Falling Chain A portion of a uniform chain of length 8ft is loosely coiled around a peg at the edge
Falling Chain A portion of a uniform chain of length 8ft is loosely coiled around a peg at the edge of a high horizontal platform, and the remaining portion of the chain hangs at rest over the edge of the platform. See the following. Suppose that the length of the overhanging chain is 3 ft, that the chain weighs 2 lb/ft, and that the positive direction is downward. Starting at t = 0 seconds, the weight of the overhanging portion causes the chain on the table to uncoil smoothly and to fall to the floor. If x(t) denotes the length of the chain overhanging the table at time t > 0, then v = dx/dt is its velocity. When all resistive forces are ignored, it can be shown that a mathematical model relating v to x is given by
xv dy/dx + v2 = 32x.
(a) Rewrite this model in differential form. Proceed as in Problems 3136 and solve the DE for v in terms of x by finding an appropriate integrating factor. Find an explicit solution v(x).
(b) Determine the velocity with which the chain leaves the platform.