Question

Imagine a piano wire of mass , under tension F, between fixed points a distance L apart. The wire at rest lies along the x axis. To simplify matters we assume that the displacement of the wire from equilibrium occurs only along the y direction. Thus y(0) = y(L) since there is no displacement at the ends of the wire. The energy of the wire for an instantaneous displacement y(3,t) is E 16+ 7) = 1 / (40+50) We write y(x, t) as the Fourier series y(pit) = { An(t) sin (92"). n=1 Show that the energy can be written as Ely(r, t)) = ༡1=1 Treat An and Àn as a set of 2.V generalized coordinates which define the instantaneous position of the wire.

(a) Calculate the average thermal energy of the wire under tension, and hence obtain its heat capacity.
(b) Determine the average value of An, of 4, and of An An' with n #n'.
(c) Express the mean square y(x, t)2 in the form of a suin of some quantity over n. From this determine the mean square displacement of the wire at I = L/2. (You may use without proof the relation, n-2 = 72/8.

Answer 1

The problem involves finding various properties related to the energy and displacement of a wire under tension using wave mechanics and Fourier series.

Step-by-step explanation:

In this problem, we are given a wire under tension and asked to calculate various properties related to its energy and displacement.

The problem involves concepts from wave mechanics and Fourier series.

We can use the given information and equations to find the average thermal energy, average values of An and Àn, mean square displacement, and so on.