Cambridge A Level Physics 9702 — 2019 Oct/Nov Paper 5 · Variant 2

1 A student is investigating the maximum height reached by a light plastic ball when it is launched vertically from a compressed spring, as shown in Fig. 1.1. final position of ball h initial position of ball spring Fig. 1.1 It is suggested that the maximum height h of the ball and the compression x of the spring are related by the equation 4rr 3tgh 1 2 = kx 3 2 where r is the radius of the ball, t is the density of the ball, g is the acceleration of free fall and k is the spring constant of the spring. Design a laboratory experiment to test the relationship between h and x. Explain how your results could be used to determine a value for t. 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 x is the independent variable and h is the dependent variable or vary x and measure/determine h 1 keep r and/or k constant 1 Methods of data collection labelled diagram of workable experiment including: • labelled spring • upper ball vertically (by eye) above the spring and at least one of the balls labelled • vertical (by eye) rule at least from top of spring to position of upper ball 1 method to determine x, e.g. use a rule/calipers to measure original and final length of spring and find the difference 1 use of micrometer/calipers/rule to measure diameter of ball 1 method to determine h described, e.g. measure the distance between the top of the ball at maximum height and the top of the ball on the spring or measure the distance between the bottom of the ball at maximum height and the bottom of the ball on the spring 1 Method of analysis plot a graph of h against x2 (or lg h against lg x) 1 relationship valid if a straight line through (0,0) (for lg h against lg x straight line with gradient = 2) 1 ρ = π × 3 3 8 gradient k r g ρ −   =   π ×   3 intercept 3 for lg against lg , 8 10y k h x gr 1 Question Answer Marks 1 Additional detail including safety considerations D1 use large box/cage 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 expression to determine k from relevant experiment, e.g. k = mg / x or gradient of F–x graph 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 x and determine average h D8 method to securely fix spring to the bench e.g. tape/G-clamp D9 experiment to determine k, e.g. place mass m on the spring and measure compression x D10 video (camera) shown level (by eye) with elevated ball and description of play back frame by frame or slow motion

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2 A student is investigating how the resistance of a thermistor varies with temperature. The thermistor is placed in water, as shown in Fig. 2.1. to electrical circuit beaker water thermistor heat Fig. 2.1 The thermistor is connected to a battery with electromotive force (e.m.f.) E and negligible internal resistance. The current I in the thermistor is measured. The resistance R of the thermistor is then determined using the expression E R = . I The experiment is repeated for different temperatures of the water. It is suggested that the resistance R of the thermistor and the thermodynamic temperature T are related by the equation R = pT q where p and q are constants. (a) A graph is plotted of lg R on the y-axis against lg T on the x-axis. Determine expressions for the gradient and the y-intercept. gradient = ............................................................... y-intercept = ............................................................... [1] (b) The value of E is 9.4 ± 0.1 V. Values of T, I and lg T are given in Fig. 2.2. T / K I / mA R / 103 Ω lg (T / K) lg (R / 103 Ω) 303 1.0 ± 0.1 2.481 313 1.6 ± 0.1 2.496 323 2.4 ± 0.1 2.509 333 3.7 ± 0.1 2.522 343 5.5 ± 0.1 2.535 353 8.7 ± 0.1 2.548 Fig. 2.2 Calculate and record values of R / 103 Ω and lg (R / 103 Ω) in Fig. 2.2. Include the absolute uncertainties in R / 103 Ω and lg (R / 103 Ω). [4] (c) (i) Plot a graph of lg (R / 103 Ω) against lg (T / K). Include error bars for lg (R / 103 Ω). [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]

Mark scheme: 2(a) and y-intercept = lg p 2(b) R / 103 Ω lg (R / 103 Ω) 9.4 or 9.40 0.97 or 0.973 5.9 or 5.88 0.77 or 0.771 or 0.769 3.9 or 3.92 0.59 or 0.591 or 0.593 2.5 or 2.54 0.40 or 0.398 or 0.405 1.7 or 1.71 0.23 or 0.230 or 0.233 1.1 or 1.08 0.04 or 0.041 or 0.033 Values of R as above. 1 Values of lg R as above. 1 Uncertainties in R from (±0.9 to ±1.2) to (±0.02 to ±0.03) and row 2 between ±0.40 and ±0.50 and row 4 between ±0.09 and ±0.10. 1 Uncertainties in lg R consistent with uncertainties in R e.g. from ±0.05 to ±0.01. 1 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 R plotted correctly. All error bars must be plotted. 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. Upper end of line should pass between (2.500, 0.70) and (2.502, 0.70) and lower end of line should pass between (2.528, 0.30) and (2.532, 0.30). Do not accept line from first to last point. 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. Gradient 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 from the line of best fit into y = mx + c. 1 2(d) p determined from y-intercept. p (= 10y-intercept) = 10(c)(iv) 1 q = answer to (c)(iii) and given to 2 or 3 significant figures. 1 Question Answer Marks 2(e) T determined from (d) or (c)(iii) and (c)(iv) with correct substitution shown. 15 q q R T p p = = or − − = = lg15 lg 1.176 lg lg p p T q q − − = = lg15 -intercept 1.176 lg gradient y T (c)(iv) (c)(iii)   −     = 1.176 10 T (c)(iv) (c)(iii) 1

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