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 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

Options

A. `0.9%`
B. `2.4%`
C. `3.1%`
D. `4.2%`

Answer

C

Explanation

(3) `LC = (0.5)/(50) = 0.01 cm`
Diameter `d = 2.5 xx (0.01) xx 20 = 2.7 mm`
`m = rho V = rho (4)/(3) pi ((d)/(3))^(3) = (pi rho d^(3))/(6)`
`rho = (6 m)/(pi d^(3))`
`(Delta rho)/(rho) xx 100 = (Delta m)/(m) xx 100 + 3 (Delta d)/(d) xx 100`
`= 2% + 3 xx (0.1)/(2.7) xx 100`
`= 2% + 1.1% = 3.1%`