Each day we encounter many kinds of oscillatory motion. Common example includes the swinging pendulum of a clock, a person bouncing on a trampoline, and vibrating guitar string. Examples on the microscopic scale are vibrating atoms in the quartz crystal of a wristwatch and vibrating molecules of air that transmit sound waves. The above cases are mechanical oscillations. We are also familiar with electromagnetic oscillations, such as electrons surging back and forth in circuits that are responsible for transmitting and receiving radio or TV signals.
Imagine an oscillating system, such as the pendulum of a clock or a mass on a spring. What must be the properties of the force that produces such oscillations?
Let us consider a simple example. Suppose we have a particle that is free to move only in the x direction, and let the particle experience a force of constant magnitude Fm that acts in the + x direction when x < 0 and in the – x direction when x > 0, as shown in Fig. a. the force, which is shown in Fig. b.
A particle of mass m at coordinate x = + xm experiences a force whose x component is – Fm and the corresponding x component of the acceleration of the particle is -am = -Fm/m. the particle moves toward its equilibrium position at x = 0 and reaches that position with velocity v = -um . when it passes through the origin to negative x, the force becomes +Fm, and the acceleration is + am. The particles slows and comes to rest for an instant at x = -xm before eventually to x = + xm. In the absence of friction and other dissipative forces, the cycle repeats endlessly.