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I have a small question.

Suppose I hold a given stick A of mass $m_A$ with attached a weight B with mass $m_B$ attached to it and that I start rotating it, eventually releasing it.

The general movement should be similar to an axe throw, with centre of mass roughly near the weight, depending on the ratio of the masses.

However, what would happen if during the rotation the stick hits an object and breaks near the point I was holding it?

Will the object immediately start to rotate around its center of mass, in our case near the weight?

What other effects will take place? I guess the rotational velocity is conserved, but what does this mean?

Is the center of the rotation different from the center of mass immediately after the break? is it moved towards the center of mass?

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What is conserved in your problem is the angular momentum. This is defined as the vector product between the momentum of each element being part of your object and its distance from the center of the rotation you are considering. In order to obtain the total quantity you have to sum over all the elements that are part of your rotating object; this is usually done with an integral.

Thus the answer to your question will heavily depend on the object that you fix on your stick, both on its mass and on its shape. As your intuition tells you the rotation of the detached part will be approximately around its center of mass, if the weight is much heavier than the stick.

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