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Suppose I want to throw an axe, so first I spin it vertically by holding the end of the axe's handle with my extended arm, and then I let go of the axe when it is at it's highest point (when the handle is perpendicularly facing the ground). Because its tangential velocity is at a 90 degrees angle, the axe will fly forwards away from me, and it will spin around its center of mass while it's in the air.

My question is: will the angular velocity of the rotation of the axe after I let it go be the same as the angular velocity that I spin it with? Am I correct in saying: while I spin the axe and hold the end of the handle, the axe's moment of inertia around this axis will be bigger than the axe's moment of inertia as it is rotating around its center of mass while it's flying. According to the conservation of angular momentum I1*ω1 = I2*ω2, and because I1 > I2, ω1 will be smaller than ω2. Therefore, when I let go of the axe, it's angular velocity will increase (the time it will take the axe to rotate around its center of mass will be shorter than the time it took me to give it one full rotation while holding it by the end of the handle). Is this correct?

Thank You

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  • $\begingroup$ You should note that the angular momentum L = r x p. And the origin is your centre of rotation. Does L change after you throw the axe? $\endgroup$ – Claudio Saspinski Jan 26 at 19:41
  • $\begingroup$ So we can only apply the conservation of angular momentum to rotations around the center of mass, therefore from the perspective of the axe's center of mass angular momentum will be conserved, and the axe's angular velocity will remain the same. I think I understand, thank you $\endgroup$ – Andrew Jan 27 at 8:04
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Up until you release the axe, any force applied by your hand can alter it's rotation and/or trajectory. After the release, it's rotation will not increase, but may slow negligibly from air resistance.

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If you swing the axe on a string attached to its handle then while it rotates about your shoulder it also rotates about its own centre of mass with the same frequency with which you swing it. (It points upward at the highest point and downwards at the lowest; all the time its direction is changing.)

Now if you cut the string at any point the axe continues rotating about its centre of mass with the same frequency as before. No torque has been applied to change its angular velocity about its centre. Cutting the string merely removed the centripetal force which kept it moving in a circle.

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