Inelastic collision and angular velocity

[Lord Crc]

Homework Statement

Hi, I would just like to know if I'm on the right track with this one.

I have two identical particles with a radius R and mass M. One is at rest and the other moves directly down with a velocity v0. The second particle hits the first particle slightly offset, so that the angle between the horizontal and the line connecting the centers of mass is theta. The collision is perfectly inelastic. The moment of inertia around the center of mass for one particle is Ic, and around the center of mass of the combined particles I.

a) what is the velocity v1 after the collision?
b) what is the angular velocity omega1 after the collision?

Homework Equations

Inelastic collision: $$m_a\vec{v}_{a0} + m_b\vec{v}_{b0} = (m_a + m_b)\vec{v}_1$$

Angular momentum: $$L = I\omega$$

Angular momentum of a particle: $$L = mvr\sin\phi$$

Conservation of angular momentum: $$L_0 = L_1$$ or $$I_0\omega_0 = I_1\omega_1$$

The Attempt at a Solution

a) I just use the equation for inelastic collision and get $$\vec{v}_1 = \frac{1}{2}\vec{v}_0$$, or?

b) Since the first particle is said to be at rest, and the assignment doesn't mention what the angular velocity of each particle is before the collision, I'll assume that neither is rotating before the collision. So, I'm thinking I can find the angular momentum pre-collision by modeling the second particle as a point mass 2R away from the center of the first, and this would be L0, so that $$L_0 = mvr sin\phi = 2mv_0R sin\phi$$. Using some trigonometric argument I'll find an expression for phi based on theta (I don't expect to have a problem with this). I then use conservation of angular momentum and solve for omega1. Does this sound right?

 

[Lord Crc]

I've now tried this approach, and it seems not to work out quite right. That is, I have to find the kinetic energy before and after the collision, and using the angular velocity I got from the above approach I get a larger kinetic energy after the collision than before. I can't find any fault in my equations when finding the kinetic energy, so I'm inclined to think the above approach is wrong.

For instance should I use the point of contact or the center of the second particle when finding the angular momentum L0?

 

[Moetasim]

Hello, your approach seems correct so far. For part a), you are correct in using the equation for inelastic collision, which states that the total momentum before the collision equals the total momentum after the collision. So, your equation of \vec{v}_1 = \frac{1}{2}\vec{v}_0 is correct.

For part b), your approach of finding the angular momentum pre-collision and using conservation of angular momentum is also correct. However, instead of modeling the second particle as a point mass, you can use the fact that the moment of inertia around the center of mass of a particle is given by I = mr^2, where m is the mass and r is the distance from the center of mass. So, the moment of inertia for the second particle would be I = 2mR^2. From there, you can use conservation of angular momentum to solve for omega1.

Overall, your understanding of inelastic collisions and angular velocity seems to be on the right track. Keep up the good work!