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Q.

A rigid body rotates about a fixed axis with variable angular velocity equal to $α - β t$ , at time t where $α, β$ are constants. The angle through which it rotates before its stops:

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a

$\frac{(α - β) α}{2}$

b

$\frac{α^{2} - β^{2}}{2 β}$

c

$\frac{α^{2} - β^{2}}{2 α}$

d

$\frac{α^{2}}{2 β}$

answer is A.

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Detailed Solution

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$ω = α - β t = 0; t = α / β$

$\int_{0}^{θ} d θ = \int_{0}^{t} ω dt; θ = \int_{0}^{α / β} (α - β t) dt$

$θ = {[α t - β t^{2} / 2]}_{0}^{α / β}$

$θ = α \times α / β - \frac{β}{2} \frac{α^{2}}{β^{2}}; θ = \frac{α^{2}}{2 β}$

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A rigid body rotates about a fixed axis with variable angular velocity equal to α−βt, at time t where α,β are constants. The angle through which it rotates before its stops: