15.4: Energy in Simple Harmonic Motion
To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to Hooke's Law, the energy stored during the compression/stretching of a string in a simple harmonic oscillator is potential energy. As the simple harmonic oscillator has no dissipative forces, it also possesses kinetic energy. In the presence of conservative forces, both energies can interconvert during oscillation, but the total energy remains constant. The total energy for a simple harmonic oscillator is equal to the sum of the potential and kinetic energy and is proportional to the square of the amplitude. It can be expressed in the following form:
The magnitude of the velocity in a simple harmonic motion is obtained by rearranging and solving the equations of the total energy.
Manipulating this expression algebraically gives the following:
Notice that the maximum velocity depends on three factors and is proportional to the amplitude. If the displacement is maximal, the velocity will also be maximal. Additionally, the maximum velocity is greater for stiffer systems because they exert greater force for the same displacement. This observation can be seen in the expression for the maximum velocity. The maximum velocity is proportional to the square root of the force constant. Finally, the maximum velocity is smaller for objects with larger masses since the maximum velocity is inversely proportional to the square root of the mass.