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A sphere of coefficient of linear expansion $α$ , mass 'm' and radius 'r' is spinning about an axis through its diameter with an angular velocity $' ω'$ ’ when it is heated such that its temperature increases by $∆ t$ , the angular velocity becomes

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a

$ωαΔt$

b

$ω (1 + αΔt)^{2}$

c

$\frac{ω}{(1 + 2 αΔt)}$

d

$ω (1 + αΔt)$

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{ΔI}{I_{1}} = 2 α Δt \Rightarrow \frac{I_{2} - I_{1}}{I_{1}} = 2 α Δt \\ I_{2} - I_{1} = 2 I_{1} α Δt \Rightarrow I_{2} = I_{1} [1 + 2 α Δt] \end{array}$
According to law of conservation of angular momentum, $I_{1} ω_{1} = I_{2} ω_{2}$
$I_{1} ω_{1} = I_{1} [1 + 2 αΔt] ω_{2} \Rightarrow ω_{2} = \frac{ω_{1}}{1 + 2 αΔt}$

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