A student investigates the extension of a spring supporting a wooden strip, as shown in Fig. 1.1. One end of the strip is at point P. The strip has length L and is at an angle θ to the bench. The spring is attached to the strip by a wire at a distance d from point P. The wire is at an angle β to the strip. The spring has extension x. It is suggested that the relationship between x and d is WL 2 cos θ = kxd sin β where k is the spring constant of the spring and W is a constant. Design a laboratory experiment to test the relationship between x and θ. Explain how your results could be used to determine a value for W. 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 2021). The mark or mark type is shown in brackets.

1. [1] θ is the independent variable and x is the dependent variable or vary θ and measure x
2. [1] keep (angle) β constant
3. [1] labelled diagram of workable experiment including: • spring attached at both ends e.g. one end connected to a clamp and stand • strip free to move • at least two labels from: clamp, stand, wire, strip, spring, bench (Do not accept extra masses added to strip.)
4. [1] use a rule to measure L and d
5. [1] use a protractor to measure θ or use a rule to measure appropriate distances to determine θ by trigonometry methods
6. [1] measure original length of spring and new length of spring using rule/calipers
7. [1] plot a graph of x against cos θ or cos θ against x (Allow log x against log (cos θ).)
8. [1] relationship is valid if a straight line passing through the origin is produced (Allow straight line with gradient = 1 for log-log graph.)
9. [1] W = for x against cos θ gradient × 2kd sin β L or W = for cos θ against x 2kd sin β gradient × L
10. [1] D1 wear goggles to prevent spring/wire/strip entering into eyes or (retort) stand used to support spring is clamped to bench
11. [1] D2 keep distance d constant
12. [1] D3 description of (separate) experiment to determine k, e.g. weigh mass and measure extension
13. [1] D4 k = weight / extension or mg / extension or gradient of weight-extension graph for candidate's workable (separate) experiment
14. [1] D5 method to prevent strip at point P sliding, e.g. use adhesive putty/hinge (Do not accept methods that prevent rotation at point P.)
15. [1] D6 use fiducial markers on spring at both ends or measure length of spring on both sides and average
16. [1] D7 method to attach wire to strip, e.g. wire wrapped around the strip/(strong) tape/drill hole and tie wire
17. [1] D8 determine x by subtracting original length of spring from new length
18. [1] D9 adjust support of spring to keep β constant
19. [1] D10 protractor correctly positioned on diagram to measure θ or correct trigonometric relationship given for θ