Question

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is $0.5$mm and there are $50$ divisions on the circular scale. The reading on the main scale is $2.5$ mm and that on the circular scale is $20$ divisions. If the measured mass of the ball has a relative error of $2\%$, the relative percentage error in the density is?

• A. $0.9\%$
• B. $2.4\%$
• C. $3.2\%$
• D. $4.2\%$

Answer 1

The correct option is C $3.2\%$

$\begin{array}{cc}{L}_{-}C=& \frac{Pitch}{\text{No. of division}C.S}\\ & =\frac{0.5mm}{50}=0.01mm\\ M\cdot S\cdot R& =2.5mm\end{array}$

$\begin{array}{cc}CSR& =20\times L\cdot C\\ & =20\times 0.01=0.02mm\\ \text{diameter}\left(d\right)& =2\cdot 5+0.02\\ & =2.52mm\\ \text{volume}\left(V\right)& =\frac{4}{3}\pi {\left(\frac{d}{2}\right)}^3\Rightarrow \frac{\pi d^3}{6}\\ \text{density}\left(\rho \right)& =\frac{\text{Mass}\left(m\right)}{V}=\frac{6m}{\pi d^3}⟶\left(i\right)\end{array}$

$\begin{array}{c}\text{Taking log}\\ \log \rho =\log 6+\log m-\log \pi -3\log d\\ \text{Differentiating;}\\ \frac{\delta \rho }{\rho }=\frac{\delta m}{m}+3\frac{\delta d}{d}\begin{array}{c}\text{(error is}\text{always added)}\end{array}\\ \begin{array}{cc}\frac{\delta \rho }{\rho }\times 100& =\frac{\delta m\times 100}{m}+3\frac{\delta d}{d}\times 100\\ & =2\%+3\times \frac{0.01}{2\cdot 52}\times 100\\ & =2\%+1.2\%\\ & =3.2\%\end{array}\end{array}$