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

Pitch of the screw gauge is 0.5 mm and there are 50 divisions on circular , hence least count of the instrument can be written as follows :
Least count LC `=(0.5)/(50)=0.01mm`
Diameter of the ball can be written as
`d= MSR+CSR xx LC= 2.5+20xx0.001= 2.5+0.20 = 2.70mm`
Error in the measurement of diameter is equal to least count and can be written as
`triangle d= 0.01mm`
Density p of the meterial can be written as follows :
`p=("mass")/("volume")=(m)/((4)/(3)pi(d/2)^(3))=(6m)/(pid^(3))`
From the above expression we can write the following :
`(trianglep)/(p)xx100=(trianglem)/(m)xx100+3(triangled)/(d)xx100`
`(trianglep)/(p)xx100=2+1.11=3.11%`