Question

. Consider a system of N particles in a uniform gravitational field. Prove that the total gravitational torque about center of mass(CM) is zero.?​

Answer 1

Explanation:

Centre of mass is a term related to the motion of the extended body, i.e. it is under rigid dynamics and it is fixed point if the shape (geometry) and mass (density) of the body remains unchanged.

Centre of mass is a representative point of the extended body where whole mass of the body assumed to be concentrated and the motion of the centre of mass actually represents the motion of the whole extended body under any external and internal forces.

Position of the centre of mass R = (Summation of product of mi and ri over all particles constitute the body) / M, where mi and ri are the mass and position of the ith particle with respect to a reference frame and M is the total Mass of the body. This Centre of mass point of body is an unique point of a body that can not be changed with the choice of different frame of reference also.

So, whatever be the displacement of the centre of mass with respect to the ground or any other frame of reference, the displacement of the centre of mass with respect to the individual particles which forms the body is always zero as it is fixed point independent of the motion of the whole body.