A string tied between `x = 0 and x = l` vibrates in fundamental mode. The amplitude `A`, tension `T` and mass per unit length `mu` is given. Find the total energy of the string.

Answer

A, B, D

Explanation

`l = lambda/2`
or `lambda = 2l, k = (2pi)/lambda = pi/l`

The amplitude at a distance x from x = 0 is given by
`A = a sin kx`
Total mechanical energy at x of length dx is
`dE = 1/2 (dm)A^2omega^2`
`= 1/2 (mudx)(a sin kx)^2 (2pif)^2`
or `dE = 2pi^2 muf^2 a^2 sin^2 kx dx` .........(i)
Here, `f = (v_2/lambda_2) = ((T/mu))/((4l^2)) and k = pi/l`
Substituting these values in Eq. (i) and integrating it from x = 0 to x = l , we get total energy of string.
`E = (pi^(2)a^(2)T)/(4l)` .