For problem 7, part (b), I know that the Galilean law of transformation holds for velocities in inertial reference frames. However, does it hold for angular momentum? If not, than how do I solve this part, because the quantity that I'm suppose to prove looks extremely similar to the Galilean law of transformations for velocities.
Your answer to part A can be written as the sum of four individual
summations (after you distribute the cross product). Two of the
summations are exactly the terms that you are asked to prove in part B.
The other two sums are equal to zero because of the definitions of center
of mass and velocity of the center of mass. I will use an underline in
the notation below to indicate a subscript:
Sum(r_i' x m_i*v_cm) = Sum(m_i*r_i' x v_cm) = Sum(m_i*r_i') x v_cm
The sum must equal zero becuase r_cm is zero in the center of mass frame.
Similiarly:
Sum(r_cm x m_i*v_i') = r_cm x Sum(m_i*v_i')
must also equal zero. Then last sum above would be equal to the total
mass times the center of mass velocity in the senter of mass frame, which
must be zero.