Q#433: [Measurement > Uncertainty] (Past Exam Paper – November 2012 Paper 11 Q6)
Quantity X varies with temperature θ as shown.
θ is determined from corresponding values of X by using this graph.
X is measured with percentage uncertainty of ±1 % of its value at all temperatures.
Which statement about the uncertainty in θ is correct?
A The percentage uncertainty in θ is least near 0 °C.
B The percentage uncertainty in θ is least near 100 °C.
C The actual uncertainty in θ is least near 0 °C.
D The actual uncertainty in θ is least near 100 °C.
Solution 433:
Answer: C.
Percentage uncertainty in X is given by (ΔX / X) x 100%.
The graph obtained is a straight line.
The general equation for a straight line is: y = mx + c (do not confuse the x-axis with the X quantity in the question)
So, the equation of the graph shown in the question is
X = mθ + c = mθ
(since the y-intercept c is zero)
Considering the percentage uncertainties,
(ΔX / X) x 100% = (Δθ / θ) x 100%
From the above equation, the percentage uncertainty in θ is equal to the percentage uncertainty in X, which is itself a constant.
So, the percentage uncertainty in θ is also constant and equal to 1%. [A and B are incorrect]
So, (Δθ / θ) x 100% = 1%
Δθ / θ = 0.01
The actual uncertainty Δθ = 0.01 (θ)
Thus, when θ is smallest, the actual uncertainty is also smallest.
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