Answer to Question #172718 in Physics for Adewole Ifeoluwa

Question #172718

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?


1
Expert's answer
2021-03-21T11:36:33-0400

The circumference can be calculated as follows:


C=πdC = \pi d

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

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


ΔC=πΔdδC=ΔCC=πΔdπd=ΔddδC=0.01cm2cm×100%=0.5%\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=πd2L4V = \dfrac{\pi d^2L}{4}

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


ΔV=πdΔd2δV=ΔVV=2Δdd=20.5%=1%\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.


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