Question

(38) Define angular momentum of a particle and obtain relation between angular mom
particle and obtain relation between angular momentum and torque for a particle
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Answer 1

Answer:

(A) Angular momentum of system of particles

Consider a system of particles made up of number of particles, moving independently to each other. Let L1, L2,L3,.............etc. be the angular momentum of different particles of the system w.r.t., to a given point.

The angular momentum of the particle system w.r.t. that given point is equal to the vector sum of angular momentum of all the particles of the system.

If L is the angular momentum of the system of particles or the body as a whole then,

see the  1st pic

(B) Torque acting on system of particles

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

see 2nd pic

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

see 3rd pic

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

see 4th pic

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

see 5th pic

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