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Applications of Differential Equations : Mechanics

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A perfectly flexible cable hangs over a frictionless peg as shown, with 8 feet of cable on one side of the peg and 12 feet on the other. The goal of this problem is to determine how long it takes the cable to slide off the peg, starting from rest.

(a) At time t 0 what proportion of the whole cable is on the left side of the peg and what proportion is on the right side?

(b) Let y = y(t) be the amount of cable (in feet) that has slid from the left side to the right side after t seconds (as shown in the picture below). Give expressions for the proportion of the cable that is on the left side and the proportion that is on the right side at time t > 0

(c) Using Newton's law F = ma, where m is mass and a is acceleration, find a differential equation for the function y(t) (a "frictionless peg" means we are ignoring any drag force). Explain your reasoning.

(d) How long does it take for the cable to slide completely off the peg?

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