A uniform string of length l, fixed at both ends is vibrating in its 2nd overtone.The maximum amplitude is 'a' and tension in string is 'T', if the energy of vibration contained between two consecutive nodes is `K/8 (a^2pi^2T)/l` then 'K' is :

Explanation

Total energy `E=int_0^l1/2dmv^2`
`=int_0^l1/2dmA_x^2omega^2=int_0^l1/2(m/l)dx.A^2sin^2kx.omega^2 =1/4mA^2omega^2`
`omega=2pif=2pi.(3v)/(2l)sqrt(T/mu)=(3pi)/l sqrt((Tl)/m)`
`therefore` Energy `=1/4ma^2. (9pi)^2/l^2.(Tl)/m`
Energy =`9/4(a^2pi^2T)/l`
So, energy between two consecutive nodes=`3/4 (a^2pi^2T)/l`