Question

Show that (dL)/(dt)=tau where L is the angular momentum and tau is the torque.​

Answer 1

We have to show that dL/dt = τ where L is the angular momentum and τ is the torque.

solution : let an arbitrary object moves with an angular velocity ω, moment of inertia of which is I about an axis passing through axis of rotation, r is the distance from the point of observation.

so, angular momentum, L = mvr = m(ωr)r = mωr²

differentiating with respect to time t,

dL/dt = d(mωr²)/dt = mr² dω/dt

we know, I = mr² [ for an arbitrary object , we can assume it]

and rate of change of angular velocity is angular acceleration

dL/dt = Iα

because torque is the product of moment of inertia and angular acceleration.

∴ τ = Iα

Therefore dL/dt = τ