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viewnote5 - PHYS 302 Unit 5 Viewing Notes Athabasca...

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PHYS 302 : Unit 5 Viewing Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 5 Viewing Notes So far, we have looked at single systems with some outside source of forcing. Most realistic physical systems have more than one part, so interaction between the parts can cause vibrations. This is known as coupling . For simplicity, we will leave damping out for now. The first example, with pendulums connected by a spring, is a good approximation of coupling without damping, since the damping is small. In the demo (2:30), oscillation seems to “move” between the two pendulums. In normal modes each object is at the same frequency, and in the case of no damping, the coupled objects can move in only two ways: either in phase or in antiphase (180º out of phase). For two objects, we expect two normal modes; their frequencies are denoted ω - and ω + . In general, motion will occur in a combination of normal modes, so for the two objects (denoted by subscripts 1 and 2) the positions are 1 0 0 cos( ) cos( ) x x t x t and 2 0 0 cos( ) cos( ) x x t x t . The constants x 0- and x 0+ show the amplitude of each normal mode and can be zero. Dr. Lewin excites first the – mode, with identical motion of the pendulums, followed by the + mode, with opposed motion (10:20). Since the spring does not do anything in the – mode, the frequency is simply that of a single pendulum 0 where 2 2 0 g l . For the opposed mode, we need a force analysis. The spring is stretched by 2 x, T=mg , and F s =-2kx .
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