Question

A smooth sphere A is moving on a frictionless horizontal plane with angular speed $\omega$ and centre of mass velocity $v$. It collides elastically and heads on with an identical sphere B at rest.
Neglect friction everywhere. After the collision, their angular speeds are ${\omega _A}$ and ${\omega _B}$respectively. Then
A: ${\omega _A} < {\omega _B}$
B: ${\omega _A} = {\omega _B}$
C: ${\omega _A} = \omega$
D: ${\omega _B} = \omega$

Answer 1

This is an example of the collision that occurs between two rigid bodies. It is also said that elastic head on collision takes place between the two spheres. We will be able to solve the question by finding the quantities that are being transferred from one sphere to another upon collision. Upon which, we get the values of related elements like velocity.

Complete step by step answer:
In the question we are given two smooth spheres that are identical. The first sphere A moves with angular speed $\omega$ and centre of mass velocity $v$. The second sphere B is at rest and head on, elastic collision takes place there.
As the given spheres are smooth, angular momentum will not be transferred between the spheres. Since this is a head on collision, sphere A transfers its velocity completely to sphere B. Further, sphere B begins to move whereas sphere A occupies the position of B and remains at rest. Hence the friction is zero, torque about their centre of mass is also having zero value.
Thus the angular velocity doesn’t change.
Hence, we can say that ${\omega _A} = \omega ,{\omega _B} = 0$

So, the correct answer is “Option C”.

Note:
The law of conservation of momentum is applicable in the case of heads on elastic collisions as only the internal forces due to collision acts in the system.
The net force will be zero as the existing forces are equal and opposite. Thus, they cancel each other.
Here , also note that in this given question, A occupies the position of B and remains at rest.