A rod of nickel of length l is clamped at its midpoint . The rod is stuck and vibrations are set up in the rod . Find the general expression for the frequency of the longitudinal vibrations of the rod . Young's modulus and density of the rod is Y and rho , respectively.

A rod of nickel of length l is clamped at its midpoint . The rod is st

1.	A rod of nickel of length l is clamped at its midpoint . The rod is stuck and vibrations are set up in the rod . Find the general expression for the frequency of the longitudinal vibrations of the rod . Young's modulus and density of the rod is Y and rho , respectively.
Answer» Solution :Case I ` l = 2 lambda_(1)//4 and v = f lambda_(1)` ` V = sqrt ((Y)/( rho)) , l = (lambda_(1))/(rho) , l = (lambda_(1))/(2)` ` f_(1) = (v)/( lambda_(1)) = (1)/( 2 l) sqrt((Y)/(rho))` Case II ` l = 2 [ (lambda_(2))/(4) + ( lambda_(2))/(2)]` ` l = ( 3lambda_(2))/( 2)` Again ` f_(2) = (v)/( lambda_(2)) = (3)/( 2 l) sqrt((Y)/(rho))` Cae III `l = 2 ( 5lambda_(3))/( 4) , lambda_(3) = ( 2l )/(5)` `f_(3) = (v)/( lambda_(3)) = (5)/( 2 l) sqrt((Y)/(rho))` Comparing EXPRESSION `f_(1) , f_(2) , f_(3) `, we have `f_(N) = ( 2N - 1)/( 2l) sqrt((Y)/( rho)) ` , where `n = 1 , 2, 3, ..`