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**1. The problem statement, all variables and given/known data**

A solid cylinder of radius

*R*is rolling without slipping on a rough horizontal surface. It collides with another identical cylinder which is initially at rest on the surface. The coefficient of restitution for the collision is 1.The coefficient of friction between the cylinders and between each cylinder and the ground is ##\mu##.If the final angular velocity of the second cylinder after it starts pure rolling is ##a/b## where

*a*and

*b*are co prime positive integers find

*a*+

*b*.

Given: Initial velocity of center of mass of the first cylinder is 17.5 m/s,

*R*=1m, ##\mu##=0.5

**2. Relevant equations**

##J=m(vf-vi)##

##L_{J}=I(\omega_{f}-\omega_{i})##

**3. The attempt at a solution**

I think the main challenge is to find the velocity of the CoM and the angular velocity of the second cylinder after collision. Then the final angular velocity can be found by conserving angular momentum about the point of contact of cylinder and ground.

I began in his way-----

I made the Free Body diagrams

Let the initial velocity of the CoM (of cylinder 1) of the cylinder and initial angular velocity of the cylinder be ##v## and ##\omega## respectively. Let ##\omega## be in clock wise direction initially.

Let the final velocity of the CoM of (of cylinder 1) the cylinder and final angular velocity of the cylinder be ##v_{0}## and ##\omega_{0}## respectively. Let ##\omega_{0}## be in clock wise direction.

Let the final velocity of the CoM (of cylinder 2) be ##v_{1}## and angular velocity be ##\omega_{1}## in anti-clock wise direction.

As net impulse is equal to change in momentum and net angular is equal to the change in angular momentum so

## J=m(v_{0}+v)##.....................(1)

##\mu JR=I(\omega_{0}-\omega)##.........................(2)

For cylinder 2

As the cylinder don't jump so

##\mu J=J_{N}## (I have neglected mg as it is very small as compared to the impulses)

As net impulse is equal to change in momentum and net angular is equal to the change in angular momentum so

##J-\mu J_{N}=mv_{1}##

##I\omega_{1}=(\mu J-\mu J_{N})R##

If we consider the time period of the collision is very small so I think we can neglect work done by friction during the collision and as the collision is elastic so I think we can conserve energy. So

##\frac { 1 }{ 2 } \left( m{ v }^{ 2 }+I{ \omega }^{ 2 } \right) =\frac { 1 }{ 2 } \left( m{ v }_{ 0 }^{ 2 }+m{ v }_{ 1 }^{ 2 }+I{ \omega }_{ 0 }^{ 2 }+I{ \omega }^{ 2 }_{ 1 } \right) ##

Are my all equations correct?