F11Physics1CLec14B

F11Physics1CLec14B - Physics 1C Lecture 14B "Shake...

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Physics 1C Lecture 14B "Shake this one, that one shakes later. The sun atom shakes; my eye electron shakes eight minutes later because of a direct interaction across." --Richard Feynman
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Outline Last time: Superposition – interference and standing waves ( f same) (action figure) Today cover: Beats - interference if f differs EM waves Optical spectrum
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Beats What happens if you try to add waves that have slightly different frequencies? Sometimes their crests will line up and sometimes one crest will overlap with the other trough. The result will be constructive interference sometimes and destructive interference others.
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Beats This phenomenon is known as beats . The time-dependent interference causes alternations in loudness of the resulting sound wave (action figure). How often you hear the “loudness” is called the beat frequency (frequencies must be close). The beat frequency will merely be the subtraction of the two frequencies that are interfering with one another: f beat = | f 2 f 1 |
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Beats: Math Model Consider two waves with equal amplitudes and slightly different frequencies f 1 and f 2 . Represent the displacement of an element associated with each wave at fixed position, x = 0: y 1 = A cos(2 p f 1 t ) and y 2 = A cos(2 p f 2 t ) Resultant position (use superposition principle): y = y 1 + y 2 = A [cos(2 p f 1 t ) + cos(2 p f 2 t ) ] Use identity: cos a + cos b = cos[( a-b )/2] cos[( a+b )/2] To get: y = 2 A cos {2 p [( f 1 - f 2 )/2] t } cos {2 p [( f 1 + f 2 )/2] t }
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Example Two strings with linear densities of 5.0g/m are stretched over pulleys, adjusted to have vibrating lengths of 50cm, and attached to hanging blocks. The block attached to string 1 has a mass of 20kg and the block attached to string 2 has mass M . Listeners hear a beat frequency of 2.0Hz when string 1 is excited at its fundamental frequency and string 2 at its third harmonic. What is one possible value for M ? Answer
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This note was uploaded on 12/11/2011 for the course PHYS 1C taught by Professor Smith during the Fall '07 term at UCSD.

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F11Physics1CLec14B - Physics 1C Lecture 14B "Shake...

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