A point mass m is suspended at the end of a massless wire of length l and cross section. If Y is the Young's modulus for the wire, obtain the frequency of oscillation for the simple harmonic motion along the vertical line.

Explanation

Let F be the force applied on the end of wire and x be the extension in the length of wire, figure,
longitudinal strain `=(x)/(L)`
Normal stress `=(F)/(A)`
Young's modulus `Y=(F//A)/(x//L) or F=YA(x)/(L)`
The restoring force due to elasticity is
`F=-(YA)/(L)x` ...(i)
Since `F prop x` and this force F will tend to bring the body si left free, it will execute linear simple harmonic motion. Comparing (i) with the equation `F=-ky`
We have , spring factor , `k=(YA)/(L)`
and inertia factor `=` mass of body `=` m .
As, frequency, `v=(1)/(2pi)sqrt((spri ng fact o r)/(i n ertia fact o r))`
so `v=(1)/(2pi)sqrt((YA)/(Lm)`