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Question

A solid body rotates about a stationary axis with an angular deceleration βω 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 ω03
Given, β is deceleration.
So, β=dωdtω
or dωdt=Kω
where K is the proportionality constant.
i.e ωω0dωω=t0K.dt
Integrating both sides and applying limits,
ω=ω0Kt2 ... (1)

When, ω=0, total time of rotation,
t=2ω0K
As we know, Average angular velocity ωav=ω.dtdt
From (1),
ω=(ω0)2+(Kt2)22ω0Kt2

Thus, ωav=2ω0K0(ω0+K2t24Ktω0)dt2ω0K
ωav=[ω0t+K2t33×4K2ω0t2]2ω0K02ω0K
ωav=ω03

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