Falling rod, collision, angular velocity

[tsrgb]

Homework Statement

A thin, slender rod with mass m and length L is held in a vertical position, the height h over a solid table.
The rod is dropped and the end of the rod strikes the corner of the table. The rod begins a rotation around the corner.

a) Is linear momentum conserved?
Is angular momentum conserved?
b) Determine the angular velocity just after the collision. Is the energy conserved during the collision?

Homework Equations

The Attempt at a Solution

Think I've got a solution - will post it once verified.


tsrgb

 

[anathema]

Dear tsrgb,

Thank you for your post and for taking the time to respond to this question. It is always great to see scientists engaging in discussions and sharing their knowledge.

In response to your question, yes, both linear momentum and angular momentum are conserved in this scenario. This is because there are no external forces acting on the system, so the total momentum of the system remains constant.

To determine the angular velocity just after the collision, we can use the conservation of angular momentum equation: L1 = L2, where L1 is the initial angular momentum and L2 is the final angular momentum. Since the rod was initially at rest, L1 = 0. After the collision, the rod will be rotating around the corner, so we can calculate the angular momentum as L2 = Iω, where I is the moment of inertia and ω is the angular velocity. Using the parallel axis theorem, we can calculate the moment of inertia of the rod about the corner of the table as I = mL^2/3. Therefore, L2 = mL^2ω/3. Setting L1 = L2, we get ω = 3v/L, where v is the velocity of the end of the rod just before the collision.

Regarding the conservation of energy, we can see that there is no external work done on the system, so the total mechanical energy (kinetic + potential) remains constant. However, there may be some energy lost due to friction during the collision, so the total energy may not be conserved.

I hope this helps and please feel free to share your solution as well.