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System Of Particles And Rotational Motion

Question

A wheel starts rotating from rest at time t = 0 with a angular acceleration of 50 radians/s². The angular acceleration ( $α$ ) decreases to zero value after 5 seconds. During this interval, a varies according to the equation
$α = α_{0} (1 - \frac{t}{5})$
The angular velocity at t = 5 s will be

10 rad/s

250 rad/s

125 rad/s

100 rad/s

Solution

125 rad/s

Given:-

$α = α_{0} (1 - \frac{t}{5})$

At t = 0, $α = α_{0}$

∴ $α_{0}$ = 50 rad/s²

$\frac{dω}{dt} = α (1 - \frac{t}{5})$

∴ $\int_{0}^{ω} dω = α_{0} \int_{0}^{5} (1 - \frac{t}{5})$

⇒ ω = $α_{0} {[t - \frac{t^{2}}{10}]}_{0}^{5}$

⇒ = 50 $(5 - \frac{25}{10})$ rad/s

⇒ ω = 125 rad/s