Question

A particle of mass m, moving with a velocity ${v_0}$, hits a stationary thin, uniform ring of equal mass m lying horizontally on a smooth horizontal surface, tangentially and sticks to it. The angular speed of system after collision is:

Answer 1

Hint: In this question, we will use the relation between the angular momentum, mass, velocity and radius; also, we will use relation between angular momentum, inertia and angular speed. Using these equations and solving for angular velocity, will give us the required result. Further, we will see the basics of angular momentum, law of conservation of momentum and collision, for our better understanding.

Formula used:
$L = mvr$
$L = {I_c}\omega $

Complete step-by-step answer:
As we know, Angular momentum is the rotational equivalent to linear momentum, given by:
$L = mvr$
Here, L is linear momentum, m is mass, v is velocity and r is the radius of an object.
Angular momentum is also the product of inertia and angular velocity.
$L = {I_c}\omega $
Angular momentum is a conserved quantity. The total angular momentum of a closed system remains constant.
Now, from the above two equations of angular momentum, we get:
$\dfrac{{mvr}}{2} = {I_c}\omega $
Inertia of ring will be given as:
${I_c} = m{r^2} + m{\left( {\dfrac{r}{2}} \right)^2}$
Inertia of bullet will be given as:
${I_c} = m{\left( {\dfrac{r}{2}} \right)^2}$
Now, the moment of inertia of the system will be given by:
${I_c} = \left( {\dfrac{{6m{R^2}}}{4}} \right)$
$ \Rightarrow {I_c} = \left( {\dfrac{{3m{R^2}}}{2}} \right)$
Putting this value in equation (1), we get:
$\dfrac{{mvr}}{2} = \left( {\dfrac{{3m{R^2}}}{2}} \right)\omega $
$\therefore \omega = \dfrac{{vr}}{3}$
Here, $(r \approx R)$
Therefore, we get the required result which gives us the angular speed of the given system after collision.

Additional Information: From the law of conservation of mass which says that mass can neither be created nor destroyed in any chemical reactions. Also, it can be defined as the mass of any one element at the beginning of a reaction will equal the mass of that element at the end of the reaction.
We know that collision means when two objects come in contact with each other for a very short period. Collision is an interaction between two masses for a very short interval where the momentum and energy of the colliding masses changes. Here are two types of collision: first id elastic collision where the energy remains same after interaction or collision and second is inelastic collision where the final energy changes after the collision of the particles or body.

Note: If angular momentum remains constant than the angular velocity of the object must increase. The resulting mass, energy or velocity after the collision of two particles depends on its direction and magnitude as well. The initial and final linear momentum is equal when there is no torque i.e., conserved angular moment.