Openphysics/MechanicsL3

Simple Harmonic Motion
All around us we observe motions that can be described as oscillating, or periodic: a child on a swing,a car bouncing on its springs, a tree swaying in the breeze. Simple harmonic motion (SHM) is the simplest kind of oscillating motion to describe mathematically.

Definition and examples
In simple harmonic motion (SHM), we observe a body moving back and forth either side of an equilibrium position. At the equilibrium position, the forces on the body are balanced. At this position, the acceleration of the body will be zero (but it won't in general be stationary). When the body is away from the equilibrium position, there is an unbalanced force on it directed back towards the equilibrium position. A child on a swing, or any simple pendulum swinging from left to right, is an example of what has been described so far. When the child is directly below the suspension point, the forces on the child are balanced. This is the equilibrium position. If the child is to the right of the equilibrium position, there will be an unbalanced force, and thus an acceleration, back to the left. This could mean the child is moving away from the centre, but slowing down, or is coming towards the centre and is speeding up. The special thing about simple harmonic motion (that means that the swing or pendulum example does not exactly qualify as SHM) is that the unbalanced force is directly proportional to the displacement from the equilibrium position.

In summary, in SHM, the unbalanced force is proportional to the displacement from the equilibrium position, and is directed towards the equilibrium position (i.e. is a restoring force - tending to restore the systen to equilibrium). We could write this equation, where F is the unbalanced force, y is the displacement and k is a positive number that depends on the actual system, but will be a constant in a system that truly moves with SHM:

$$F=-ky$$

The vertical oscillations of mass hanging on a spring are simple harmonic motion, if we disregard the gradual loss of energy with time, resulting in the oscillations dying away. The k in the equation above will actually be the spring constant of the spring, although it is not entirely starightforward to show this. There are two forces acting: the force from the spring and the force of gravity. We would have to show by experiment or analysis that the resultant of these two varies as described above...but it does!

The movement of a liquid, slopping back and forth in a U-tube, is another example. Once again, we would have to disregard the loss of energy due to friction and the like.

The bob, or mass at the end of simple pendulum moves approximately with SHM. The motion is very close to SHM for small oscillations, i.e. when the bob does not move very far from the centre.

We can determine if a system will move with SHM by carefully analysing the forces acting on the moving mass to see if the unbalanced force varies with displacement in the way described above.