Precession of ellipsoid Question

[the keck]

Homework Statement

A uniform symmetric ellipsoid (Mass M) has a large semi axis c and small semi axis a. A particle of mass m<<M is moving along a straight line parallel to the x-axis with speed v(i). Its y-coordinate is a/2 and its z-coordinate it c/2. After an inelastic collision, it sticks to the ellipsoid and then it (The ellipsoid) starts to move and rotate. One can assume the moment of inertia of the composite system equals to that of the ellipsoid.

Find the precession of the ellipsoid c-axis around direction of the angular momentum

The Attempt at a Solution

There are a couple of initial things I worked out:

- By linear momentum conservation, v(f)=m*v(i)/M

- The angular momentum of the composite system is L=M*v(f)*a + (2/5*M*a^2)*(v(f)/a) = 7/5*M*v(f)*a

- From what I guess, the ellipsoid rotates and precesses about the z-axis, but I'm still sort of learning precession, so I'm not sure how to go about doing this.

Thanks

Regards,
The Keck

 

[Jhero]

Dear The Keck,

Thank you for your post. I will try my best to help you solve this problem.

Firstly, your initial calculations are correct. The linear momentum conservation equation gives us the final velocity of the composite system after the collision. And the angular momentum of the composite system can be calculated using the moment of inertia of the ellipsoid and the velocity of the composite system.

To determine the precession of the ellipsoid around the z-axis, we can use the conservation of angular momentum. The initial angular momentum of the particle is zero since it is moving in a straight line parallel to the x-axis. After the collision, the particle sticks to the ellipsoid, so the initial angular momentum of the composite system is equal to the angular momentum of the ellipsoid before the collision. Therefore, we have:

L(i) = I*w(i)

where L(i) is the initial angular momentum of the composite system, I is the moment of inertia of the ellipsoid, and w(i) is the initial angular velocity of the ellipsoid.

After the collision, the composite system starts to rotate and precess around the z-axis. The angular momentum of the composite system is now given by:

L(f) = I*w(f) + L(c)

where L(f) is the final angular momentum of the composite system, w(f) is the final angular velocity of the ellipsoid, and L(c) is the angular momentum of the particle that is now attached to the ellipsoid.

Using the conservation of angular momentum, we can equate the initial and final angular momenta and solve for w(f):

I*w(i) = I*w(f) + L(c)

w(f) = (I*w(i) - L(c))/I

To determine the precession of the ellipsoid, we can use the formula:

ω(p) = w(f)/cosθ

where ω(p) is the precession angular velocity, w(f) is the final angular velocity of the ellipsoid, and θ is the angle between the angular momentum vector and the z-axis.

In this case, since the ellipsoid is rotating and precessing around the z-axis, the angle θ is equal to 90 degrees. So we have:

ω(p) = w(f)/cos90 = w(f)

Substituting the value of w(f) we calculated earlier, we get:

ω(p) = (I*w(i) - L(c))/(I*cos90)