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julienl07
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Hello. This has been bothering me.
A point mass is on a rotating flywheel that has a constant initial angular velocity, ω0. The object (point mass), initially at some distance r0 from the axis of rotation, now moves out to a further distance rf, and then stops. Say the wheel has a moment of inertia Iw, and the mass is m.
If the system includes the wheel and object, its total moment of inertia is
I = Iw+mr2
By conservation of angular momentum
ωf=ω0[itex]\frac{I_{0}}{I_{f}}[/itex]
If the initial rotational kinetic energy of the system was [itex]\frac{1}{2}I_{0}ω_{0}^{2}[/itex], the final is [itex]\frac{1}{2}I_{f}ω_{f}^{2}[/itex] = [itex]\frac{1}{2}I_{0}ω_{0}^{2}\frac{I_0}{I_f}[/itex] = [itex]E_{rk_0}\frac{I_0}{I_f}[/itex]. Ignoring the gory details, the rotational kinetic energy of the system has decreased, but no external work has been done.
In my ideal situation, I imagine that the object is attached to the axis of rotation by a cord, and that there are 'guide walls' forming an 'aisle' that goes radially outward from the axis. The tension in the cord is momentarily lowered, and the object moves radially outward through this 'aisle', before the cord is held still. The only interaction that I can think of (other than from the cord) is a normal force between the guide wall and the object. This force both increases the speed of the object, and slows the angular speed of the wheel by a torque, so that the angular velocity of the object and wheel remain equal as it moves outward.
So, sure internal work has been done from the torque, but no external work has been done.
How is it that the total rotational/kinetic energy of the system has changed, but no external work has been done? No heat was generated or anything.
Is there something that I have not seen? Can anyone explain this?
Thanks.
A point mass is on a rotating flywheel that has a constant initial angular velocity, ω0. The object (point mass), initially at some distance r0 from the axis of rotation, now moves out to a further distance rf, and then stops. Say the wheel has a moment of inertia Iw, and the mass is m.
If the system includes the wheel and object, its total moment of inertia is
I = Iw+mr2
By conservation of angular momentum
ωf=ω0[itex]\frac{I_{0}}{I_{f}}[/itex]
If the initial rotational kinetic energy of the system was [itex]\frac{1}{2}I_{0}ω_{0}^{2}[/itex], the final is [itex]\frac{1}{2}I_{f}ω_{f}^{2}[/itex] = [itex]\frac{1}{2}I_{0}ω_{0}^{2}\frac{I_0}{I_f}[/itex] = [itex]E_{rk_0}\frac{I_0}{I_f}[/itex]. Ignoring the gory details, the rotational kinetic energy of the system has decreased, but no external work has been done.
In my ideal situation, I imagine that the object is attached to the axis of rotation by a cord, and that there are 'guide walls' forming an 'aisle' that goes radially outward from the axis. The tension in the cord is momentarily lowered, and the object moves radially outward through this 'aisle', before the cord is held still. The only interaction that I can think of (other than from the cord) is a normal force between the guide wall and the object. This force both increases the speed of the object, and slows the angular speed of the wheel by a torque, so that the angular velocity of the object and wheel remain equal as it moves outward.
So, sure internal work has been done from the torque, but no external work has been done.
How is it that the total rotational/kinetic energy of the system has changed, but no external work has been done? No heat was generated or anything.
Is there something that I have not seen? Can anyone explain this?
Thanks.
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