Question

# A solid body rotates about a stationary axis with an angular deceleration β∝√ω where ω is its angular velocity. If at the initial moment of time, its angular velocity was equal to ω0, then the mean angular velocity of the body averaged over the whole time of rotation till it comes to rest is,

A
ω03
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B
ω03
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C
ω02
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D
ω02
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Solution

## The correct option is B ω03Given, β is deceleration. So, β=−dωdt∝√ω or −dωdt=K√ω where K is the proportionality constant. i.e −ω∫ω0dω√ω=t∫0K.dt Integrating both sides and applying limits, ⇒√ω=√ω0−Kt2 ... (1) When, ω=0, total time of rotation, t=2√ω0K As we know, Average angular velocity ωav=∫ω.dt∫dt From (1), ω=(√ω0)2+(Kt2)2−2√ω0Kt2 Thus, ωav=2√ω0K∫0(ω0+K2t24−Kt√ω0)dt2√ω0K ωav=[ω0t+K2t33×4−K2√ω0t2]2√ω0K02√ω0K ⇒ωav=ω03

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