A student is investigating the motion of a small steel ball in cooking oil. A measure of the oil's resistance to the ball's motion is called viscosity. Viscosity has the units pascal second (Pa s). The student drops a ball into a cylinder of oil as shown in Fig. 2.1. [Figure 2.1] The velocity of the ball is measured when it becomes constant and then the viscosity of the oil is determined. The experiment is repeated for different temperatures of oil. It is suggested that the viscosity η and the Celsius temperature θ are related by the equation η = ρθ^q where p and q are constants.

Mark scheme

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

1. [1] gradient = q
2. [1] y-intercept = lg p
3. [1] all lg(θ/°C) and lg(η/10⁻³ Pas) values correctly calculated and recorded in Fig. 2.2 (e.g. lg(θ/°C): 1.58 or 1.580, 1.66 or 1.663, 1.74 or 1.740, 1.81 or 1.806, 1.86 or 1.857, 1.90 or 1.898; lg(η/10⁻³ Pas): 1.61 or 1.613, 1.51 or 1.505, 1.40 or 1.398, 1.30 or 1.301, 1.23 or 1.230, 1.15 or 1.146)
4. [1] absolute uncertainties in lg η: ± 0.01 to ± 0.03 (based on propagation of uncertainty from ±1 in η/10⁻³)
5. [1] six points plotted correctly must be accurate to the nearest half small square diameter of points must be less than half a small square
6. [1] error bars in lg η plotted correctly all error bars to be plotted total length of bar must be accurate to less than half a small square and symmetrical
7. [1] line of best fit drawn points must be balanced do not allow line from top plot to bottom plot if points are plotted correctly then lower end of line should pass between (1.820, 1.275) and (1.835, 1.275) and upper end of line should pass between (1.640, 1.525) and (1.650, 1.525)
8. [1] worst acceptable line drawn steepest or shallowest possible line mark scored only if all error bars are plotted
9. [1] gradient determined with clear substitution of data points into Δy / Δx; distance between data points must be at least half the length of the drawn line must be negative
10. [1] uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient)
11. [1] y-intercept determined by substitution of correct point into y = mx + c
12. [1] y-intercept of worst acceptable line determined by substitution into y = mx + c; uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept); no ECF from false origin method
13. [1] p = 10^(y-intercept) and given to 2 or 3 sf
14. [1] q = gradient and q and p have correct power of ten from (c)(iii) and (c)(iv)
15. [1] absolute uncertainty in p = |10^(y-intercept_best_fit) - 10^(y-intercept_worst_line)| (or similar expression involving powers of 10 and max/min y-intercept values) absolute uncertainty in q = uncertainty in gradient correct substitution of numbers must be seen
16. [1] θ = (100 / p)^(1/q) or lg θ = (lg(100) - lg p) / q or lg θ = (2 - y-intercept) / gradient; correct substitution of numbers must be seen