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Question

Two straight metallic strips each of thickness t and length L are riveted together. Their coefficients of linear expansion are α1 and α2. If they are heated through temperature Δθ, the bimetallic strip will bend to form an arc of radius


A
t(α1+α2)Δθ
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B
t(α2α1)Δθ
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C
t2(α1+α2)Δθ
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D
t2(α2α1)Δθ
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Solution

The correct option is B t(α2α1)Δθ
Given:
Thickness of metallic strips =t
Length of metallic strips =L
Coefficient of linear expansion of strip 1 =α1
Coefficient of linear expansion of strip 2 =α2
Temperature change =Δθ
To find:
Radius of arc of bimetallic strip =r

Before heating -


After heating -


Let the angle subtended by the arc formed at the centre be θ.
We know,
θ=lR, where l is the length of arc and R is radius of arc.
For strip 1,
θ=L(1+α1Δθ)r.......(1)
[ from formula of linear expansion ]
Similarly, for strip 2,
θ=L(1+α2Δθ)r+t.......(2)
Angle subtended by both the strips at the centre will be equal, so from (1) and (2),
L(1+α1Δθ)r=L(1+α2Δθ)r+t
r=t(1+α1Δθ)(α2α1)Δθ
As 1>>α1Δθ [because α<<1]
r=t(α2α1)Δθ

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