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 $0.5mm$ and there are $50$ divisions on the circular scale. The reading on the main scale is $2.5mm$ and that on the circular scale is $20$ divisions. If the measured mass of the ball has a relative error of $2\%$, then the relative percentage error in the density is:

The correct option is D $3.1\%$

Given

Screw gauge readings

Pitch $=0.5mm$

Circular scale division $=50$

Main scale reading $=2.5mm$

Least count =$\frac{0.5}{50}=0.01mm$

Circular scale division reading $=20divisions$

Relative error $=2$%

So,

Then dimensions reading from screw gauge= Diameter of a solid ball

S.G. Reading = main scale reading $+\left(\frac{pitch}{circularscaledivision}\right)\times$ circular scale division reading

Diameter of solid ball $=2.5+\frac{0.5}{50}\times 20=2.7mm$

$Asdensity\rho =\frac{mass}{Volume}=\frac{M}{\frac{4\pi }{3}{\left(\frac{D}{2}\right)}^3}$

The relative percentage error in the density is

$\frac{\Delta \rho }{\rho }\times 100=(\frac{\Delta M}{M}+\frac{3\Delta D}{D})\times 100$

% error $=(\frac{\Delta M}{M}\times 100+\frac{3\Delta D}{D})\times 100$

$=(2+3\times \frac{0.01}{2.7})\times 100$

$=3.1$ %