The tension in the rope will vary with radius r. The tension at a distance r from the center must supply the force to keep the mass of the rope that is further out than r accelerating inward. The mass of this piece in m , and its center of mass moves in a circle of radius , and so

T(r) = w 2 = (L 2 – r 2 ).

An equivalent method is to consider the net force on a piece of the rope with length dr and mass dm = dr m/L. The tension must vary in such a way that

T(r) – T(r + dr) = – ω 2 r dm, or = –(mω 2 /L)rdr. This is integrated to obtained

T(r) – (mω 2 /2L)r 2 + C, where C is a constant of integration. The tension must vanish at r = L, from which C = (mω 2 L/2) and the previous result is obtained.

The speed of propagation as a function of distance is

v(r) = = = = ,

where > 0 has been chosen for a wave traveling from the center to the edge. Separating variables and integrating, the time t is

t = = .

The integral is done explicitly by letting r = L sin θ, dr = L cos θ d θ, = L cos θ, so that

= θ = arc sin , an

t = arc sin (1) =