CAIE A-Level Physics 1.3.3 Uncertainty in Derived Quantities

Practise propagating absolute and percentage uncertainties through sums, differences, products, quotients and powers, then reporting matching precision.

Exam points

Combine absolute uncertainties for sums and differences.
Combine percentage uncertainties for products, quotients and powers, applying the relevant power multiplier.
Convert between combined percentage and absolute uncertainty and quote the final value to matching precision.

[Maximum number: 6]

(a)

The density $\rho$ of the rock from which the pebble in (b) is composed is given by

\rho=\frac{M r^{n}}{k L}

where n is an integer and k is a constant, with no units, that is equal to 2.094 .

[ 6 ]

(i)

Calculate the percentage uncertainty in $\rho$ .

percentage uncertainty =% [3]

[ 3 ]

$(\Delta \rho / \rho)=(\Delta M / M)+2(\Delta r / r)+(\Delta L / L)$

C1

percentage uncertainty $=[(0.001 / 1.072)+2 \times(0.0004 / 0.0420)+(0.0001 / 0.1242)](\times 100)$

C1

$\begin{aligned} =0.09 \%+2 \times 0.95 \%+0.08 \% =2 \% \end{aligned}$

A1

(ii)

Determine $\rho$ with its absolute uncertainty. Give your values to the appropriate number of significant figures.

[ 3 ]

$\begin{aligned} \rho =\left(1.072 \times 0.0420^{-2}\right) /(2.094 \times 0.1242) =2337\left(\mathrm{~kg} \mathrm{~m}^{-3}\right) \end{aligned}$

C1

$\begin{aligned} \Delta \rho =0.021 \times 2337 =49\left(\mathrm{~kg} \mathrm{~m}^{-3}\right) \end{aligned}$

C1

$\rho=(2340 \pm 50) \mathrm{kg} \mathrm{m}^{-3}$

A1

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1.3.3—Assess the uncertainty in a derived quantity by simple addition of absolute question 1

Question (a)

Question (i)

Question (ii)