Show that dl/dt = τ, where L is the angular momentum and τ is the torque.
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As we know, l = r × p, differentiating on both the sides,
⇒ \frac{dl}{dt}
dt
dl
= \frac{d}{dt}
dt
d
( r × p)
⇒ \frac{dl}{dt}
dt
dl
= \frac{dr}{dt}
dt
dr
× p + r × \frac{dp}{dt}
dt
dp
⇒ \frac{dl}{dt}
dt
dl
= v × p + r × \frac{dp}{dt}
dt
dp
⇒ \frac{dl}{dt}
dt
dl
= v × mv + r × \frac{dp}{dt}
dt
dp
since, v = 0
⇒ \frac{dl}{dt}
dt
dl
= r × \frac{dp}{dt}
dt
dp
= r × F = τ
⇒ \frac{dl}{dt}
dt
dl
= τ
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