To cause a displacement of a body controlled by elastic forces, work has to be done. For example, work has to be done in stretching a spring against the elastic forces. This work will be stored in the body as potential energy. When the deforming force is released, this potential energy manifests in the form of kinetic energy which makes the body to move. Thus, a body in SHM, has both potential energy and kinetic energy.
The work done in displacing a body through a distance y is given by force X displacement. In SHM, the force is not a constant but proportional to the displacement. Therefore, an average force
can be assumed to produce a displacement y.
But F = Ky in magnitude
Thus, the total energy is a constant.
At the equilibrium position, y = 0
At the extreme position, y = A,
As the body oscillates, there is a transformation between kinetic energy and potential energy, maintaining the total energy constant. The loss in potential energy appears as a gain of kinetic energy and vice versa.
The constant interchange of energy between potential and kinetic forms is essential for producing and maintaining any type of oscillation. In the case of the oscillating bob of a simple pendulum, it loses kinetic energy after passing through the middle of the swing and then stores the energy as potential energy, as it rises to the top of the swing. The reverse occurs as it swings back. In the case of oscillating layers of air, when a sound wave passes, kinetic energy of the moving air molecules is converted to potential energy when the air is compressed. In the case of electrical oscillations a coil L and a capacitor C in the circuit constantly exchange energy, this is stored alternately in the magnetic field of L and the electric field of C.
The figure shows graphs of kinetic energy and potential energy versus displacement. Each of them is a parabola. The sum of the potential energy and the kinetic energy values remains constant as represented by the horizontal line. At the points where the two curves cross, we can write
Equation (2.53) can also be written as
Thus, the energy of a body executing SHM is directly proportional to
- the square of the frequency
- the square of the amplitude
