The centre of mass of a body is a point at which the entire mass of the body is supposed to be concentrated. The \(\vec{r}\) of c.m of the system of two particles of masses m₁ and m₂ with position vector \(\vec{r}_{1} \text { and } \vec{r}_{2}\) is given by 
\(\vec{r}=\frac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}\) 
for an isolated system, where no external force is acting \(\overrightarrow{v_{c m}}\) = constant
Under no circumstance, the velocity of centre of mass of an isolated system can undergo a change.
(i) What should be the position of the centre of mass of a system of two particles of unequal masses?
(ii) An electron and a proton move towards each other with velocities v₁ and v₂ respectively. What is the velocity of their centre of mass?
(iii) Two bodies of masses 1kg and 2 kg are located at (1, 2) and (-1, 3) respesctively, determine the coordinates of the centre of mass.
(iv) A bomb dropped from an aeroplane in level flight explodes in the middle. How would be the motion of centre of mass of the fragments?
(v) Two blocks of masses 5 kg and 2 kg are placed on a frictionless surface and connected by a spring. An external kick gives a velocity of 14 ms^-1 to the heavier block in the direction of the lighter one. Determine the velocity gained by the centre of mass.
(vi) Can centre of mass of a body lie where there is absolutely no mass? Give example.
(vii) Can centre of mass of a body coincide with the geometrical centre of the body?

Moment of inertia of a body about a given axis is the rotational inertia of the body about that axis. It is represented by 1= MK², where M is mass of body and K is radius of gy ration of the body about that axis. it is a scalar quantity, which is measured in kg m².
When a body rotates about a given axis and the axis of rotation also moves, then total K.E of body = K.E of translation + kinetic energy of rotation.
\(K=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2}\)
(i) Is the M.I of a body about a given axis is vector or scalar quantity?
(ii) On what factors does M.I of a body depend?
(iii) Determine the moment of inertia of circular disc and circular ring of same mass and radius about an axis perpendicular to plane.
(iv) A 40 kg flywheel in the form of a uniform circular disc of diameter 1 m is making 120 rpm. What is the M.I about a transverse axis through its centre?
(v) Determine kinetic of rotation of the flywheel in the above case.
(vi) Calculate radius of gyration of a cylindrical rod of mass m and length L about an axis of rotation perpendicular to its length and passing through its centre,
(vii) Determine the ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring.

A hollow sphere of mass M and Radius R is initially at rest on a horizontal rough surface. It moves under the action of constant horizontal force F as shown in Fig.

(i) Determine the direction of frictional force between the sphere and the surface. 
(ii) Determine the linear acceleration of the sphere.
(iii) What is the frictional force between the sphere and the surface?
(iv) State the relationship between angular momentum and torque of rotating body about an axis.
(v) Solid cylinder of mass m and radius r rolling down an inclined plane of \(\theta\)  inclination e without slipping. Determine the acceleration of the cylinder down the inclined plane.

Angular momentum of a rotating body is measure of quantity of motion in rotational motion about an axis which is measured by product of moment of inertia and angular velocity i.e., L = \(I \omega\) . It is moment of linear momentum about axis of rotation. Being a vector quantity its direction is along the axis of rotation. In the absencse of an external torque, angular momentum vector remains constant.
(i) If angular momentum is conserved in a system whose moment of inertia is decreased, will its rotational kinetic energy be also conserved? Explain.
(ii) Why spin angular velocity of a star is greatly enhanced when it collapses under gravitational pull and becomes a neutron star?
(iii) A Person sits near the edge of a circular platform revolving with a uniform angular speed. What will be the change in the motion of the platform?
(iv) What would happen if the person starts moving from the edge toward the centre of the platform?
(v) Why are there two propellers in a helicopter?
(vi) A thin uniform circular disc of mass M and Radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity \(\omega \text { . }\) Another disc of same dimensions but of mass \(\frac{\boldsymbol{M}}{\mathbf{4}}\) is placed gentally 4 on the first disc coaxially show that angular velocity of the system is \(\frac{4}{5} \omega.\)