Physics - Unit & Dimension Question with Solution | TestHub

As density$=\frac{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}$

$\therefore \rho =\frac{M}{\frac{4}{3}\pi R^3}=\frac{3}{4}\frac{M}{\pi R^3}$

$\therefore$The percentage error in density is

Taking "log" both sides we get$\ln \rho =\ln \frac{3}{4}+\ln M-\ln \pi -3\ln R$

differentiating, we get$\frac{\Delta \rho }{\rho }=\frac{\Delta M}{M}-\frac{3\Delta R}{R}$

But as error is always added$\Rightarrow \frac{\Delta \rho }{\rho }=\frac{\Delta m}{m}+\frac{3\Delta R}{R}$

$\Rightarrow$$\frac{\mathrm{\Delta }\rho }{\rho }\times 100\%=\left(\frac{\mathrm{\Delta }M}{M}+3\frac{\mathrm{\Delta }R}{R}\right)\times 100\%$

$=3\%+3\left(2\%\right)=3\%+6\%=9\%$