Rotation doubt

1 vote
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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 0.5 .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.
My attempt
calculated impulse for the collision and used angular impulse and linear impulse equations but got wrong answer , i would like to see what answer you get and how ( complete answer needed !)

edited Dec 3, 2016
Shubham : Thank you for posting your question here after it got closed on Physics Stack Exchange. However, please note that (as I said there) "you are still expected to show some effort".

Please would you edit your question to show what you have tried and explain what difficulty you are having.
Your approach should work. Please could you provide more details of your calculation? How did you take account of friction?
plz , can i ask you this , why do't you do the problem completely and tell what answer are you getting if we get the same i may be able to that the answer given may be wrong ! i was getting a+b = 38 ! what do you get ?
You need to state your work, as in the details of your calculation. Sorry, but otherwise this will be closed.
i have stated my work already , i am very disappointed that such website are just a wastage of time and no one worthy to even do this question is present here. you may delete if you want i will do it myself , if possible, otherwise i will let this question go ( i'll forget this question !)

1 vote

This is a tough problem. I don't have an answer yet.

If you know the velocity $v_2$ of the CoM and the angular velocity $\omega_2$ which the 2nd cylinder ($C_2$) acquires from the collision, then it is relatively easy to work out the angular velocity when $C_2$ stops slipping and is purely rolling. The difficulty for me is finding $v_2$ and $\omega_2$. Specifically, how to deal with the friction from the ground during the collision.

Since the collision is elastic (coefficient of restitution $e=1$), the velocity of the CoM of $C_1$ will be $0$ after the collision, while that of $C_2$ will be the amount lost by $C_1$. The collision may also transfer some angular momentum. In the absence of external torques, this would also be conserved, so the angular velocity lost by $C_1$ would equal that gained by $C_2$. However, there is also friction from the ground acting on both cylinders during the collision. It is not clear to me what effect this will have.

The friction force from $C_1$ on $C_2$ acts downward. It tends to give $C_2$ some backspin, and it increases the normal reaction from the ground, thereby increasing this friction force on $C_2$. The friction force from $C_2$ on $C_1$ tends to reduce or reverse its forward angular velocity and decrease the normal reaction with the ground, reducing this friction force on $C_1$.

What is not clear to me is how to calculate the friction impulses on the cylinders from the ground. In Satvik Pandey's solution (see below) the weight of the cylinders is ignored, but I do not think this is justified.

It seems to me that increased friction with the ground will prevent $C_2$ from rotating during the collision, so that $C_2$ will emerge with no angular velocity.

The problem appears on the Brilliant website where the initial velocity of $C_1$ is given as $17.5m/s$ and its radius as $R=1m$. The correct answer is stated to be $a+b=503$. One solution has been posted by Satvik Pandey, but his answer is $a+b=20+3=23$. The person who posted the question (Karthik Kannan) has not posted a solution nor any comments. Possibly KK is a pseudonym for Deepanshu Gupta who apologises for getting the question wrong.

Satvik Pandey discussed the problem in Physics Forums where he provided the above answer.

In post #16 Satvik says this answer is wrong (presumably he learnt this from posting on the Brilliant site) but he is convinced by his friend's solution, which has the correct answer of $455+48=503$. The Homework Helpers both think this answer of $503$ is wrong and $23$ is correct. Satvik claims that in this solution the 1st cylinder rolls over the 2nd, but this is not consistent with coefficient of restitution $e=1$ ; the rolling contact point implies $e=0$.

I shall keep working on this problem and add to my answer when I make progress.

answered Dec 8, 2016 by (28,466 points)
edited Dec 8, 2016
the problem is tough tht's why i'm skig it  , you cn see my profile too  nd the level of  uestions i solve nd this ws the only uestion i pst month i wsn't ble to solve, sorry for
writing like this , some of my keybord keys re ot workig ( you cn guess which one re ot ) nd thnks for seeing to my problem , nd you cn see the solution in the reports section below tht uestion  on brillint , here is my profile too ... https://brilliant.org/profile/shubham-zhez3v/ ( ll the uestions reposted re lredy solved by me , :)
Thank you for responding, Shubham. How did you get 38? What did you do differently from Satvik Pandey? How did you handle friction with the ground during the collision?