A body of mass m is suspended from a weightless wire of length l. If Y is the Young's modulus of the material of the wire, calculate the frequnecy of vibration in the vertical direction.

A body of mass m is suspended from a weightless wire of length l. If Y

1.	A body of mass m is suspended from a weightless wire of length l. If Y is the Young's modulus of the material of the wire, calculate the frequnecy of vibration in the vertical direction.
Answer» Solution :If the mass m causes an elongation x of the WIRE, then LONGITUDINAL stress = Young.s modulus `xx` longitudinal strain = `Y*x/l`. By definition, longitudinal stress = REACTION force acting per UNIT area `therefore` Reaction force = longitudinal stress `xxalpha" "[alpha="cross-sectional area of the wire"]` = `(Yx)/lxxalpha=(Yxalpha)/l`. This reaction force acts as the restoring force. So, ACCELERATION of mass m, `a=("restoring force")/m=(Yxalpha)/(ml)` `therefore` Frequency, `n=1/(2pi)sqrt(("acceleration")/("displacement"))=1/(2pi)sqrt(alpha/x)=1/(2pi)sqrt((Yalpha)/(ml))`.