Simple Harmonic Motion for a Horizontally Stretched String

The tension in a stretched string is usually expressed in terms of it's modulus of elasticityThis is in fact an expression of Hooke's Law – Force is proportional to extension.

Suppose that a string of length l is stretched between two points A and B. A particle P is attached to the string at C between the points A and B.

Suppose the lengths AC and CB are in the ratiothen the natural length of AC is and the extension of AC isSimilarly the natural length of CB isand the extension of CB is

Suppose the particle P is pulled an extra distance x to the right.

The extension of AC is nowso


It will experience a net force to the left, towards the equilibrium position C of magnitude

Now applywithwhich has the form of simple harmonic motionwith