# Impulse Problem -- A rolling cylinder collides with a cylinder at rest

1. Mar 19, 2015

### Satvik Pandey

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?

2. Mar 19, 2015

### haruspex

Are you measuring v and v0 in opposite directions?
For energy, it's not the time that matters but the integral of the force over the distance. $\Delta E = \int F.ds = \int F.v.dt$, where v is the relative speed of the surfaces in contact. Taking v to be constant we get $\Delta E = v\int F.dt = v\Delta J$. So the question is, is the relative velocity zero?

3. Mar 19, 2015

### TSny

Coefficient of restitution is often defined as the ratio of relative velocity of separation after the collision to relative velocity of approach before the collision. The velocities of separation and approach are components of velocity along the line of impact. If this definition is applicable here, then you don't need to worry about energy equations.

See for example http://www.brown.edu/Departments/Engineering/Courses/En4/notes_old/oblique/oblique.html

4. Mar 20, 2015

### Satvik Pandey

In case of extended bodies we have to consider the velocities of the point of contact of bodies. Right?
As the collision is elastic so we can equate the velocity of approach and separation in common normal direction.
So $v=v_{0}+v_{1}$

I have a confusion. In this situation impulses are acting not only due to interaction between the particles but also from external sources like impulse from the ground and friction. Can we use this concept even in these cases?

Last edited: Mar 20, 2015
5. Mar 20, 2015

### Satvik Pandey

Yes they are in opposite direction. Sorry, I didn't mention it.

In this (https://www.physicsforums.com/threads/just-a-confusion-a-mass-falling-onto-a-pivoting-rod.790098/) thread we solved the question by conserving angular momentum about the point O even if net external torque was acting on it. We equated the angular momentum of the system just before and after the collision. And we assumed that net torque didn't bring much change in the angular momentum of the system during collision. Why can't we do same thing with energies?

Why can't I equate the total energy of the system before and after collision and assume the work done by the friction to be very small?

6. Mar 20, 2015

### haruspex

In that case you considered a limited, finite torque (MgL) acting for an infinitesimal time. The integral is zero. The torque of the impact ($mL\frac{\Delta v}{\Delta t}$)was unlimited, in the sense that the shorter the time it was considered to act over the larger the torque. In that case the integral is known and nonzero.
In the present case, looking at the integral wrt distance, you have the same issue - a force which is unlimited. So it is not clear whether its integral over an infinitesimal distance will be zero.
In attempting to solve the question myself, I used the same principle that TSny suggests. However, oblique impacts of elastic bodies is a complicated matter. Ever played with a superball? A spun ball dropped on concrete will not only bounce up at an angle, its direction of spin reverses. So not only does the linear motion bounce, the rotational motion does too. But in that case there is no slipping. It is not made clear, but it seems to me we are supposed to assume static friction is overcome.

7. Mar 20, 2015

### Satvik Pandey

Thank you I got it now.

8. Mar 20, 2015

### Satvik Pandey

I am getting ${ v }_{ 1 }=\frac { 6\times 17.5 }{ 7 }$
and ${ \omega }_{ 1 }=\frac { 4\times 17.5 }{ 7 }$

and final angular velocity when it starts pure rolling as
${ \omega }_{ final }=\frac { 20 }{ 3 }$

Am I right?

9. Mar 20, 2015

### TSny

That's what I get using the relative velocity interpretation of the coefficient of restitution.

Last edited: Mar 20, 2015
10. Mar 20, 2015

### Satvik Pandey

Just a last question:
Can we use the concept of coefficient of restitution in any case if value of $e$ is provided? I am asking this because I used to think that we can not use this when impulses acts from external sources like impulse due to friction and normal in this case.

11. Mar 20, 2015

### TSny

I have never seen a textbook problem where you cannot use $e$ in terms of relative velocities, even when there are external impulses during the collision. But I don't have the knowledge to make a general claim.

12. Mar 20, 2015

### haruspex

Usually the friction is orthogonal to the relative velocity, so it's ok. But here I think there's a problem. The friction between the second ball and the ground is also horizontal.
It would be interesting to see what happens with an energy approach. I believe I can show that if an impulsive frictional torque (i.e. sudden change in angular momentum) JFr is applied to a body with MoI of I then the total work done is $\frac{(J_Fr)^2}{I}$, but only half ends up as rotational KE.

13. Mar 20, 2015

### Satvik Pandey

Thank you!

14. Mar 20, 2015

### Satvik Pandey

Could you please tell me how you arrived at that result? Is $J_F$ is frictional force between cylinders?

15. Mar 21, 2015

### haruspex

I was using JF as a generic frictional impulse.
But on second thoughts, I doubt my result is right, since it ignores the concurrent force from the impact of the first ball.
I'll try again, but I'm a bit busy at the moment.

16. Mar 21, 2015

### Satvik Pandey

No problem.
My solution is wrong. In this I have assumed that limiting value of friction acts between cylinder and cylinder 1 slips over cylinder 2. However we can't do so. One of my friend has shown that friction between cylinders is sufficient for cylinder 1 to roll over cylinder 2. He got final $\omega=455/48$. I am convinced with his solution. What do you guys think?

17. Mar 21, 2015

### TSny

It seems to me that the coefficient of friction given in the problem must be for kinetic friction. You need the coefficient of kinetic friction to determine how far cylinder 2 travels before it starts rolling without slipping. A coefficient of restitution greater than zero implies that the cylinders separate after the collision. So, I don't see how cylinder 1 is going to roll up over cylinder 2. Maybe I'm overlooking something.

18. Mar 21, 2015

### haruspex

As TSny has posted, your friend's solution is almost surely wrong.
I've come to the conclusion that as far as the elastic collision is concerned the effect of the impulsive friction from the ground on the second cylinder is to make it seem more massive. Since the ratio of the relative speeds equation does not depend on the masses, it is still valid. So I agree with your answer in post #8.

19. Mar 22, 2015

### Satvik Pandey

As there is no information so I think we should use $\mu_{k}=\mu_{s}=0.5$.
The e is greater than 0. So this means that they do stick with each other. But it may be possible that time period in which they(cylinders) remain in contact, they might roll over each other instead of sliding on one other. I think they can roll over each other for the small time in which they remain in contact with each other.
What do you think guys?

20. Mar 22, 2015

### haruspex

It is quite impossible for the first cylinder to roll over the second. The instant after impact, it cannot have a greater horizontal speed than the second since that would mean the two intersect. Since it will become airborne, the two therefore separate.
It is true that the second cylinder will now have backspin, so will lose some of its horizontal speed, so perhaps the first can catch it up, but I very much doubt it.