# The Physics Philes, lesson 80: Oscillation, Conservation, Physics-ification!

We’ve learned a lot about simple harmonic motion over the past several weeks. We’ve learned how to find the velocity, acceleration, and displacement of an object in SHM. Now we need to integrate energy into our understanding.

Let’s think about our object oscillating at the end of a spring.

The only force that does any work on the oscillating body is the horizontal spring force. Any vertical forces that are acting on the body don’t do any work. (Remember that we’re using “work” in the physical sense.) Because of this we can say that the total mechanical energy is conserved.

To make this work for us, we need to dig way back into our memory banks and pull out our knowledge of kinetic and potential energy. Don’t worry. I’m here to help.

Kinetic energy is half of the product of the mass of the object times the square of its velocity. Potential energy is half of the force constant k times the displacement x:

If, as we stated above, the total mechanical energy of the oscillating system is conserved, then the sum of the kinetic and potential energy should be constant. So, if the system has total mechanical energy E, the potential and kinetic energy should always add up to that value. Or, in math:

Since x is the displacement of the oscillating body, the total mechanical energy is directly related to the amplitude A. When the body reaches maximum displacement, it stops for a moment before it snaps back. At this point, the velocity is zero, and energy of the system is entirely potential. In this case, the total mechanical energy is

That means that, whenever the oscillating body is at it’s maximum amplitude A (or -A), the potential energy at that point is equal to kinetic energy and potential energy at any other spot along the displacement.

So, that was a quick lesson this week. Sorry, but that’s how it goes. Next week, though, we will explore some applications of simple harmonic motion. See you then!

*Featured image credit: jar () via Flickr*

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