Answer to Question #63458 in Classical Mechanics for Harry

Question

2. A string of uniform tension T is initially at rest along the negative half of the x–axis. A train of travelling waves of frequency ! and having wavenumber of magnitude k is incident in the direction of increasing x. At x = 0, the string is attached to a ring of mass m free to move without friction vertically up and down a vertical runner. The mass density of the string itself plays no part in this problem, and there is no string to the right of the ring.
a. [5 points] Use a labelled diagram in the region of x = 0 to show that for small displacements of the string, y(x, t), the boundary condition at x = 0 is

m(d^2y/dt^2) = T (dy/dx)