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. [Figure 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 4πr³pgh = 1kx2 3 where r is the radius of the ball, p 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 p. 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.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [1] x is the independent variable and h is the dependent variable or vary x and measure/determine h
2. [1] keep r and/or k constant
3. [1] 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
4. [1] method to determine x, e.g. use a rule/calipers to measure original and final length of spring and find the difference
5. [1] use of micrometer/calipers/rule to measure diameter of ball
6. [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
7. [1] plot a graph of h against x2 (or Ig h against lg x)
8. [1] relationship valid if a straight line through (0,0) (for Ig h against lg x straight line with gradient = 2)
9. [1] ρ = 3k / 8πr³g × gradient for Igh against lg x, ρ = 3k / 8πgr³ × 10-y-intercept
10. [1] 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
11. [1] D2 expression to determine k from relevant experiment, e.g. k = mg / x or gradient of F-x graph
12. [1] D3 stand on bench with clamped rule vertically to measure vertical distance
13. [1] 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
14. [1] D5 r = d/2 when diameter measured
15. [1] D6 repeat diameter measurement in different directions and find average
16. [1] D7 repeat experiment for each value of x and determine average h
17. [1] D8 method to securely fix spring to the bench e.g. tape/G-clamp
18. [1] D9 experiment to determine k, e.g. place mass m on the spring and measure compression x
19. [1] D10 video (camera) shown level (by eye) with elevated ball and description of play back frame by frame or slow motion