Irodov Problem 1.197

Let there be n particles in the system and let their masses be m1,m2,...,mn. Let their position vectors as seen from the center of inertia of the system be r1,r2,...,rn and their velocity vectors as seen from the center of inertia be v1,v2,...,vn. From the very definition of moment of inertia we have,






Suppose that the velocity vector of the center of inertia of the system of particles with respect to frame K is given by Vc. Then the velocities of the individual particles in the system as seen from the frame K will be the vectors v1+Vc,v2+Vc,...,vn+Vc. The momentum of the system of particles as seen from frame K will then be,







The position vectors of the particles as seen from frame K will be r1+rc,r2+rc,...,rn+rc. So, the angular momentum of the system of particles as seen from K will be,

Irodov Probem 1.196

Let there be n particles in the system with masses m1,m2,...,mn. Let their position vectors with respect to O be the vectors r1,r2,...,rn and their velocities be the vectors v1,v2,...,vn. Thus, we have,






The position vector of the particles with respect to point O' will be r1-r0,r2-r0,...,rn-r0. So the angular momentum of the system of particles with respect to point O' will be,










Clearly if p=0, the angular momentum will not be altered.

Irodov Problem 1.195

The sphere is acted upon by two forces i) the force of gravity mg whose component along the inclined plane is and ii) the force of friction acting at the point of contact between the inclined plane and the sphere. As seen from the initial point of contact O there is however only one torque acting on the system - that due to the force of gravity given by . The force of friction f acting on the sphere does not induce a torque on the sphere as seen from O since it passes through the point O so that moment arm for the friction force is 0.

Torque is the rate of change of angular momentum and so if the angular momentum of the sphere at any point is M, then,






Clearly the absence or presence of friction does not alter anything since it has no net torque on the sphere as seen from point O.

Irodov Problem 1.194

The instantaneous velocity of the mass at any time t will be v = gt (since its falling freely under gravity).
The angular momentum as seen by an observer at the center of the pulley will be m(r x v) = mRgt since the arm of the moment will always be R here.