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Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor
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As the damping on an oscillator is increased there comes a

Chapter 5, Problem 5.27

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QUESTION:

As the damping on an oscillator is increased there comes a point when the name "oscillator" seems barely appropriate. (a) To illustrate this, prove that a critically damped oscillator can never pass through the origin x = 0 more than once. (b) Prove the same for an overdamped oscillator.

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QUESTION:

As the damping on an oscillator is increased there comes a point when the name "oscillator" seems barely appropriate. (a) To illustrate this, prove that a critically damped oscillator can never pass through the origin x = 0 more than once. (b) Prove the same for an overdamped oscillator.

ANSWER:

Step 1 of 3

Part (a)

For the critical damping, the decay constant would be equal to the natural frequency, 

The general solution for the critical damping

Here,is the distance, t is time and  is the decay constant.

If  is zero for the different value of the time then damped oscillator pass through the origin more then one time but if  is zero for the one value then damped oscillator can never pass through the origin more than one time.

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