Simple Harmonic Motion
A swinging pendulum. An oscillating spring. These are two examples of oscillations. There is something common to all oscillations. To see this, think of a body hanging on a spring.
When you pull the body down a bit and release it, the body would oscillate up and down. If you pull the body down further, you can feel the spring pulling back with a larger force. It is this force that causes the oscillation. When the body is displaced from the rest position, the spring produces a force that tries to pull it back to the rest position. We call this the restoring force. The larger the displacement, the larger is the restoring force.
So the thing that is common to all oscillations is this: The larger the displacement, the larger is the restoring force.
Think of a pendulum at rest. Pull the bob to one side and release it. It will swing back towards the rest position. That is, the restoring force acts towards the rest position.
So another thing that is common to all oscillations is this: The restoring force acts towards the rest position.
In physics, we often try to understand things by making up a simple explanation. This could be a guess or an assumption. For oscillations, we often assume that the restoring force is directly proportional to the displacement. The force would be opposite to the displacement, of course, so as to pull the body back to the rest position.
Suppose I displace the bob from the rest position by 1 cm. There is a certain restoring force. Suppose I double this displacement to 2 cm. According to the above assumption, the force would now be doubled. This is because we assume that the force is proportion to the displacement. In real life, this is only approximately true. It gets even less true if the bob gets further from the rest position.
However, although this assumption is too simple, it is often enough to help us understand and estimate the velocities and periods of the oscillation fairly accurately. For this reason, we are going to learn more about how to do such calculations.
First, let us summarise the above ramblings. An oscillatory motion that obeys the above assumption is called a simple harmonic motion. An oscillation is simple harmonic if:
The restoring force is directly proportional to the displacement from the rest position, and acts in the opposite direction to this displacement.
This is also taken as the definition of simple harmonic motion.
We shall only look at the simple case when the motion is along a straight line. Let the force be F and the displacement be x. Then the definition can be written as
where k is a constant. The k is there because if two quantities are proportional, this is how they are related. The minus sign just means that F is in the opposite direction to the x. Recall Newton's second law:
where m is the mass of the oscillating body and a is the acceleration. Equating the right hand sides of these two equations and rearranging, we get
As it turns out, it is possible to show, using some mathematics that I shall not go into, that (k/m) is related to the angular frequency ω by
So the above expression for the acceleration may also be written as
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 27 March 2011.