Conservation of angular momentum

[eprparadox]

Homework Statement

We have a rod of length l and mass M. It is lying on a frictionless surface and an impulse, I, is delivered to this rod a distance D above the center of mass. What is the angular velocity (omega, w) about the center of mass point.

Homework Equations

net external torque = time derivative of angular momentum
torque = r x F
angular momentum = I*w = r x p

The Attempt at a Solution

Essentially, I need to find the angular velocity (w) about the center of mass of this rod. I can do it if I choose as my reference point the center of mass of the rod. But I wanted to choose a point that was in line with the impulse which would eliminate the external torque on the system and thus allow me to use conservation of angular momentum to find the same value of w, just using this new reference point.

I get w(cm) = 12*I*D/(M*l^2) for the angular velocity about the center of mass. I want to find this same answer picking as my reference point a point, P, which is parallel to the impulse point so that r X F = 0 and so Li = Lf. I do know that Li must be zero since, of course, the rod isn't moving before the impulse. Afterwards, I also know that Lf = I(about that new point P) * w. I'm not sure if I am missing something in that final Lf term or not.

Any help would be greatly appreciated. Thank you very much.

 

[Doc Al]

When using reference point P, don't forget that the angular momentum will be the sum of the angular momentum about the center of mass plus the angular momentum of the center of mass.

 

[nlightner]

The conservation of angular momentum states that the total angular momentum of a system remains constant if there is no external torque acting on it. In this case, the impulse applied to the rod creates a torque about the center of mass, causing it to rotate. However, if we choose a reference point that lies on the line of action of the impulse, there will be no external torque acting on the system and the angular momentum will be conserved. Therefore, the angular velocity about this point can be found using the equation L = I*w, where I is the moment of inertia and w is the angular velocity.

In your attempt at a solution, you correctly identified that the initial angular momentum is zero and the final angular momentum is equal to I*w. However, it is important to note that the moment of inertia will be different for different reference points. Therefore, you will need to calculate the moment of inertia about the new reference point P in order to find the correct value of w. This can be done using the parallel axis theorem, which states that the moment of inertia about a parallel axis is equal to the moment of inertia about the center of mass plus the mass of the object times the square of the distance between the two axes.

I hope this helps clarify your understanding of conservation of angular momentum in this scenario. Keep up the good work!