A student is investigating the oscillations of a mass attached to an arrangement of springs. Fig. 2.1 shows a mass attached to two springs connected in series. [Figure 2.1] The student determines the spring constant k for the arrangement of the springs. A stopwatch is used to measure the time t for 20 oscillations. The measurement of t is repeated and the average period T is determined. The experiment is repeated for different arrangements and different numbers of springs. It is suggested that T and k are related by the equation T = 2π√(M/k) where M is the mass.

Mark scheme

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

1. [1] gradient = 4π²M
2. [1] Values of T as above. (1.12 or 1.120; 0.950 or 0.9500; 0.815 or 0.8150; 0.655 or 0.6550; 0.570 or 0.5700; 0.47 or 0.470)
3. [1] Values of T² as above. (1.25 or 1.254; 0.903 or 0.9025; 0.664 or 0.6642; 0.429 or 0.4290; 0.325 or 0.3249; 0.22 or 0.221)
4. [1] Uncertainties in T increase from ±0.01 to ±0.02.
5. [1] Uncertainties in T² about ±0.02.
6. [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.
7. [1] Error bars in T² plotted correctly. All error bars to be plotted. Length of bar must be accurate to less than half a small square and symmetrical.
8. [1] Line of best fit drawn. If points are plotted correctly then lower end of line should pass between (0.048, 0.5) and (0.052, 0.5) and upper end of line should pass between (0.098, 1.0) and (0.104, 1.0).
9. [1] Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). All error bars must be plotted.
10. [1] Gradient determined with clear substitution of points from the line of best fit into Δy/Δx. Distance between points must be at least half the length of the drawn line.
11. [1] uncertainty = gradient of line of best fit – gradient of worst acceptable line or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient)
12. [1] M determined from gradient and given to 2 or 3 significant figures and with correct unit. M = gradient / 4π² or M = (c)(iii) / 39.478
13. [1] % uncertainty in M = % uncertainty in gradient
14. [1] k calculated. Correct substitution of numbers required. k = 4π²M / T² or k = 4π²(d)(i) / 2.5² or 6.3165×(d)(i) or k = gradient / T² or k = (c)(iii) / 2.5² or (c)(iii) / 6.25
15. [1] Absolute uncertainty in k. Correct substitution of numbers required. Using M: uncertainty in k = (ΔM/M + 2×ΔT/T) × k or uncertainty in k = ((d)(ii)/100 + 0.008) × k or max k = 4π² × max M / min T² or min k = 4π² × min M / max T². Using gradient: uncertainty in k = (Δgradient/gradient + 0.008) × k or max k = max gradient / min T² or min k = min gradient / max T²