If the temperature of a uniform rod is slightly increased by Δt, its moment of inertia I about a perpendicular bisector increases by -
2αI∆t
We know that moment of inertia of a road along its perpendicular bisector is -
I = ml212.
As Δl is small, ΔI will be a much smaller quantity,
∴ ΔI ∼ dI.
And,
dI = dl2(m12)
⇒ ΔI ≈ dI = 2 × l × Δl × m12. ..........(1)
Now,
Δll = αΔt
⇒ Δl = αlΔt. ................(2)
From (1) & (2),
ΔI = 2 × (ml212) × αΔt
⇒ ΔI = 2αIΔt