SHM Energies

Imagine a very smooth table. Attach a spring to the edge. Fix a piece of wood to the other end, pull it, and let go. The wood moves left and right. There is obviously kinetic energy, since there is motion. As the wood to the right, it slows. When it reaches the rightmost point, the velocity is zero for an instant, and then it moves left. Clearly, when the velocity is zero, the kinetic energy must also go to zero. What happened to the energy?

We know that energy is conserved, so it cannot just disappear. If the kinetic energy vanished, then it must have changed into another type of energy. In this case, it has become potential energy. What is this potential energy about?

We have seen what gravitational potential energy is. If you lift up a ball and hold it there, the ball does not move. It does not look like it has any energy. When you let go, it falls and gains more and more kinetic energy. It is as if there is an energy hidden inside that we cannot see, and that reappears when you drop the ball. This hidden energy is called the potential energy. Actually, this hidden energy exists because when you lifted up the ball, you did work against gravity. The work produces this potential energy.

Going back to the oscillating wood, notice that each time it moves left or right, it compresses or stretches the spring. If the wood moves to the right, it pulls at the spring and makes it longer. So the wood does work on the spring. It is this work that becomes potential energy. This energy is stored in the stretched spring, in the tension. In doing this work, the ball uses up its kinetic energy, so it slows down. When it stops, all the kinetic energy has changed into potential energy of the spring. This shows itself as tension in the spring, which pulls at the wood. So the wood starts moving to the left, and the potential energy is changed back to kinetic energy.

This type of potential energy is called elastic potential energy. From the above description, we see that the kinetic energy and potential energy keeps changing into each other. Lets summarise the main features:

  1. At the maximum displacement x0, the kinetic energy is zero and the potential energy at its biggest.
  2. At the rest position, the potential energy is zero since the spring is neither stretched nor compressed.
  3. At the rest position, the kinetic energy is biggest, since the velocity v0 is biggest.
  4. The kinetic energy mv02/2 at the rest position is equal to the potential energy at kx02/2 the maximum displacement, since one is completely changed into the other.
  5. If there is no friction, the total of the kinetic and potential energies at any time does not change.
  6. So the total energy is equal to kx02/2, which is also equal to mv02/2.



Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 5 May 2011.