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Question

A solid body rotates with deceleration about a stationary axis with an angular deceleration |α|=kω; where 'k' is a constant and ω is the angular velocity of the body. If the initial angular velocity is ω0, then mean angular velocity of the body averaged over the whole time of rotation is

A
ω0
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B
ω02
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C
ω03
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D
ω04
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Solution

The correct option is C ω03
dωdt=kω1/2
ωω0dωω1/2=kt0dt
2[ω]ωω0=kt
2ω02ω=kt
2ω=2ω0kt;ω=(ω012kt)2
The body comes to a stop (ω=0) in time
t0=2ω0k
ω=(ω0kt2)2
ω=ω0+k2t24ktω0
Since <ωar>=t00ωdt00dt=ω0t0+k24t303kω0t202t0
<ωav>=ω0+k24t203kω0.t02
<ωar>=ω0+k2124ω0k2kω02ω02k
<ωav>=ω0+ω03ω0
<ωav>=ω03.

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