Question

define angular momentum and torque .derive relation b/w them.

Answer 1

Angular momentum(L):- It is an measure of the rotation of body, taking into account its mass , shape and speed.

L = r ×​ p
where r is distance from the axis of rotation and the p is the linear momntum of the body.

Torque(τ): It is an moment of force i.e. tendency of a force to rotate an object about an axis.

τ = r × f
where r is the perpendicular distance from the axis of the rotation and f is the applied force.

For rotational motion torque is given by:
τ = I$\alpha$ ----- (1)
where I is the moment of inertia of the body about an axis of rotation.
$\alpha$ is the angular acceleration.
We know for the rotational body angular momentum is conseverd. L = I$\omega$ ----- (2)

where $\omega$ is the angular velocity.

As we know
$\alpha = d\omega /dt$ ----(3)

taking derivative of equqtion no. (2) with respect to time, we get.
dL/dt = I$d\omega /dt$ ----(4)

using (3) &(4), we get.
dL/dt = I$\alpha$ ---- (5)

compare equation no (1) & (5), we get.

$\tau =dL/dt$
This is the relationship between torque and angular momentum.