There are many situations where we can observe the special kind of oscillations called simple harmonic motion (s.h.m.). Some are more obvious than others. For example, the vibrating strings of a musical instrument show s.h.m. When plucked or bowed, the strings move back and forth about the equilibrium position of their oscillation. The motion of the tethered trolley in Figure 18.3 and that of the pendulum in Figure 18.4 are also s.h.m. (Simple harmonic motion is defined in terms of the acceleration and displacement of an oscillator – see topic 18.5 Representing s.h.m. graphically.)
Here are some other, less obvious, situations where simple harmonic motion can be found:
- When a pure (single tone) sound wave travels through air, the molecules of the air vibrate with s.h.m.
- When an alternating current flows in a wire, the electrons in the wire vibrate with s.h.m.
- There is a small alternating electric current in a radio or television aerial when it is tuned to a signal in the form of electrons moving with s.h.m.
- The atoms that make up a molecule vibrate with s.h.m. (see, for example, the hydrogen molecule in Figure 18.11a).
Oscillations can be very complex, with many different frequencies of oscillation occurring at the same time. Examples include the vibrations of machinery, the motion of waves on the sea and the vibration of a solid crystal formed when atoms, ions or molecules bond together (Figure 18.11b). It is possible to break down a complex oscillation into a sum of simple oscillations, and so we will focus our attention in this chapter on s.h.m. with only one frequency. We will also concentrate on large-scale mechanical oscillations, but you should bear in mind that this analysis can be extended to the situations already mentioned, and many more besides.

Figure 18.11: We can think of the bonds between atoms as being springy; this leads to vibrations, a in a molecule of hydrogen and b in a solid crystal.

The requirements for s.h.m.

If a simple pendulum is undisturbed, it is in equilibrium. The string and the mass will hang vertically. To start the pendulum swinging (Figure 18.12), the mass must be pulled to one side of its equilibrium position. The forces on the mass are unbalanced and so it moves back towards its equilibrium position.
The mass swings past this point and continues until it comes to rest momentarily at the other side; the process is then repeated in the opposite direction. Note that a complete oscillation in Figure 18.12 is from right to left and back again.
The three requirements for s.h.m. of a mechanical system are:
- a mass that oscillates
- a position where the mass is in equilibrium
- a restoring force that acts to return the mass to its equilibrium position; the restoring force F is directly proportional to the displacement x of the mass from its equilibrium position and is directed towards that point.

The changes of velocity in s.h.m.

As the pendulum swings back and forth, its velocity is constantly changing. As it swings from right to left (as shown in Figure 18.12) its velocity is negative. It accelerates towards the equilibrium position and then decelerates as it approaches the other end of the oscillation. It has positive velocity as it swings back from left to right. Again, it has maximum speed as it travels through the equilibrium position and decelerates as it swings up to its starting position.
This pattern of acceleration–deceleration–changing direction–acceleration again is characteristic of simple harmonic motion. There are no sudden changes of velocity. In the next topic, we will see how we can observe these changes and how we can represent them graphically.