Ring Threaded on Elasic String Suspended From the Ceiling

When a ring is threaded on an elastic string which is attached to the ceiling at two points, then ignoring friction, the particle will come to equilibrium at the midpoint of the string and the tension will be the same throughout the string. Suppose the modulus of elasticity of the string isand the natural length of the string is

The length of the string with the ring attached is

The tension in the string is then

Resolving vertically gives

In general we can measurethen findby writing

oris given, thencan be found using a numerical method.