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Question:

In the determination of Young's modulus (Y=4MLg/πld²) by using Searle's method, a wire of length L=2m and diameter d=0.5mm is used. For a load M=2.5kg, 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. They have the same pitch of 0.5mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement:

due to the error in the measurement of d is twice that due to the error in the measurement of l.

due to the errors in the measurements of d and l are the same.

due to the error in the measurement of d is four times that due to the error in the measurement of l.

due to the error in the measurement of l is twice that due to the error in the measurement of d.

Solution:

L.C=0.5/100=0.005
ΔY/Y=Δl/l+2Δd/d
Δl=0.005 × 10⁻³ / (0.25 × 10⁻³) = 0.02
Δd=2 × 0.005 × 10⁻³ / (0.5 × 10⁻³) = 0.02
ΔY/Y = 0.02 + 2(0.02) = 0.06
Therefore, the contributions to the maximum probable error of the Y measurement due to the errors in the measurements of d and l are the same.