Motion of a Particle Attached to Two Springs on a Smooth Surface

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A particle between two springs on a table can execute simple harmonic motion if pulled to one side.

A particle of mass m is attached to two strings of natural lengthsandand force constants andrespectively. The ends of the springs are attached to endpoints A and B, so that and both springs are initially in tension. Suppose that the mass is pulled to the right from its equilibrium position so that the length of spring 1 isand the length of spring 2 is

The tension in spring 1 isand the tension in spring 2 is

The resultant force on the mass isUsinggives

(1)

When the system is in equilibrium,so that

LetthenandSubstitute this into (1) to give

Henceand the motion is simple harmonic with period

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