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Say we have a solid bar in space. It is on a pivot, the pivot being right at the bar's center of mass

There is a massless rocket pushing on one end of it, making it spin, faster and faster.

Suddenly, the pivot disappears, and at the same time the rocket disappears, and thus the mass does not have a force on it anymore and is also free to move about.

enter image description here

What would happen? Would the mass just spin in place, or would it go off in the last direction the rocket pushed it?

I have a theory for each, but they are contradictory:

1. The mass stays in place spinning around its center of mass:

Momentum from the top spinning part balances out with momentum from the bottom spinning part, meaning the bar only has angular momentum but no translational momentum overall.

This makes sense, and if there was a rocket in the top part providing momentum in the opposite direction, I would accept it without further question.

enter image description here

However, the rocket is only on one side of the rod, so that leads me to think...

2. The mass keeps on spinning, but now has translational momentum as well

The reason for this is because the part with the rocket attached got the last "piece of momentum" before the rocket disintegrated, and thus will move the mass forwards...?

Also, I have a sub-questions to add:

1. What if the pivot was further up on the rod and then the same thing happened? Or, equivalently center of mass wise, what if the top part without the rocket was more massive than the bottom part?

Thank you!

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Your first theory is correct. As long as the pivot is in place, any translational force is cancelled out by the pivot. The center of mass of the bar is not moving. If both disappear at the same time, then there are no more forces acting on the center of mass of the bar, so Newton's first law tells us there is no change to its motion.

But you are smart to ask the second question. If the pivot is not on the center of mass, then the center of mass is constantly moving in a circle as the rocket fires. When the pivot disappears, the center of mass will continue to move in whatever direction it was currently moving, sending the bar away from its initial position.

The translational motion of rigid bodies can be analyzed by considering their center of mass.

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The rocket and the pivot disappear at the same time, so there is no chance for an unbalanced force on the center of mass, meaning that the center of mass of the rod does not pick up any momentum.

If we start analyzing the system from the moment right after desintegration, let’s call it $t=0$, then we would have a massive bar with no linear momentum, but only angular momentum, as you have said it yourself, and there is no reason for this to change.

If the pivot was not located at the center of mass, then this center of mass would have some momentum, and after the pivot disappears this momentum would be conserved, so the bar would have linear momentum, and the direction of it would depend on when the pivot disappeared.

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