Average kinetic energy of a damped oscillator

In summary, for a damped mechanical oscillator with energy given by $$E = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}k x^2$$, it is commonly observed that the average kinetic energy ##\langle T \rangle## is half of the total energy of the system. This can be explained by the concept that the energy oscillates between kinetic and potential energy. A more formal way to show this is true is through the average value of a function, where for ##y=f(x)## the average value of ##y## in the interval ##a<x<b## is given by $$y_{avg}=\frac{1}{b-a}\int_a^b
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MuIotaTau
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For a damped mechanical oscillator, the energy of the system is given by $$E = \frac{1}{2}m \dot{x}^2 + \frac{1}{2}k x^2$$ where ##k## is the spring constant. From there, I've seen it dictated that the average kinetic energy ##\langle T \rangle ## is half of the total energy of the system. This makes sense, since the energy sort of "sloshes" back and forth between kinetic and potential energy, but is there a more formal way to show this is true?
 
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  • #2
if you have ##y=f(x)## then the average value of ##y## in ##a<x<b## is the height of a rectangle with the same area as the graph of ##y## vs ##x## in that region. $$y_{avg}=\frac{1}{b-a}\int_a^b f(x)\;\text{d}x$$
 
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Oh, that's simple! Thank you!
 

Related to Average kinetic energy of a damped oscillator

What is the definition of average kinetic energy of a damped oscillator?

The average kinetic energy of a damped oscillator refers to the average amount of energy that a damped oscillator system possesses due to its motion. It is a measure of the amount of energy that is in the form of kinetic energy and is calculated by taking the average of the kinetic energy at different points in time.

How is the average kinetic energy of a damped oscillator calculated?

The average kinetic energy of a damped oscillator can be calculated by using the equation KEavg = (1/2)mω2A2e-2βt, where m is the mass of the oscillator, ω is the angular frequency, A is the amplitude, β is the damping coefficient, and t is the time.

What factors affect the average kinetic energy of a damped oscillator?

The average kinetic energy of a damped oscillator is affected by several factors, including the mass of the oscillator, the amplitude of the oscillations, the damping coefficient, and the frequency of the oscillations. Changes in any of these factors can result in a change in the average kinetic energy of the oscillator.

What is the relationship between the average kinetic energy and the amplitude of a damped oscillator?

The average kinetic energy of a damped oscillator is directly proportional to the square of the amplitude of the oscillations. This means that as the amplitude increases, the average kinetic energy also increases, and vice versa.

Can the average kinetic energy of a damped oscillator ever reach zero?

No, the average kinetic energy of a damped oscillator can never reach zero because there will always be some amount of energy present due to the motion of the oscillator. However, as time goes on, the average kinetic energy will decrease and approach zero as the system loses energy due to damping.

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