In the determination of Young’s modulus by using Searle’s method, a wire of length and diameter is used. For a load , an extension in the length of the wire is observed. [2012]

Question

In the determination of Young’s modulus $(Y = \frac{4 M L g}{π l d^{2}})$ 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. They 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 [2012]

Smart Achievers · Accepted Answer

(1)

$The maximum possible error in Y due to l and d$

$\frac{Δ Y}{Y} = \frac{Δ l}{l} + \frac{2 Δ d}{d}$

$Least count = \frac{Pitch}{No. of divisions on circular scale}$

$= \frac{0.5}{100} mm = 0.005 mm$

$Here, Δ d = Δ l = 0.005 mm$

$Error contribution of l = \frac{Δ l}{l} = \frac{0.005 mm}{0.25 mm} = \frac{1}{50}$

$Error contribution of d = \frac{2 Δ d}{d} = \frac{2 \times 0.005 mm}{0.5 mm} = \frac{1}{50}$

$Hence contribution to the maximum possible error in the measurement of y due to l and d is the same.$

Solution

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

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

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

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

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Solution

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

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

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

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

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

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

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

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