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Question

When the temperature of a rod increases from t to t+Δt, its moment of inertia about a perpendicular axis passing through its centre increases from I to I+ΔI. If α be the coefficient of linear expansion of the rod, then the value of ΔII is

A
2αΔt
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B
αΔt
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C
3αΔt
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D
4αΔt
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Solution

The correct option is A 2αΔt
Given:
Temperature change =Δt
Change in moment of inertia =ΔI
Coefficient of linear expansion of the rod =α
To find:
Ratio, ΔII=?
We know that, moment of inertia of a rod about a perpendicular axis passing through its centre is given by
I=mL212 ........(1)
where m is mass and L is the length of the rod.
Differentiating w.r.t. L, we get
dIdL=112×2mL
ΔI=2×112×mLΔL
ΔI=2×112m×L2L×ΔL=2IΔLL
[ from (1) ]
ΔI=2IαLΔTL
[ from formula of linear expansion ]
ΔII=2αΔT

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