Coefficient of restitution in rotational motion

[iitjee10]

Suppose A ball of mass m moving with a speed v collides with a rod of mass M and length L placed horizontally on a smooth floor. The coefficient of restitution is 0.5.

In this case how do we utilise the information of COR.

If it were given COR is 1 then we could conserve kinetic energy. But in this case how do we how about it?

 

[chrisk]

A coefficient of restituion of 0.5 is used to relate the final relative velocities to the initial relative velocities: V2f - V1f = e(V1i - V2i) where V= velocity, f = final, i = initial, and e = COR. Using the conservation of momentum with the preceding relation can determine the final velocities of the objects for head on collisions. A lot is unkown; the radius of the ball? the collision point on the bar? the initial angular orientation of the bar? is the ball rolling or sliding? the diameter of the bar? Since the floor is smooth it can be assumed it's frictionless and the ball would be sliding, not rolling (provided the initial release of the ball did not produce a rotation), and it can be treated as a head on collision.

 

[iitjee10]

assume the ball hits the rod at an end perpendicularly, the ball is sliding, smooth horizontal floor.
now, how do we use v2f - v1f = e(v1i - v2i) as there is an angular velocity of the rod after the collision.

 

[Lucien1011]

Can anyone tell if I am correct or not. I am really not sure.
First by v2f - v1f = e(v1i - v2i) and conservation of momentum, v1f and v2f is known.
then by conservation of angular momentum (axis taken at the cg of the rod), the angular velocity of the rod is known. And that does it.

 

[chrisk]

Lucien1011 is correct. The final linear velocities of the ball and center of mass of the rod are found using the COR equation and the conservation of linear momentum. Then using the conservation of angular momentum, the angular velocity of the rod can be found. The moment of inertia of the rod must be computed about an axis of rotation through the center of the rod because the rod will rotate about it's center of mass after the collision, and the center of mass will have a translational velocity. The initial anglular momentum is mvL/2 (ball) + 0 (rod). The final angular momentum must equal the initial momentum.

 

[iitjee10]

solve both linear momentum and angular momentum and please show, i think the answers come out to be different

 

[chrisk]

The linear momentum is different from the angular momentum. Again, solve for the final velocities of the ball and the center of mass of the rod using the COR equation and the conservation of linear momentum (initial momentum equals final momentum). Then use the conservation of angular momentum. Below is a web page that shows the method. Refer to Example 2 at the bottom of the web page.

http://dept.physics.upenn.edu/courses/gladney/mathphys/java/sect4/subsection4_1_6.html