# The circle is horizontal such that gravity can be

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ignore any effects related to the curvature. The circle is horizontal such that gravity can be ignored. a. Find equations of motion for the three masses in terms of the small displacements from the equilibrium position of each mass. b. Determine the frequencies and the relative amplitudes for each of the normal modes. Make a simple sketch of the motion of the masses for each of the normal modes. How many different frequencies are there in this system? c. Because the masses are connected in a circle some of the results of normal mode calculations do not correspond to oscillatory motion. Explain why. 2

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k k m m h(t) y (t) y (t) 1 2 Figure 3: Two Masses Hanging from an Oscillating Support Problem 3.4 (20 pts) Consider two identical masses m connected together with a spring and attached with another spring to a moving support (see Figure 3). The support is oscillating vertically and its position is given by h ( t ) = A cos( ωt ). The Hooke constant of the two identical springs is k . Ignore effects of damping. a. Find coupled differential equations that govern displacements from equilibrium of the masses y 1 ( t ) and y 2 ( t ). Express your results in terms of ω 2 0 = k m . Note that the effect of gravity results in a shift of equilibrium position but it does not affect the harmonic motion.
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• Spring '14
• Massachusetts Institute of Technology, normal modes

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