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Physics Lesson 10.2.1 - Damped and Sustainable Simple Harmonic Oscillations
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Welcome to our Physics lesson on Damped and Sustainable Simple Harmonic Oscillations, this is the first lesson of our suite of physics lessons covering the topic of Pendulums. Energy in Simple Harmonic Motion, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Damped and Sustainable Simple Harmonic Oscillations
If we make a system oscillate, the amplitude of oscillations would decrease until it becomes zero if energy is not provided continuously to the system. In this case, the oscillations fade away with time. They are known as damped oscillations whose amplitude decreases with time. Look at the graph below.
The "envelope" that surrounds the damped simple harmonic motion graph shows a decreasing exponential function of type x(t) = A0 × e - γ × t where A0 is the initial amplitude and γ is a constant. This is still a simple harmonic motion because the other quantities such as period and frequency don't change with time. This kind of SHM has the equation
As you see, a cosine function is used here instead of the sine function we have used in the previous tutorial because we are interested to point out the initial amplitude A0. Given that cos 0 = 1, it is better to use a cosine function here as it takes the maximum value at the initial instant (t = 0).
On the other hand, in sustainable SHM, the amplitude does not change with time. The envelope shows a horizontal function of the type x(t) = A0.
Therefore, the constant γ is zero as
As a result, we have
and therefore, the equation of SHM becomes
This is the same equation as the one obtained when studying sustainable SHM in the previous tutorial. Cosine function is used only for convention but it can be replaced by sine function as well by making the proper arrangements.
Thus, we can say that sustainable SHM are a special case of damped SHM.
You have reached the end of Physics lesson 10.2.1 Damped and Sustainable Simple Harmonic Oscillations. There are 5 lessons in this physics tutorial covering Pendulums. Energy in Simple Harmonic Motion, you can access all the lessons from this tutorial below.
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