## Wednesday, December 3, 2014

### Energies of Body in Simple Harmonic Motion

Force on a body  in simple harmonic motion

The body is in simple harmonic motion because there is a force acting towards the mean position. Generally this force could be called as a restoring force. As we have derived a equation for acceleration we can derive the equation for the force using the Newton’s second law of motion as shown below. The constant is called as force constant. Basing on the derivation we can again show that the acceleration is directly proportional to displacement but in the opposite direction.

Potential energy of your body in simple harmonic motion

We can derive the equation for the potential energy of a body in simple harmonic motion using the basic concept of integration. Let us consider a particle having a displacement from the mean position. To displace the body by a small value we shall do some small amount of the work. The total displacement is some of this kind of so many displacements therefore we have to calculate the work and for each part and add all of them to get the total work done in this process.A mathematical phenomenon called integration has to be used to do this job.

The equation that we have derived is actually equation for the displacement of body by a smaller value. Any way after the work is done as per the law of conservation of energy, the work can not disappear. Work and energy are simialar kind the terms like you’re having energy, then you can do the work and vice versa.

Therefore after doing the work, this work that will be stored in the format of energy and that energy is called potential energy of the body in simple harmonic motion. We can derive the equation for the potential energy basing on the concept of integration as shown below.

It could be easily noticed that potential energy is maximum at the extreme position and intreme position and it is minimum at the mean position.

Kinetic energy of your body in simple harmonic motion

As the body in simple harmonic motion is having some velocity that will automatically process kinetic energy also. The derivaetion for the kinetic energy is comparatively quite simple as shown below. It could be easily noticed that kinetic energies is maximum at the mean position and as it is going from mean position to extreme position it will be keep on decreasing and finally becomes zero.

We shall notice that at one point if the potential energies maximum at the same point kinetic energies minimum and vice versa. This clearly satisfy the conservation of  energy that the total sum of the energy always remains constant. If one energy increases another energy decreases, but the total energy of the system always remains constant. Energy is neither created nor destroyed it just converts from one format to another format. This is called law conservation of energy which is valid for the entire universe in all conditions.

In the following diagram we have shown how to derive the question for kinetic energy and the total energy.

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