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When you unbalance a rapidly spinning object, it's going to be a lot different than letting something fall from your hand.
Part of the problem constructing the device will be to deal with the resulting forces on an unbalanced rapidly rotating object. With strong enough materials and special construction it's theoretically possible, though I still don't know if there are any actual physical materials strong enough to allow the device as specified to be built. One way of constructing the device (at least in principle) is found by copying the desing of the household washing machine - the motor and rotor are connected together rigidly, but the whole assembly is suspended from cables, allowing them to move. Without some such special arrangement, the connection between the motor and rotor will break.
An analysis of the forces is still a complicated job. For starters,c onsider the following digram
left---------------right^
With "up" towards the top of the page, and "down" towards the bottom of the page, consider the sense of the rotation such that the right side is moving "up".
The clearest way to analyze the problem is to consider a non-rotating frame of reference, IMO - that has the added advantage of being the standard approach.
We can then ask the quesiton, with no other forces applied, what is the resulting motion when we detach a piece of the rotor on the right side?
The answer is that the detached piece will continue to move upwards. When we sum the total momentum of the rotating piece in the up down direction, we see that the rotating piece moves downwards. This momentum must remain conserved. The general motion of any rigid body will be a sum of rotation around it's center of mass, plus translation. So the motion of the main rotor will be that it will continue to rotate around its new center of mass (which will be displaced from the old center of mass), but it will also move downwards.
[add]
Note that the new center of mass was already moving downwards to start with, because it is located to the left of the original center of mass. It's original downwards velocity was w*dr, where dr is the amount the center of mass shifts to the left.
The shift of the location of the center of mass of the rotor is the key to solving the problem and getting the correct answer.
Anyway, that's the clearest way I can think of to describe the solution (it's a bit difficult with only limited diagrams and ascii text).
I've also simplified the problem somewhat - a real system would not only have a rotor, but a motor of non-negilgible mass to consider. I've omitted the motor in the simplified analysis.
It may help to draw a digram like the one above, with the velocities of all the parts shown as arrows. (I won't attempt to do this in ascii). You can then see visually how the rotational motion decomposes on the main rotor to the sum of a new rotational motion plus a translational motion downwards - you add up the arrows for the rotational motion around the new axis of rotation, plus the arrows representing the linear motion, and see that they add up to the original velocity vectors of the rotating bar.
Part of the problem constructing the device will be to deal with the resulting forces on an unbalanced rapidly rotating object. With strong enough materials and special construction it's theoretically possible, though I still don't know if there are any actual physical materials strong enough to allow the device as specified to be built. One way of constructing the device (at least in principle) is found by copying the desing of the household washing machine - the motor and rotor are connected together rigidly, but the whole assembly is suspended from cables, allowing them to move. Without some such special arrangement, the connection between the motor and rotor will break.
An analysis of the forces is still a complicated job. For starters,c onsider the following digram
left---------------right^
With "up" towards the top of the page, and "down" towards the bottom of the page, consider the sense of the rotation such that the right side is moving "up".
The clearest way to analyze the problem is to consider a non-rotating frame of reference, IMO - that has the added advantage of being the standard approach.
We can then ask the quesiton, with no other forces applied, what is the resulting motion when we detach a piece of the rotor on the right side?
The answer is that the detached piece will continue to move upwards. When we sum the total momentum of the rotating piece in the up down direction, we see that the rotating piece moves downwards. This momentum must remain conserved. The general motion of any rigid body will be a sum of rotation around it's center of mass, plus translation. So the motion of the main rotor will be that it will continue to rotate around its new center of mass (which will be displaced from the old center of mass), but it will also move downwards.
[add]
Note that the new center of mass was already moving downwards to start with, because it is located to the left of the original center of mass. It's original downwards velocity was w*dr, where dr is the amount the center of mass shifts to the left.
The shift of the location of the center of mass of the rotor is the key to solving the problem and getting the correct answer.
Anyway, that's the clearest way I can think of to describe the solution (it's a bit difficult with only limited diagrams and ascii text).
I've also simplified the problem somewhat - a real system would not only have a rotor, but a motor of non-negilgible mass to consider. I've omitted the motor in the simplified analysis.
It may help to draw a digram like the one above, with the velocities of all the parts shown as arrows. (I won't attempt to do this in ascii). You can then see visually how the rotational motion decomposes on the main rotor to the sum of a new rotational motion plus a translational motion downwards - you add up the arrows for the rotational motion around the new axis of rotation, plus the arrows representing the linear motion, and see that they add up to the original velocity vectors of the rotating bar.
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