A student investigates springs made of metal wire, as shown in Fig. 1.1. [Figure 1.1] The student constructs several springs from wire of thickness t. Each spring has a different cross-sectional area A. The student investigates how the spring constant k varies with A. It is suggested that the relationship between k and A is k = Bpt⁴ / (A³/²N) where p is the density of the metal, N is the number of turns of wire on the spring and ẞ is a constant. Design a laboratory experiment to test the relationship between k and A. Explain how your results could be used to determine a value for ẞ. 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 (May/June 2020). The mark or mark type is shown in brackets.

1. [1] A is the independent variable and k is the dependent variable or vary A and measure k
2. [1] keep N constant
3. [1] labelled diagram of workable experiment including: • spring fixed at one end to a support • load attached to the other end of the spring • labelled load
4. [1] method to measure mass or weight of load: use top-pan balance to measure mass or newton meter to measure weight
5. [1] use of a micrometer/calipers to determine t and rule/calipers to measure the diameter of the spring
6. [1] method to measure extension, e.g. labelled ruler drawn parallel to spring, equilibrium position and displaced position indicated and x indicated or description of use of ruler to measure equilibrium position and displaced position and difference determined
7. [1] plot a graph of k against 1/A^3/2 (or A⁻³ᐟ²) or equivalent e.g. lg k against lg A
8. [1] relationship valid if a straight line passing through the origin is produced (for lg k against lg A, relationship valid if a straight line with gradient -3/2)
9. [1] β = gradient × N / pt⁴ [for Ig k against Ig A, B = 10y-intercept × N / (pt)³]
10. [6] (D1) use safety goggles/safety screen to prevent injury to eyes from (moving) spring/load or use cushion/sand box in case load falls
11. (D2) keep t constant
12. (D3) k = mg/x or F/x
13. (D4) use of set square when taking measurements to determine extension of spring
14. (D5) repeat measurement of t along wire/spring and average
15. (D6) repeat measurement of diameter D of spring (to determine A) in different directions and average
16. (D7) use of A = πD²/4
17. (D8) method to ensure clamped rule to measure extension 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
18. (D9) method to determine the density of the wire or additional detail on construction of coil
19. (D10) method to determine the mean diameter of the spring, e.g. subtract t from external diameter of spring