SHM Acceleration
A body hanging on a spring oscillates up and down. Its displacement changes. Its velocity changes. Its acceleration also changes.
Lets try and get a better idea as to how they change. Suppose that at some point in time, that body has reached it highest position. At this position, the displacement is at its maximum. It is just about to turn back and start downwards. So at this instance in time, the velocity is zero. What about the acceleration? Is it zero also?
If you are new to this, it is easy to think that because the velocity is zero, the acceleration is also zero. If you think so, then you must quickly banish the thought. The acceleration at a particular time has nothing to do with the velocity itself at that same time. The acceleration is related to the change in that velocity. That is to say, the acceleration depends on the difference between the velocity now, and the velocity a bit later. So if the velocity does not change, the acceleration is zero. If the velocity changes a lot, the acceleration is very big. This is regardless of whether the velocity now is big or small. So from the velocity alone, it is difficult to tell what the acceleration is. There is a simpler way.
The acceleration is proportional to the force, according to Newton's second law. In simple harmonic motion, the force is in turn proportional to the displacement. So the acceleration is proportional to the displacement:
where
and T is the period of oscillation. This means that when the displacement is at a maximum, so is the acceleration. This is because the restoring force is also at its largest. Furthermore, since the force points towards the rest position, so does the acceleration. So the acceleration is in the opposite direction to the displacement, which explains the minus sign in the above relation between a and x. Plotting both on the same graph would give a wave like curve for x, and a similar curve for a that is upside down with respect to x.
In the following graphs, various combinations of x, v and a are plotted on the same graphs. You should convince yourself that you understand how they are related to each other.
For example, the velocity and acceleration graphs do not peak at the same time because the velocity is biggest at the rest position, whereas the acceleration is biggest at the maximum displacement.
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 1 April 2011.