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 rac{1}{8} feet. The ball is started in motion from the equilibrium position with a downward velocity of 2 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

mu'' + gamma*u' + ku = 0

where m is weight/acceleration due to gravity (4/32) or 1/8 gamma is the damping coefficient (it's given) and k = spring constant or weight/displacement = 4/(1/8) =32

(1/8)y'' + 4y' + 32u = 0

y = Ae^(-bt/2m) sin(' t + ),
where A and are constants a

nd ' = [k/m - b/(2m)^2] with k/m > (b/2m)^2, i.e., k > b^2/4m

--->• (0) *exp((-16)*t) + (6)*t *exp((-16)*t)

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