One end of a long metallic wire of length L tied to the ceiling. The other end is tied with a massless spring of spring constant K . Amass hangs freely from the free end of the spring. The area of cross section and the young's modulus of the wire are A and Y respectively. If the mass slightly pulled down and released, it will oscillate with a time period T equal to:

Step-by-step solution

Calculation of spring constant of wire Young's modulus of wire, ( Y = F / A L / L = F L L A ) Due to elasticity of the wire, it behaves as equivalent spring ( ( K _ spring = F / x ) ). Hence, Spring constant for Wire, ( K _ w = F L = Y A L ) Calculation of Equivalent spring constant Let a force ( F ) be applied to the end of the spring. The wire will extend. Total extension ( S = x _ wire + x _ spring = F _ w K _ w + F _ S K ) But ( F _ S = F _ w ) (By Newton's, third Law, Both will apply equal and opposite forces