Question 19: A metal wire has mass (0.4 \pm 0.002)\,\mathrm{g}, radius (0.3 \pm 0.001)\,\mathrm{mm} and length (5 \pm 0.02)\,\mathrm{cm}. The maximum possible percentage error in the measurement of density will nearly be:
(1) 1.6\%
(2) 1.4\%
(3) 1.2\%
(4) 1.3\%
Answer: Option (1)
Explanation:
Density of a wire is given by
\rho=\frac{m}{\pi r^{2}l}The maximum fractional error in density is equal to the sum of fractional errors of all measured quantities, with powers taken into account.
\frac{\Delta \rho}{\rho}=\frac{\Delta m}{m}+2\frac{\Delta r}{r}+\frac{\Delta l}{l}Fractional error in mass is
\frac{\Delta m}{m}=\frac{0.002}{0.4}=0.005Fractional error in radius is
\frac{\Delta r}{r}=\frac{0.001}{0.3}\approx0.00333Fractional error in length is
\frac{\Delta l}{l}=\frac{0.02}{5}=0.004Substituting these values,
\frac{\Delta \rho}{\rho}=0.005+2(0.00333)+0.004 \frac{\Delta \rho}{\rho}=0.01566Percentage error in density is
0.01566\times100\approx1.6\%Hence, the maximum possible percentage error in the measurement of density is nearly 1.6\%.