Conservation laws - Ball hitting a rod about its end

[mintsnapple]

Homework Statement

Homework Equations

v = wr
K_total = 1/2Mv^2 + 1/2Iw^2
L = Iw
p = mv

The Attempt at a Solution

a. No friction or other outside forces are acting on the system, so linear momentum is conserved.

The collision is elastic, so kinetic energy is conserved.

There are no external torques so angular momentum is also conserved.

b. We can now write our conservation laws:

Linear momentum: Mv_0 = Mv_f + Mv_f
Energy: 1/2Mv_0^2 = 1/2Mv_f^2 + 1/2Iw_f^2
Angular momentum: MRv_0 = Iw + MRv_f

Is my thinking correct so far?

 

[Simon Bridge]

So far so good.
Take conservation of angular momentum at the instant of impact to make the maths easy.

 

[mintsnapple]

I or a rod rotating about its center = ML^2/12

From linear momentum, the velocity of the ball and center of rod is given by Mv_0 = 2Mv_f, or v_f = v_0/2

To find the angular velocity, we can use our energy equation and plug in v_f:
1/2Mv_0^2 = 1/2M*(v_0^2 /4) +1/2*(ML^2/12)*w_f^2
1/2Mv_0^2 = 1/8Mv_0^2 + ML^2w_f^2/24
3/8v_0^2 = L^2w_f^2/24
9v_0^2/L^2 = w_f^2
w_f = 3v_0/L

How does that look?

 

[Simon Bridge]

The reasoning works OK.
So long as you are clear about the "before" and "after" situations.

Did you account for energy stored in the rotation (my eyes cross when there are so many symbols and numbers)?
I'm guessing you are assuming the ball is sliding and not rolling?

What you really need is some way to check your own results instead of relying on someone else to tell you you've got it right or not.

One way to do that is the sketch the situation before and after, putting little arrows on to indicate what you've calculated, and looking to see if it makes sense. If you have a smooth surface handy - vinyl table top say - you can even do a mini experiment.

 

[HalfLostAndSearching]

Yes, your thinking is correct so far. By stating that there are no external forces or torques acting on the system, you are applying the principle of conservation of momentum and energy. This means that the total momentum and energy of the system before and after the collision will remain the same. By using the equations for linear momentum, kinetic energy, and angular momentum, you are correctly applying the conservation laws to this specific scenario.