A student is investigating the oscillations of a mass attached to two springs connected in series, as shown in Fig. 2.1. springs mass Fig. 2.1 A stopwatch is used to measure the time t for 10 oscillations. The measurement of t is repeated and the average period T is determined. The experiment is repeated for different masses. It is suggested that T and mass M are related by the equation T= (2πM^q) / √k where k is the spring constant of the two springs in series and q is a constant.

Mark scheme

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

1. [1] gradient = q
2. [1] y-intercept = lg(2π/√k) or lg 2π - 1/2 lg k
3. [1] Values of T as above. (1.56, 1.79, 1.97, 2.14, 2.31, 2.45)
4. [1] Values of lg T as above. (0.193 or 0.1931, 0.253 or 0.2529, 0.294 or 0.2945, 0.330 or 0.3304, 0.364 or 0.3636, 0.389 or 0.3892)
5. [1] Uncertainties in T all ±0.04.
6. [1] Uncertainties in lg (T/s) consistent with uncertainties in T e.g. from ±0.011 to ±0.007.
7. [1] 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.
8. [1] Error bars in lg T plotted correctly. All error bars must be plotted. Length of bar must be accurate to less than half a small square and symmetrical.
9. [1] Line of best fit drawn. Lower end of line should pass between (2.22, 0.22) and (2.25, 0.22) and upper end of line should pass between (2.49, 0.34) and (2.51, 0.34). Do not accept line from first to last plot.
10. [1] Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted.
11. [1] Gradient determined with clear substitution of data points from the line of best fit into Δy / Δx. Distance between data points must be greater than half the length of the drawn line.
12. [1] uncertainty = gradient of line of best fit – gradient of worst acceptable line or uncertainty = 1/2 (steepest worst line gradient – shallowest worst line gradient)
13. [1] y-intercept determined by substitution of correct point from the line of best fit into y = mx + c.
14. [1] k determined from y-intercept. k = (2π / 10^(y-intercept))^2 = (2π / 10^(c(iv)))^2
15. [1] q = answer to (c)(iii) and given to 2 or 3 significant figures.
16. [1] M determined from (d) or (c)(iii) and (c)(iv) with correct substitution shown. M = (T√k / 2π)^(1/q) = (T√k / 2π)^(1/(c(iii))) or M = 10^((lg T - y-intercept) / q) = 10^((lg 1 - lg(2π/√k)) / q)