Question

1. The total kinetic energy of the vibrating string is given by the integral (a) Why does this integral give kinetic energy? (b) The potential energy V(t) of a stretched string originates in the work done against the tension T when as a result of its displacement, the string becomes longer. Show that (c) Use the series expression (3.16b if Nettel) to show that the total energy can be written as a sum over normal mode terms, with no cross-terms linking different modes

Answer 1

The kinetic energy k = 1/2 (m/l middot dx) u^2 y = 1/2 (rho dx) (partial differential y/ partial differential t)^2 The K.E in the fixed string for the n^th mode: K.E(t) = integral^L_0 1/2 rho (partial differential y/ partial differential t)^2 dx K.E(t) = 1/2 integral^L_0 rho (partial differential y/ partial differential t)^2 dx The potential energy is the work done by deformation. P.E = V(t) = T (ds - dx) ds^2 = dx^2 + dy^2 = dx^2 (1 + (partial differential y/ partial differential x)^2) acc to Binomial series expansion, ds almostequalto dx (1 + 1/2 partial differential y/ partial differential x)^2 + ...) Therefore v(t) = 1/2 T (partial differential y/ partial differential x)^2 dx The P.E in the fixed string: V(t) = integral^L_0 1/2 T (partial differential y/ partial differential x)^2 dx V(t) = T/2 integral^L_) [partial differential y (x, y)/ partial differential x]^2 dx \r\n Total energy of normal