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. 3.1%
• D. 4.2%

Answer 1

Answer: (C)

Least count of screw gauge $=\frac{0.5}{50}=0.01 \,mm =\Delta r$
Diameter $r=2.5\, mm +20 \times \frac{0.5}{50}=2.70\, mm$
$\frac{\Delta r}{r}=\frac{0.01}{2.70}$
or $ \frac{\Delta r}{r} \times 100=\frac{1}{2.7}$
Now, density $d=\frac{m}{V}=\frac{m}{\frac{4}{3} \pi\left(\frac{r}{2}\right)^{3}}$
Here, $r$ is the diameter.
$\therefore \frac{\Delta d}{d} \times 100=\left\{\frac{\Delta m}{m}+3\left(\frac{\Delta r}{r}\right)\right\} \times 100$
$=\frac{\Delta m}{m} \times 100+3 \times\left(\frac{\Delta r}{r}\right) \times 100$
$=2 \%+3 \times \frac{1}{2.7}=3.11 \%$