R.Fizpatricks' problem from his online Lectures on CM

[gainwmn]

R.Fizpatricks' problem from his online "Lectures on CM"

In those "Lectures on classical mechanics" in problem 2.4. for 'Newtonian mechanics' section the author gets the expression for angular frequency of small oscialations as the square root from the ratio of potential energy and moment of inertia in equilibrium, based on the virial theorem formula, presented in the text of the problem. Now, the latter formula I have easily proven, but could not figure out how he gets the angular frequency expression mentioned above from it.
Any takers? Thanks in advance.

 

[gtoguy97]

The answer to this question can be found in the solution to the problem given in the "Lectures on Classical Mechanics" by R. Fizpatrick. In the solution, it is explained that the angular frequency of small oscillations is obtained by taking the square root of the ratio of the potential energy to the moment of inertia in equilibrium. This follows from the fact that the virial theorem formula (which is derived in problem 2.4) states that the time-averaged kinetic energy is equal to half of the potential energy. Therefore, the total energy of the system is equal to the potential energy, and the angular frequency can be determined by taking the square root of the ratio of the potential energy to the moment of inertia in equilibrium.