Question

what is the relation between torque and angular momentum

Answer 1

Tipper Rumpf gives an acceptable definition of angular momentum and torque, but this question specifically asks about the relationship between the two.

Angular moment is the canonical momentum conjugate to rotational displacement. Torque is the generalised force associated to rotational displacement.

In the Lagrangian formulation of a system, the Euler-Lagrange equation of momentum stipulates that the generalised force (dL/dq) is the time derivative of the generalised momentum (dL/d q dot). Therefore, the relationship between angular momentum (l) and torque (T) is that dl/dt = T. Torque equals the time derivative of angular momentum.

Answer 2

Hey !!

We know that,

Torque,           τ = r × F

Angular momentum,     L = r × p

On differentiating both sides w.r.t time t, we get

         dL/dt = d/dt (r × p) = dr/dt × p × r × dp/dt

          = v × p + r × F                      [∴ dp/dt = F]

            = 0 + τ

∵              τ = dL/dt

Thus, the torque acting on a particle is equal to its rate of change of angular momentum.

Good luck !!