Question

define the angular momentum and torque? prove that the time rate of charge of angular momentum is equal to the applied external torque?

Answer 1

Angular momentum-the quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity.
Torque-Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point, and will label 'O'. We will call the force 'F'.
Torque acting on system of particles

Torque acting on the system of particles from equation above angular momentem of the system of particles is given as


When particles of the system are in motion then their motion is due to external and interaction due to internal forces so force acting on any particle of the system is given by

 
Here Fiext is the external force acting on the ith particle and ∑Fij is the sum of the force acting on the particle due to internal interaction of different particles.Putting the value of Fi in the equation we get 


RHS of the equation 2 shows that summantion of the moment of interacting force (internal) .Here internal interaction forces balance each other so torque due to internal forces adds to zero hence 
 

Above equation proves that total torque acting on a system of particles is equal to the vector sum of the torque acting on the different particles due to external force on the particle and its value is also equal to the rate of change of angular momentum