# Rolling on a plank which is resting on friction less surface

• Krushnaraj Pandya
In summary, the problem involves a long plank and a thin circular ring with initial velocity v(i) on a smooth horizontal surface. The ring slips without rotation on the plank with a coefficient of friction C. At time t, the ring stops slipping and pure rolling starts. The goal is to find the value of t and the velocity of the center of the wheel at t. The equations used for solving this problem include conservation of momentum in the horizontal direction, torque, and translational motion of the plank and ring. The direction of friction and the final angular velocity need to be determined in order to apply the no slip condition. It is important to specify the direction of velocity and acceleration, both linear and rotational,f

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## Homework Statement

A long plank of mass M rests upon a smooth horizontal surface. A thin circular ring (m, R) slips (without rotation) upon the plank with initial velocity v(i). The coefficient of friction between the wheel and the plank is C. at time t, the ring stops slipping and pure rolling starts, find value of t and velocity of center of wheel at t.

## Homework Equations

taking v1 as final velocity of ring, v2 as final velocity of plank, a1 as acceleration of ring and a2 as acceleration of plank, (alpha) as angular acceleration of ring, w as final angular velocity of ring and friction=Cmg. I wrote the following equations-
1) mv1+Mv2=mv(i) (conservation of momentum in horizontal direction)
2) from torque=M.I*(alpha), alpha=Cg/R
3) from translational motion of plank Cmg=Ma2
4) from translational motion of ring Cg=a1
5) w=(Cg/R)*t

## The Attempt at a Solution

I can't figure out the direction of friction and therefore of final angular velocity to use in the no slip condition
v1+Rw=v2, or the directions of a1 and a2 to use in the equations of motion v1=v(i)+a1t. Intuitively, for ring to start rolling, w has to be in a clockwise direction but if plank is moving forward then it can also be in the anticlockwise direction, please shed some light on this and let me know if any of the equations I wrote are incorrect

v1+Rw=v2
It is important to specify which direction is considered positive for each velocity and acceleration, linear and rotational. Without that, the correctness of any individual equation cannot be determined.
for ring to start rolling, w has to be in a clockwise direction
If the linear motion is left to right, yes.

By the way, if the question does not require you to find the time until rolling or the accelerations involved there is a much easier way. Can you find a stationary point such that none of the horizontal forces on the disk have any moment about that point?

It is important to specify which direction is considered positive for each velocity and acceleration, linear and rotational. Without that, the correctness of any individual equation cannot be determined.

If the linear motion is left to right, yes.

By the way, if the question does not require you to find the time until rolling or the accelerations involved there is a much easier way. Can you find a stationary point such that none of the horizontal forces on the disk have any moment about that point?
I see that I can conserve angular momentum and that'd be fruitful but my textbook explicitly states to use the theory written up till the problem given to better understand the concepts and get used to applying them

I agree the directions are very important, which is why I separated and numbered the equations I'm sure about; also- I am getting confused between two intuitions. Since the point of contact has to be stationary relative to the plank, it can do so in two ways. either the plank can be moving left and the ring to the right, compelling w to be clockwise, but if the plank will move to the right, w will have to be anticlockwise. I can't determine which is correct. I'm thinking the direction of friction would clarify things but I can't determine that either without any doubts.

either the plank can be moving left and the ring to the right, compelling w to be clockwise, but if the plank will move to the right, w will have to be anticlockwise.
This isn’t right. What if the ring moves to the right at the same rate as the plank? What about faster or slower?

To know which way the plank moves, think about which way friction is directed. (There is an equal and opposite frictional force on the ring and plank.)

Since the point of contact has to be stationary relative to the plank
That is ambiguous. It depends what you mean by point of contact.
If you mean the relative motions of the plank and that point on the wheel's circumference instantaneously in contact with it, yes, they must have the same velocity when rolling is established. But not if you are thinking of point of contact as the locus over time of the position inspace where the wheel touches the plank.
either the plank can be moving left and the ring to the right, compelling w to be clockwise
I don't see that. Surely the plank could be moving to the right, the wheel a bit faster to the right and rotating clockwise.

@haruspex I meant the point in contact instantaneously...I see your point, I did not think of that before @Nathanael mentioned it. From what I understand now, the friction force on the plank will be directed towards the right while it will be towards the left for the ring giving it a clockwise angular velocity
is that correct?

what if the ring moves slower than the plank, w would have to be anticlockwise then. Does this mean we cannot assume any directions for such a given problem??
or is this the same case as the velocity of plank is towards the left instead too...making it a symmetry of the first case where w is clockwise

Also, when do we apply v1+Rw=v2 and when do we apply a1+R(alpha)=a2...this question has me totally confused

Where did everybody go?

Where did everybody go?
To sleep I’m back

From what I understand now, the friction force on the plank will be directed towards the right while it will be towards the left for the ring giving it a clockwise angular velocity
is that correct?
Yes I agree.

what if the ring moves slower than the plank, w would have to be anticlockwise then.
I’m not sure what to tell you other than it can’t happen here. We know the ring slides right, so we know friction on it is left, so we know the torque on it is clockwise. There’s just no way for things to start spinning the other way.

Also, when do we apply v1+Rw=v2 and when do we apply a1+R(alpha)=a2...this question has me totally confused
I don’t agree with that acceleration equation. A linear acceleration of R(alpha) would happen for a ball which is changing speed but always rolling. It doesn’t work here because rolling only occurs once accelerations are finished.

Anyway I don’t think we need to consider accelerations. I personally like to think in terms of impulses. Over the whole time t there is some frictional impulse, J, which acts on the ring and plank (oppositely). This impulse slows down the ring but gets it rotating, and speeds up the plank, and it must do all that in just the right way. (J can just be written in terms of friction and time, but I still prefer J.)

(I also like the angular momentum solution, too bad your book didn’t cover that.)

To sleep I’m back

Yes I agree.

I’m not sure what to tell you other than it can’t happen here. We know the ring slides right, so we know friction on it is left, so we know the torque on it is clockwise. There’s just no way for things to start spinning the other way.

I don’t agree with that acceleration equation. A linear acceleration of R(alpha) would happen for a ball which is changing speed but always rolling. It doesn’t work here because rolling only occurs once accelerations are finished.

Anyway I don’t think we need to consider accelerations. I personally like to think in terms of impulses. Over the whole time t there is some frictional impulse, J, which acts on the ring and plank (oppositely). This impulse slows down the ring but gets it rotating, and speeds up the plank, and it must do all that in just the right way. (J can just be written in terms of friction and time, but I still prefer J.)

(I also like the angular momentum solution, too bad your book didn’t cover that.)
Pardon, I live in India so didn't consider the time difference. Conserving Angular momentum can solve this question in a matter of seconds. It has been given in the textbook although they want us to solve this question so that we learn to apply the other concepts...which is why I am asking all this. I understand what you're trying to say and I'll apply this again and try to get the answer...

Got it! Finally! Even though I had to write 7 equations for solving in 7 variables, it was worth it, now I know I can solve this problem in more than one way. Thanks a lot everyone! I hope one day I'll have the same amount of intuition in physics as you people do, amazing aptitudes :D