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# Damped harmonic motion

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This problem is an example of critically damped harmonic motion.

A hollow steel ball weighing 4 pounds (mass = 1/8 slugs) is suspended from a spring. This stretches the spring 1/8 feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second) . Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t.

Take as the gravitational acceleration 32 feet per second per second. (Note that the positive y direction is down in this problem.)

When using English units (lb, ft, etc.) you need to be a bit careful with equations involving mass. Pounds (lb) is a unit of force, not mass. Using mg=F and g=32 ft/sec^2 we see that if an object weights L pounds it will have a mass of L/32 lb*sec^2/ft (or slugs).