This problem is an example of critically damped harmonic motion. A hollow steel

Question

This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds 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 6 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.)

Explanation / Answer

Mass is weight/(gravitational acceleration), so

m = 4/32 = 1/8 slugs.

The spring constant, assuming Hooke's Law, is determined by the given displacement

F = 4 lb = k(1/8) ft ==> k = 32 lb/ft.

The damping constant is given as 4 (slug/sec). The IVP you have to solve, with the "down is positive" orientation is

1/8y'' + 4y' + 32y = 0
y'' + 32 y' + 256y = 0 , y(0) = 0, y'(0) = 6

Solving this should be straight forward.

y'' + 32*6 = 0
y '' = - 192

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