Question

A metal cylinder has a length of 4 cm and diameter 2 cm. What percent uncertainty is introduced into the calculated circumference by an uncertainty of 0.01 cm in the diameter measurement? What percent uncertainty does this introduce in the calculated volume?

Answer 1

The circumference can be calculated as follows:

$$C = \pi d$$

where $d = 2cm$ is the diameter. Let the uncertainty in the diameter measurement be $\Delta d = 0.01cm$.

Taking the differential, and dividing by $C$ in order to find percent uncertainty, obtain:

$$\Delta C = \pi\Delta d\\ \delta_C = \dfrac{\Delta C}{C} = \dfrac{\pi\Delta d}{\pi d} = \dfrac{\Delta d}{d} \\ \delta_C = \dfrac{0.01cm}{2cm}\times 100\% = 0.5\%$$

The volume can be calculated as follows:

$$V = \dfrac{\pi d^2L}{4}$$

where $L = 4cm$ is the length of the cylinder. Taking the differential and dividing by $V$, obtain:

$$\Delta V = \dfrac{\pi d\Delta d}{2}\\ \delta_V = \dfrac{\Delta V}{V} = \dfrac{2\Delta d}{d} = 2\cdot 0.5\% = 1\%$$

Answer. 0.5% and 1% respectively.