Point mass strikes thin rod: angular momentum

[nelkypie]

Homework Statement

A point mass "m" moving at speed "v" strikes the end of a stationary uniform thin rod of length "l" and the same mass "m" at a right angle. The point mass does not stick to the rod; the collision is elastic. There are no pivot or anchor points. The system is floating freely in empty space.

a) What point will the rod rotate about after the collision, and why?
b) How would I set up relations to determine the various velocities of the point and rod after the collision?
c) In what case would the collision between a point mass and a thin rod be inelastic? How would I have to change the initial and final states of the point and rod to maximize loss of kinetic energy?

Homework Equations

L=Iω
L=r x mv
v=ωr
conservation of angular momentum equation
conservation of kinetic energy equation

The Attempt at a Solution

Intuitively speaking, I think that the rod should rotate about its center of mass because there is no external torque on the system to displace the axis of rotation elsewhere.

I set up a conservation of kinetic energy formula with the initial KE of the point mass on one side and the sum of the point mass' KE and the rod's KE (two separate terms for translation and rotation) on the other. I also set up a conservation of angular momentum formula, but two equations aren't enough to solve for three unknowns. What component am I missing?

In order to maximize the loss of kinetic energy, the point would have to stick to the rod such that they share the same final angular velocity. However, I'm not sure if that's feasible in this scenario. I'm wondering whether the rod would have to be attached to a pivot of some sort for the point mass to stick.

 

[haruspex]

Intuitively speaking, I think that the rod should rotate about its center of mass because there is no external torque on the system to displace the axis of rotation elsewhere.

The impacting mass has a moment about the centre of the rod.

I set up a conservation of kinetic energy formula with the initial KE of the point mass on one side and the sum of the point mass' KE and the rod's KE (two separate terms for translation and rotation) on the other. I also set up a conservation of angular momentum formula, but two equations aren't enough to solve for three unknowns. What component am I missing?

Conservation of linear momentum.

In order to maximize the loss of kinetic energy, the point would have to stick to the rod such that they share the same final angular velocity. However, I'm not sure if that's feasible in this scenario. I'm wondering whether the rod would have to be attached to a pivot of some sort for the point mass to stick.

It certainly could stick - just have it made of putty, say.