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. 0.009
B. 0.024
C. 0.0311
D. 0.042

Answer

(c )

Explanation

Least count of screw gauge `=("pitch")/("no.of divisions on cricular scale")`
`=(0.5mm)/(50) = 0.01mm.`
Diameter of ball, D = MSR + CSR xx LC
=2.5 mm + (20)xx0.01 mm = 2.7 mm
As density, `rho = (mass)/(volume) = (M)/((4pi)/(3)((Delta)/(2))^3)`
Relative error in the density, `(Delta rho)/(rho) = (Delta M)/(M) + (3 Delta D)/(D)`
Relative percentage error in the density is
`=(Delta rho)/(rho)xx100 = ((Delta M)/(M) + (3 Delta D)/(D))xx100`
`=(DeltaM)/(M)xx100 + (3 Delta D)/(D)xx100`
`=2% + 3xx(0.01)/(2.7)xx100`
= 2% + 1.11 % = 3.11 %