Average potential and kinetic energy of SHO

In summary, the conversation discusses finding the average potential and kinetic energy of a standard driven damped oscillator. The speaker provides a trick for solving integrals of cos²(t) and explains how to account for the added energy from the driver and loss of energy from damping when finding the average kinetic energy.
  • #1
mewmew
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I am using Frenches book on waves and have a question. You have a standard driven damped oscillator and I am suppose to find the average potential and kinetic energy of it. They are both similar so I will just use the potential for example. I took [tex] \frac{1}{T}\int^T_0 \frac{1}{2}KX^2 dt[/tex] Where [tex]X=Cos(\omega t-\rho)[/tex] I solved the resulting integral with half angle formulas but found my answer to be a bit messy, is that the correct way of getting the average kinetic energy or am I missing something here? Thanks
 
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  • #2
There's a trick to doing integrals of cos²(t). First, you should know that:

sin²(t)+cos²(t) = 1 for all t.

Therefore,

[tex]\int _a ^b (\cos ^2 (t) + \sin ^2 (t))dt = b-a = \int _a ^b \cos ^2 (t)dt + \int _a ^b \sin ^2 (t)dt[/tex]

Also, you should know that

[tex]\int \limits_T \cos ^2 (t)dt = \int \limits_T \sin ^2 (t)dt[/tex]

where T is one period. Therefore, since your average potential energy will be taken over one period (T = b-a), you can conclude that:

[tex]\int _a ^b (\cos ^2 (t) + \sin ^2 (t))dt = \int \limits_T (\cos ^2 (t) + \sin ^2 (t))dt = \int \limits_T (\cos ^2 (t) + \cos ^2 (t))dt = 2\int \limits_T \cos ^2 (t)dt=T[/tex]
[tex]\int _a ^b \cos ^2 (t)dt = (1/2)T[/tex]

Now about your question concerning average kinetic energy. Your system is driven, so you have to account for the added energy from the driver and also the loss of energy from the damping. What I think you should do is take your differential equation of motion and find the function of velocity wrt time and find the average kinetic energy using a similar method.
 
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What is SHO?

SHO stands for Simple Harmonic Oscillator. It is a system in which a particle oscillates back and forth around a fixed point with a constant frequency.

What is potential energy?

Potential energy is the energy stored in an object due to its position or configuration. In the case of SHO, it is the energy stored in the system when the particle is at its maximum displacement from the fixed point.

What is kinetic energy?

Kinetic energy is the energy that an object possesses due to its motion. In SHO, it is the energy that the particle has when it is in motion, either towards or away from the fixed point.

How is the average potential energy of SHO calculated?

The average potential energy of SHO can be calculated by taking the average of the potential energy at the maximum displacement and the potential energy at the equilibrium point. This is because the potential energy in SHO is directly proportional to the square of the displacement from the equilibrium point.

How is the average kinetic energy of SHO calculated?

The average kinetic energy of SHO can be calculated by taking the average of the kinetic energy at the maximum displacement and the kinetic energy at the equilibrium point. This is because the kinetic energy in SHO is directly proportional to the square of the velocity of the particle.

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