(a) Let us consider a system of n particles of masses
m1 m2 ..........
mn. Let their velocities wrt ground be
→V1g →V2g, ......→VngThe moment of the system of the particles is →p=m1→V1g+m2→V2g+.......+mn→Vng.
⇒→p=m1(→V1c+→Vg)+m2(→V2c+→Vg)+.......... where →V1c, →V2c are velocities if individual masses with respect to the centre of mass and Vg is the velocity of the centre of mass of the system wrt ground.
After adding up all of the term we get,
→p=p′i+mi→V
(b) The kinetic energy of n- particle of the systme
K=12m1V21g+12m2V22g+.....
On simplification we get
K=K′+12MV2 where V=V−g
(c) The angular momentum of the sphere about O is L=I0ω
But according to the perpendicular axes theorem
I0=IC+MR2
∴L=(IC+MR2)ω
L=ICω+MR2ω
L=ICω+M(Rω)R
L=ICω+MVR
L=→L′+→R×M→V
(d) The angular momentum of the system wrt centre of mass
→L′=∑r′i×p′i
Differentiating wrt time, we get
dL′dt=∑→ri′×dp′dt
We know that dL′dt=→τext
where →τext is the sum of the external torques acting on the system about the centre of the mass.