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A sphere is spinning about its diameter. Its temperature is gradually increased by $Δ T$ . The coefficient of linear expansion of sphere is $α$ . Match the columns:
Column - I Column - II
A) Fractional change in angular momentum p) $α Δ T$
B) Fractional change in moment of inertia q) $- 2 α Δ T$
C) Fractional change in angular velocity r) $2 α Δ T$
D) Fractional change in kinetic energy s) zero

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A - r , B - s , C - p , D - p

A - q , B - p , C - r , D - s

A - s , B - p , C - r , D - p

A - s , B - r , C - q , D - q

answer is D.

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Detailed Solution

$A \to s; B \to r; C \to q; D \to q$

$I = \frac{2}{5} m R^{2}$

$∴ \frac{Δ I}{I} = \frac{Δ M}{M} + \frac{2 Δ R}{R}$

$= 0 + 2 α Δ T = 2 α Δ T$

In the process, angular momentum is constant and so,

$\frac{Δ L}{L} = 0$

As $L = I ω$

$∴$ $\frac{Δ L}{L} = \frac{Δ I}{I} + \frac{Δ ω}{ω}$

or $0 = 2 α Δ T + \frac{Δ ω}{ω}$

or $\frac{Δ ω}{ω} = - 2 α Δ T$

Also $\frac{Δ k}{k} = - 2 α Δ T$ .

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