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 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% the relative %error in density is
a)0.9% b)2.4% c)3.1% d)4.2%

Answer 1

$$\begin{gathered} sothediameteris=2.5+(\frac{0.5}{50}\times 20)=2.7 \\ theerrorindiameterestimationis\delta r=0.01mm \\ \frac{\delta D}{D}=\frac{0.01}{2.7}=0.0037 \\ now\rho =\frac{m}{v} \\ \ln \left(\rho \right)=\ln m+\ln \left(\frac{4}{3}\pi {\left(D/2\right)}^3\right) \\ \frac{\delta \rho }{\rho }=\frac{\delta m}{m}+3\frac{\delta D}{D} \\ \frac{\delta \rho }{\rho }=3.1\% \end{gathered}$$