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 $0.5mm\&$ there are $50$ divisions on the circular scale. The reading on the main scale is $2.5mm\&$ 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____(Approximate the answer to the nearest integer.)

Answer 1

Least count $=\frac{0 . 5}{50}mm$
diameter of ball $=2.5mm+20\times \frac{0 . 5}{50}$
$=2.7mm$
$\rho =\frac{M}{V}=\frac{M}{\frac{4}{3} \pi R^{3}}$
$ln\rho =lnM-ln\left(\frac{4}{3} \pi R^{3}\right)$
$\frac{\Delta \rho }{\rho }\times 100=\frac{\Delta M}{M}\times 100+3\frac{\Delta R}{R}\times 100$
$=3.1\%$