) E determined using gradient and 1 E and Z given to 2, 3 or 4 sf. 3 E = gradient Z determined using y-intercept and 1 E and Z given with SI units with correct powers of ten E  y -intercept 3  y -intercept Z = or Z = 4 4  gradient Unit of E: V or A  Unit of Z:  2(d)(ii) Percentage uncertainty in Z with method shown. 1  gradient y -intercept  %uncertainty =  +   gradient y -intercept  or Correct substitution for max/min methods. 2(e) R determined to a minimum of 2sf from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and correct powers of ten. 1 0.1 mA = 0.1  10–3 A and 1 −−3 y -intercept 0.10  10 R = or gradient E 4Z R = − 3  0.10  10 −3 3

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