Three identical masses of mass m are connected by four springs. The springs between rigid walls...

 

 

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Three identical masses of mass m are connected by four springs. The springs between rigid walls and masses have spring constant of k, and the other two, springs among masses, have K'as shown in Figure The masses vibrate along the line joining their centers without friction. Figure 1 (a) [2 points] Find the equation of motion for each masses. You may label them from the left, i.e. the leftmost mass is the first one. (b) [2 points] Find the normal frequencies of the system. (c) [3 points] Suppose that all four springs are identical, i.e. k . What are the normal frequencies? Describe the normal modes of vibration. (d) [3 points] Suppose that N identical masses are connected by N+1 identical springs of spring constant k between two rigid walls. Initially all masses are equally spaced and springs are in their natural lengths. Show that the normal frequencies of the system are given by - sin 2 (N+1) (3) Wn = 21 where n = 1,2, 3,, N. ... Three identical masses of mass m are connected by four springs. The springs between rigid walls and masses have spring constant of k, and the other two, springs among masses, have K'as shown in Figure The masses vibrate along the line joining their centers without friction. Figure 1 (a) [2 points] Find the equation of motion for each masses. You may label them from the left, i.e. the leftmost mass is the first one. (b) [2 points] Find the normal frequencies of the system. (c) [3 points] Suppose that all four springs are identical, i.e. k . What are the normal frequencies? Describe the normal modes of vibration. (d) [3 points] Suppose that N identical masses are connected by N+1 identical springs of spring constant k between two rigid walls. Initially all masses are equally spaced and springs are in their natural lengths. Show that the normal frequencies of the system are given by - sin 2 (N+1) (3) Wn = 21 where n = 1,2, 3,, N. ...

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