Simple Harmonic Motion

A pendulum moving backwards and forwards repeatedly is an example of an oscillation. The displacement of an object is measured from the equilibrium position (the lowest point in which the object would usually come to standstill).

Consider a pendulum: in one oscillation (when released from above the equilibrium position) the displacement decreases as the pendulum returns to equilibrium and then increases and reverses as the pendulum moves away from equilibrium in the opposite direction. The displacement decreases again as the pendulum returns to equilibrium and finally increases once more as it moves away from equilibrium back to its starting point.

THIS CAN BE REPRESENTED BY A COSINE GRAPH
where displacement is on the y axis and time is on the x axis.

SHM is defined as an oscillating motion when the acceleration is

• proportional to the displacement
• always in the opposite direction to the displacement

These two conditions regarding acceleration and displacement must be kept if shm is to occur. The time period is independent of the amplitude of the oscillations.

As a result of this the acceleration is always in the opposite direction to the displacement, hence can be calculated by:

acceleration (a) = - (2Πf)² × displacement (x)

The gradient of the displacement/time graph gives the velocity. The velocity is therefore zero when the gradient is zero.

An object that has a constant amplitude has no friction acting on it, hence is said to be oscillating freely. Friction acts as a damping force hence gradually reduces the amplitude over a period of oscillations.

The potential energy Ep changes with displacement from equilibrium. As Potential Energy (Ep) lost = Kinetic Energy (Ek) gained
we can see that as the kinetic energy increases, the potential energy decreases.

There are three types of damping; light, heavy and critical.
Light damping is when there is only a minor force reducing the amplitude (i.e. friction) meaning the oscillation continues whilst the amplitude gradually reduces.
Heavy damping is when the damping force is so strong that no oscillating motion occurs but takes more time than critical damping.
Critical damping is just enough to stop the system oscillating in the shortest possible time.

Resonance is when the periodic driving force matches the natural frequency, making the amplitude of the oscillations noticeably increase.

The lighter the damping in the system, the larger the amplitude becomes hence the statement above is only true for systems with little or no damping.

An example of resonance in context is when the Tacoma Narrows Bridge collapsed as the wind along the bridge span exerted a periodic driving force that matched the natural frequency of the bridge span, meaning its amplitude increased causing it to collapse.

applied frequency of periodic driving force = natural frequency
for an oscillating system experiencing little or no damping