The time period of a mass suspended by a spring (force constant K) is T. If the spring is cut into three equal pieces, what will be the force constant of each part? If the same mass be suspended from one piece what will be the periodic time?

Explanation

Consider the spring be made of a combination of three springs in series each of spring constant k. The effective spring constant K is given by
` 1/K = 1/k + 1/k + 1/k = 3/k`
or `K = k/3 " or " k = 3k`
`therefore ` Time period of vibration of a body attached to the end of this spring,
` T = 2k sqrt(m/k) = 2pi sqrt( (m)/(k/3) ) = 2pi sqrt( (3m)/(k) ` .....(1)
When the spring is cut into three pieces, the spring constant = k, time period of vibration of a body attached to the end of this spring,
`T_1 = 2pi sqrt(m/k)`
From eqns (1) and (2)
`(T_1)/(T)1 = (1)/(sqrt3) " or " T_1 = (T)/(sqrt3)`