A particle at rest starts moving with constan | vedclass

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Question

(C) Given: Angular acceleration = $\alpha$, Radius = $r$, Initial angular velocity $\omega_0 = 0$.At any time $t$, angular velocity $\omega = \omega_0

Vedclass · Accepted Answer

$V = r\sqrt{\frac{\alpha}{3}}$

Vedclass · Answer

(C) Given: Angular acceleration = $\alpha$, Radius = $r$, Initial angular velocity $\omega_0 = 0$.At any time $t$, angular velocity $\omega = \omega_0 + \alpha t = \alpha t$.Centripetal acceleration $a_c = \omega^2 r = (\alpha t)^2 r = \alpha^2 t^2 r$.Tangential acceleration $a_t = r\alpha$.According to the problem, $a_c = \frac{1}{3} a_t$.Substituting the values: $\alpha^2 t^2 r = \frac{1}{3} (r\alpha)$.$\alpha^2 t^2 = \frac{\alpha}{3} \implies t^2 = \frac{1}{3\alpha} \implies t = \frac{1}{\sqrt{3\alpha}}$.Linear speed $V = \omega r = (\alpha t) r$.Substituting $t$: $V = \alpha \left( \frac{1}{\sqrt{3\alpha}} \right) r = r \sqrt{\frac{\alpha^2}{3\alpha}} = r \sqrt{\frac{\alpha}{3}}$.

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