Two equal masses on an effectively frictionless horizontal air track are held be

Question

Two equal masses on an effectively frictionless horizontal air track are held between rigid supports by three identical springs. The displacements from equilibrium along the line of springs are described by coordinates xA and xB. If either of the masses is clamped, he period T(=2Pi/omega) for one complete vibrations of the other is 3 seconds.

(a) If both masses are free, what are the periods of the two normal modes of the system? At t = 0, mass A is at its normal resting position and mass B is pulled aside a distance of 5 cm. The masses are released from rest at this instant.

(b) Write an equation for the subsequent displacement of each mass as a function of time.

(c) What length of time (in seconds) characterizes the periodic transfer of the motion from B to A and back again? After one cycle, is the situation at t=0 exactly reploduced? Explain.

Explanation / Answer

a. the effective spring constant is 2k so the frequency is sqrt(2k/m)

b.Imagine when the two masses move side to side together as if they were connected by a rigid pole. The frequency is sqrt(2k/2m) = sqrt(k/m)

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