Question

derive an exprsssion for the total energy of a harmonic oscillator and show that it is constant and propotional to the square of amplitude.​

Answer 1

Energy and the Simple Harmonic Oscillator

To study the energy of a simple harmonic oscillator, we need to consider all the forms of energy. Consider the example of a block attached to a spring, placed on a frictionless surface, oscillating in SHM. The potential energy stored in the deformation of the spring is

U=1/2kx^2.

In a simple harmonic oscillator, the energy oscillates between kinetic energy of the mass

K=1/2mv^2

and potential energy

U=1/2kx^2

stored in the spring. In the SHM of the mass and spring system, there are no dissipative forces, so the total energy is the sum of the potential energy and kinetic energy. In this section, we consider the conservation of energy of the system. The concepts examined are valid for all simple harmonic oscillators, including those where the gravitational force plays a role.