Show that the virial theorem is correct for a particle of...

Show that the virial theorem is correct for a particle of mass $m$ free to move in a plane, and attached to one end of a Hooke's-law spring exerting force $F=-k r$, if the particle is in (a) a circular orbit, (b) an elliptical orbit.

Key Concepts

Elliptical Orbit Dynamics

Elliptical orbits are more general closed trajectories where the distance between the particle and the center of force changes with time. Despite these variations, when averaged over a complete cycle of motion, the relationship between the kinetic and potential energies still obeys the virial theorem. In the case of a quadratic potential, the time-averaged kinetic energy over one period equals the time-averaged potential energy, demonstrating the robustness of the virial theorem even when the instantaneous values fluctuate.

Circular Orbit Dynamics

In the case of a circular orbit, the radius remains constant throughout the motion. This simplifies the calculations because the kinetic and potential energies remain constant with time. For a circular orbit in a quadratic potential, one can directly verify that the kinetic energy equals the potential energy, which is consistent with the prediction of the virial theorem for a harmonic oscillator.

Virial Theorem

The virial theorem relates the time-averaged kinetic energy of a bound system with its potential energy. In systems obeying a potential that is a homogeneous function of the coordinates, the theorem provides a fixed proportional relationship between these energies, independent of the specific details of the motion. For instance, when the potential energy is quadratic in displacement, as in Hooke’s law, the virial theorem predicts that the average kinetic energy equals the average potential energy.

Hooke's Law Potential

Hooke's law describes a restorative force that is proportional to the displacement from an equilibrium position. The force is given by F = -kr and results in a potential energy of V = (1/2)kr², which is a homogeneous function of degree 2. This quadratic nature of the potential means that the application of the virial theorem leads to a simple relation between the time-averaged kinetic and potential energies, making it an ideal example to illustrate the theorem.

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