A particle executing simple harmonic motion possesses both kinetic energy and potential energy and the total energy of particle executing simple harmonic motion at any point is equal to the sum of kinetic energy and potential energy i.e.Let at any instant, the particle be at P at a distance y from mean position and v be the velocity of particle at P.
Kinetic energy: The kinetic energy of particle executing simple harmonic oscillation at any instant is given byThe velocity of particle at a distance y from the mean position is,Potential energy: The potential energy stored in the particle is equal to the work done in displacing the particle from mean position to y. Let the particle be displaced through a distance x from mean position. The restoring force F acting on particle is,Therefore, work done against this restoring force in moving the particle through a small distance dx is,Therefore, total work done by restoring force in displacing the particle from mean position to P is,The potential energy stored in the body is equal in magnitude and opposite in sign of the work done by restoring force. ThusTotal energy is,E = K + UIt is clear from the above equation that total energy is independent of the position of particle during its motion. Thus, total energy is constant.