15.4: Énergie du mouvement harmonique simple
To study 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 deformation of a simple harmonic oscillator is a form of potential energy, and because it has no dissipative forces, it also possesses kinetic energy. Most of the time, both the energies become interconverted with each other during oscillation, and the total energy remains constant. Thus, the total conservation of energy for a simple harmonic oscillator is equal to the sum of potential energy and kinetic energy and is proportional to the square of the amplitude. It can be given by the expression
By rearranging and solving the equations of total energy, the magnitude of velocity in a simple harmonic motion is obtained.
Manipulating this expression algebraically gives:
where
Here, notice that the maximum velocity depends on three factors and is directly proportional to the amplitude. If the displacement is maximum, the velocity will also be maximum. Also, maximum velocity is greater for stiffer systems because they exert greater force for the same displacement. This observation is seen in the expression for maximum velocity; it is proportional to the square root of the force constant. Finally, the maximum velocity is smaller for objects that have larger masses, since maximum velocity is inversely proportional to the square root of the mass.
This text is adapted from Openstax, College Physics, Section 16.5: Energy and the Simple Harmonic Oscillator.
