Simple Harmonic Motion for a Particle Suspended on an Elastic String

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Suppose we suspend a particle of mass m from an elastic spring. The tension in the string isThe particle will be in equilibrium when the tension is equal to the weight.

We havewhere x-e labels the equilibrium extension.

If the particle is pulled down a further small distanceso that the resultant force on the particle will be upwards and is given by

We are taking downwards as positive (sinceis assumed positive), but the acceleration will be upwards and negative. Applying gives

ButDifferentiating this twice givesso

This is the equation of simple harmonic motionwith

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