General Physics Ques 28
- 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 $0.5 \mathrm{~mm}$ and there are 50 divisions on the circular scale. The reading on the main scale is $2.5 \mathrm{~mm}$ 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
(2011)
(a) $0.9 %$
(b) $2.4 %$
(c) $3.1 %$
(d) $4.2 %$
Show Answer
Answer:
Correct Answer: 28.( c )
Solution:
- Least count of screw gauge $=\frac{0.5}{50}=0.01 \mathrm{~mm}=\Delta r$
Diameter, $r=2.5 \mathrm{~mm}+20 \times \frac{0.5}{50}=2.70 \mathrm{~mm}$
$ \frac{\Delta r}{r}=\frac{0.01}{2.70} \text { 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.
$ \begin{aligned} \therefore \frac{\Delta d}{d} \times 100= & {\frac{\Delta m}{m}+3(\frac{\Delta r}{r})} \times 100 \\ & =\frac{\Delta m}{m} \times 100+3 \times(\frac{\Delta r}{r}) \times 100 \\ & =2 %+3 \times \frac{1}{2.7}=3.11 % \end{aligned} $
BETA
