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Question

A steel scale measures the length of a copper rod as L cm when both are at 20C, the calibration temperature for the scale. If the coefficients of linear expansion for steel and copper are αs and αc respectively, what would be the scale reading (in cm) when both are at 21C?

A
L(1+αc)(1+αs)
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B
Lαcαs
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C
Lαsαc
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D
L
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Solution

The correct option is A L(1+αc)(1+αs)
Given, the length of steel scale and copper rod at 20C =L .
For a change in temperature (ΔT), the length of a material changes to the new length given by,
L=L0(1+α ΔT)
For steel scale,
New length due to change in temperature is given by
Lsteel=L(1+αs ΔT) ......(1)
For copper rod,
New length due to change in temperature is given by
Lcopper=L(1+αc ΔT) ......(2)
Hence, the relative expansion of copper rod w.r.t. steel scale after the rise in temperature is given by
Lapp=LcopperLsteel=[L(1+αcΔT)L(1+αsΔT)]
[ Using (1) and (2)]
Lapp=L(αcαs)ΔT .....(3)
Given,
Initial temperature T1=20C
Final temperature T2=21C
Change in temperature (ΔT)=1C
From (3), we can write that
Lapp=L(αcαs)
Therefore, measured length at 21 C
=L+Lapp=L(1+αcαs)

By applying binomial expansion on option (a),
L(1+αc)(1+αs)=L(1+αc)(1+αs)1
=L(1+αc)(1αs)
=L(1+αcαsαcαs)
[α value is very small, hence we can neglect αcαs]
L(1+αcαs)=Lapp
Thus, option (a) is correct.

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