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

`f_(n) = ( 2n - 1)/( 2l) sqrt((Y)/( rho)) , where n = 1 , 2 , 3 ….`

Explanation

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, ..`