First slide

Elastic Modulie

Question

One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to a mass less spring of spring constant K. A mass m 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. Find the time period with which mass m will oscillate if it is slightly pulled down and released

Difficult

A

$2 π \sqrt{\frac{A Y K}{m (A Y + K L)}}$
B

$2 π \sqrt{\frac{m (A Y + K L)}{A Y K}}$
C

$2 π \sqrt{\frac{m Y K}{(A Y + K L)}}$
D

$2 π \sqrt{\frac{m A Y L}{(A Y + K L)}}$

Solution

$e l o n g a t i o n i n t h e w i r e e = \frac{F L}{A Y} F = \frac{A Y}{L} e e q u i v a l e n t s p r n g c o n s \tan t k_{1} = \frac{A Y}{L} a l o n g w i t h t h i s a s p r i n g o f f o r c e c o n s \tan t K i s c o n n e c t e d i n s e r i e s t h e n \frac{1}{K_{e}} = \frac{1}{k_{1}} + \frac{1}{K} = \frac{L}{A Y} + \frac{1}{K} t i m e p e r i o d T = 2 π \sqrt{\frac{1}{K_{e}}}$

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