Falling Chain A portion of a uniform chain of length 8 ft 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 Figure 2.4.2. 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 x v d v d x + v 2 = 32 x . (a) Rewrite this model in differential form. Proceed as in Problems 31–36 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. FIGURE 2.4.2 Uncoiling chain in Problem 45
Falling Chain A portion of a uniform chain of length 8 ft 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 Figure 2.4.2. 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 x v d v d x + v 2 = 32 x . (a) Rewrite this model in differential form. Proceed as in Problems 31–36 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. FIGURE 2.4.2 Uncoiling chain in Problem 45
Solution Summary: The author explains how to simplify the given mathematical model by comparing it with the standard form of exact differential equation Mdx+Ndv.
Falling Chain A portion of a uniform chain of length 8 ft 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 Figure 2.4.2. 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
x
v
d
v
d
x
+
v
2
=
32
x
.
(a) Rewrite this model in differential form. Proceed as in Problems 31–36 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.
FIGURE 2.4.2 Uncoiling chain in Problem 45
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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