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Q.

A ceiling fan starts rotating from rest with angular acceleration varying as $α = \sqrt{θ}$ , $' θ'$ being angular displacement magnitude. Then $(ω$ is the angular velocity)

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a

$ω^{2} \propto θ^{3 / 2}$

b

$θ \propto ω^{2 / 3}$

c

$ω \propto θ$

d

$ω \propto θ^{2}$

answer is B.

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Detailed Solution

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$α = \frac{d ω}{d t} = \frac{d ω}{d θ} \times \frac{d θ}{d t} = \frac{d ω}{d θ} \times ω$

$α \propto \sqrt{θ}$ (given)

$\frac{d ω}{d θ} \times ω = K θ^{1 / 2}$

$\int ω d ω = \int K θ^{1 / 2} d θ$

$\frac{ω^{2}}{2} = K \frac{θ^{3 / 2}}{3 / 2}$

$ω^{2} \propto θ^{3 / 2}$

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A ceiling fan starts rotating from rest with angular acceleration varying as α=θ , 'θ' being angular displacement magnitude. Then ( ω is the angular velocity)