Consider six equal masses constrained to move on a circle of fixed radius and connected by identical springs of spring constant k.
(a) Find the normal mode frequencies of the system for small displacement of the masses.
(b) Find the (time dependent) displacement of the masses from each normal mode. Give a physical description of the motion of the masses for normal modes with the highest and lowest frequencies.
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A set of coupled masses is constrained to move on a circular path. The chain consists of four light masses m alternating with four heavy masses M, joined by identical springs with force constant k. The equilibrium spacing (measured along the circumference of the circle), between two adjacent masses is a/2.
(a) Calculate the (coupled) equations of motion for the nth light mass m
and the nth heavy mass M. Consider forces due to adjacent masses only.
(b) Solve these equations of motion for the oscillatory normal modes of the system.
(c) Briefly indicate the motion associated with each of the allowed frequencies.
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