The density of a solid metal cylinder…

The density of a solid metal cylinder is to be found. Its mass is measured as 8.42 g with a balance of least count 0.01 g. Its diameter is measured…

Numerical answer

The density of a solid metal cylinder is to be found. Its mass is measured as

8.42\ \mathrm{g}

with a balance of least count

0.01\ \mathrm{g}

. Its diameter is measured with a screw gauge of pitch

1\ \mathrm{mm}

and

100

circular divisions: the main scale reads

4\ \mathrm{mm}

and the circular scale reads

36

divisions, while with the jaws closed the circular scale reads

5

divisions (the zero lies below the reference line, so the gauge reads

+5

divisions when closed). Its length is

62.4\ \mathrm{mm}

measured with vernier calipers of least count

0.1\ \mathrm{mm}

. Using least counts as the uncertainties, find the maximum percentage error in the calculated density. Give the answer correct to two decimal places.

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Answer: 0.74

Screw gauge least count

=1/100=0.01\ \mathrm{mm}

. The corrected diameter is

d=4+36\times0.01-5\times0.01=4.31\ \mathrm{mm}

(positive zero error is subtracted). Density

\rho=\dfrac{m}{\pi d^2l/4}=\dfrac{4m}{\pi d^2l}

, so the fractional uncertainty is

\dfrac{\Delta\rho}{\rho}=\dfrac{\Delta m}{m}+2\dfrac{\Delta d}{d}+\dfrac{\Delta l}{l}

. Substituting:

\dfrac{0.01}{8.42}+2\times\dfrac{0.01}{4.31}+\dfrac{0.1}{62.4}=0.001188+0.004640+0.001603=0.007430

. The maximum percentage error is

0.74\%

(the density itself is

\approx9.25\ \mathrm{g\,cm^{-3}}

). Using

d=4.36

mm without the zero correction changes the error only slightly (

0.7377\%\approx0.74\%

) but gives a wrong density (

\approx9.04\ \mathrm{g\,cm^{-3}}

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Original question written in the JEE Advanced 2025 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake