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16.8 Forced oscillations and resonance

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  • Observe resonance of a paddle ball on a string.
  • Observe amplitude of a damped harmonic oscillator.
The figure shows the panel of the piano containing the strings, which are visibly in horizontal lines. Just below the strings is the wooden block of the piano containing the different type string handle bars and blocks.
You can cause the strings in a piano to vibrate simply by producing sound waves from your voice. (credit: Matt Billings, Flickr)

Sit in front of a piano sometime and sing a loud brief note at it with the dampers off its strings. It will sing the same note back at you—the strings, having the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them. Your voice and a piano’s strings is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency. In this section, we shall briefly explore applying a periodic driving force acting on a simple harmonic oscillator. The driving force puts energy into the system at a certain frequency, not necessarily the same as the natural frequency of the system. The natural frequency    is the frequency at which a system would oscillate if there were no driving and no damping force.

Most of us have played with toys involving an object supported on an elastic band, something like the paddle ball suspended from a finger in [link] . Imagine the finger in the figure is your finger. At first you hold your finger steady, and the ball bounces up and down with a small amount of damping. If you move your finger up and down slowly, the ball will follow along without bouncing much on its own. As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude. When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive it. The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance    . A system being driven at its natural frequency is said to resonate    . As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball.

The given figure shows three pictures of a horizontal viewed single finger containing a string, suspended downward vertically, being tied to a paddle ball at its downward end. In the first figure the ball is stretching up and down very slowly having less displacement, the displacement shown in the figures as faded shades of the ball and is depicted as 2X. Whereas in the second figure the movement of the ball is highest, while in the third the movement is least. In all the three figures the ball is at its equilibrium with respect to its movement. The frequency, f, for the first figure is very low, for the second figure as f not, while for the third figure it is highest.
The paddle ball on its rubber band moves in response to the finger supporting it. If the finger moves with the natural frequency f 0 size 12{f rSub { size 8{0} } } {} of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically. At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with lower-amplitude oscillations.

[link] shows a graph of the amplitude of a damped harmonic oscillator as a function of the frequency of the periodic force driving it. There are three curves on the graph, each representing a different amount of damping. All three curves peak at the point where the frequency of the driving force equals the natural frequency of the harmonic oscillator. The highest peak, or greatest response, is for the least amount of damping, because less energy is removed by the damping force.

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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