# Is Conservation of Energy maintained when the orbit of a rotating mass increases in diameter?

I read an example where someone was explaining how the law of conservation of energy does not have to be maintained within a rotating mass even though angular momentum is maintained. The given example is an ice skater who spins faster as she brings her arms inward. The energy used bringing her arms inward gets transferred to her total rotational energy. Therefore, her total rotational energy increases while angular momentum was maintained.

My question is about the opposite movement. Assume you have a weight positioned on a rotating wheel and the weight is held in place by a lock on a radial slide. Now assume the wheel is rotating in motion. If you unlock the weight (via electronic control for example), the weight will move outwardly toward the perimeter of the wheel due to its own inertia (centrifuge effect). When this occurs, does the kinetic energy of the wheel decrease?

I actually performed an experiment of this very thing, and my crude setup seemed to confirm that the available energy in the wheel DOES indeed decrease. If this complies with the mathematical laws, can someone please confirm and explain where the energy goes? It's certainly not being lost to vibration or heat. How is the energy in the wheel decreasing simply because the weight travels outward to the perimeter?

• Why do you claim, "It's certainly not being lost to vibration or heat."? – Mark Eichenlaub Nov 2 '15 at 1:36
• I'm curious: what experiment did you do to confirm this? – garyp Nov 2 '15 at 1:38

Yes, the rotational kinetic energy decreases. The extra energy is converted to thermal energy in the wheel and environment.

If you imagine letting the weight go, it will slide across the surface of the wheel as it moves towards the edge. This sliding is motion against friction, so energy is lost there. Then the weight might bang into whatever holds it at the edge of the wheel, dissipating more energy. So the energy is simply thermalized just like it is when you drop a lump of clay to the ground.

It would be possible to construct a device so that the energy goes somewhere else. For example, you could put a hole in the center of the wheel and tie a rope from the weight, through the hole, to another weight. Then as the weight on the wheel moved towards the edge it would lift the weight on the rope, and at least some of the energy would be stored as gravitational potential energy. Similarly, you could make the weight on the wheel compress a spring as it moved towards the edge, and then some of the energy would be stored as potential energy in the spring, etc.

Wait...wait... what? No, no, no, no... No. If a body is spinning, supposing non frictional surfaces and all of that, the energy would not decrease (if there is no external force). The energy is always constant. In that kind of problems there is only rotational energy (there is no other energy):

$$$E_{rot}={1\over 2}I\,\omega^2,$$$

where $I$ is the moment of inertia which depends on the geometrical form of the body, and $\omega$ is the angular velocity.

Take for example the case of the ice skater. Yes, if she place her arms against her body, the angular velocity would increase... that doesn't mean that also the energy would increase. The factor of moment of inertia in the expression of the rotational energy will ensure that this doesn't happen. If the angular velocity increase (decrease) is because the moment of inertia decrease (increase), and it does in such a way that keeps constant the total energy.

The same goes to your experiment. As the mass moves away from the center of rotation, its moment of inertia would increase, which translate in an decrease in angular velocity... because nature wants its energy constant (when there aren't external forces acting on it).

The energy is dissipated when the mass stops at the perimeter. Work has to be done to stop the radial motion of the mass.