Answer :

A. Take a system of I moving particles:


Mass of the ith particle = mi


Velocity of the ith particle = vi


momentum of the ith particle, pi = mivi


Velocity of the canter of the mass = V


Thus, velocity of the ith particle with respect to canter of the mass, vi’ = vi – V …..(1)


Multiplying mi throughout the equation (1) we get,


mivi’ = mivi – miV


pi’ = pi - miV


Where,


pi’ = momentum of the ith particle with respect to the canter of the mass.


Taking the summation of the momentums of the all the particle with respect to the canter the mass of the body,


…..(2)


Where,


ri’ = position of the ith particle with respect to the canter of mass and,


As per the definition of the canter of the mass, we have,


from equation 2,


B. We have the relation for velocity of the particle as,


…..(3)


Taking the dot product with itself will give,


Here, For the canter of mass of the system of the particle,


Therefore,


C. Total angular momentum of the system of the particles,


The las two terms vanish for both contain the factor which is equal to zero from the definition of the center of the mass. Also,


So that,


D. From the previous solution,


( ) Total toque,


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