In the determination of Young's modulus `(Y = (4MLg)/(pi// d^2) )` by using Searle's method, a wire of length L= 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively. The have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement

Options

A. due to the errors in the measurement of d and l are the same
B. due to the error in the measurement of d is twice that due to the error in the measurement of l
C. due to the error in the measurement of l is twice that due to the error in the measurement of d
D. due to the error in the measurement of d is four times that due to the error in the measurement of l

Answer

A

Explanation

Least count of the screw gauge and micrometer both are same and can be written as follows :
Least count LC `=("pitch")/("number of divisions on CS")=(0.5)/(100)`
Error in the measurement of d and I will be equal to least count.
`triangled= triangle l= 0.005 mm`
From the given relation `(Y=(4 MLg)/(pi l d^(2)))` we can write the following :
`(triangleY)/(Y)=(triangle l)/(l)+2(triangled)/(d)`
Contribution to total relative error due I can be written as
`(trianglel)/(l)=(0.005)/(0.25)=0.02`
Contribution in total error due to d can be written as
`2(triangled)/(d)=2(0.005)/(0.5)=0.02`
We can see that both the contributions are equal and thus option a is correct.