Finding angular velocity after asteroid collision

[Cpt. DeMorgan]

Homework Statement

In the figure a spherical non spinning asteroid of mass M = 4E16 kg and radius R = 1.5E4 m moving with speed v1 = 2.4E4 m/s to the right collides with a similar non spinning asteroid moving with speed v2 = 5.9E4 m/s to the left, and they stick together. The impact parameter is d = 1.4E4 m. Note that I_sphere = 2/5*M*R^2.

After the collision, what is the velocity of the center of mass and the angular velocity about the center of mass? (Note that each asteroid rotates about its own center with this same angular velocity. Assume that the asteroids move in the x-y plane, and that the asteroid of speed v1 moves in the positive x direction.)

Homework Equations

L_A,f = L_A,i
L_Rot = Iω, L_Rot = r1cm x p1 + r2cm x p2
L_tran = r_a,cm x p_tot

The Attempt at a Solution

I have found v_cm. Now I am looking for the angular velocity. I have considered both asteroids to be included in the system and the surroundings to be nothing. Because there are no surroundings dL_A/dt is 0. Therefore, L_Af = L_Ai. If this is the case, the angular velocity should be the same in the initial and final conditions. Is this true?

I then said that L_rot = Iω, and L_rot = r_cm1 X p_1 + r_cm2 X p_2. So I had the following equation:

Iω = r_cm1 X p_1 + r_cm2 X p_2.

Then I solved for ω,

ω = (r_cm1 X p_1 + r_cm2 X p_2)/I.

Is this correct reasoning? Are the initial and final angular velocities different? Should I consider a system with just one asteroid instead?

Thank you

 

[jhosamelly]

Use the conservation of angular momentum equation.

 

[Cpt. DeMorgan]

Okay. If I use the conservation form I will need to change my system, correct? So now my system is just one asteroid and my surroundings are the other asteroid.

Can I still use the equations I included or will I need to use another definition L_rot and L_trans?

And to add to my many questions I have, I watched a lecture video on youtube with a similar problem.



The professor explained that the center of mass changed throughout the collision. Will I need to take this into account?

 

[haruspex]

Okay. If I use the conservation form I will need to change my system, correct? So now my system is just one asteroid and my surroundings are the other asteroid.

No, it should be simplest and safest to take the mass centre of the system as reference.
You didn't define the variables in your equations, so I cannot say whether they're right.
What is the moment of each about the common mass centre before collision?

 

[Nickie]

for your question. Your reasoning and approach are correct. The initial and final angular velocities will be the same in this case, since there are no external torques acting on the system. Therefore, the angular momentum of the system is conserved.

However, I would like to clarify a few things in your solution. First, the equation L_rot = Iω should be written as L_rot = I*ω, since it is a multiplication of two quantities. Also, the equation for the angular velocity should be ω = (r_cm1 X p_1 + r_cm2 X p_2)/(I_1 + I_2), where I_1 and I_2 are the moments of inertia of the two asteroids about their respective centers of mass.

Additionally, I would like to point out that your equation for L_rot is not entirely correct. The correct equation for angular momentum is L_rot = r_cm X p, where r_cm is the distance from the point of rotation to the center of mass and p is the linear momentum. So, the equation should be L_rot = (r_cm1 X p_1) + (r_cm2 X p_2).

Finally, I would like to address your last question about considering a system with just one asteroid. It is not necessary to do so, as the equation for angular velocity will be the same regardless of whether you consider one or both asteroids in the system. However, if you do consider only one asteroid, you will need to use the moment of inertia of that asteroid about its own center of mass, rather than the combined moment of inertia of both asteroids.

I hope this helps clarify your solution. Keep up the good work in your studies of physics and science!