Differential Equations Mass Springs Question: Under some circumstances when two

Question

Differential Equations Mass Springs Question:

Under some circumstances when two parallel springs, with constants k1 and k2, support a single mass, the effective spring constant of the system is given by

A mass weighing 20 pounds stretches one spring 3 inches and another spring 9 inches. The springs are attached to a common rigid support and then to a metal plate. As shown in the figure, the mass is attached to the center of the plate in the double-spring arrangement.

A) Determine the effective spring constant of this system. K = ??? lb/ft

B) Find the equation of motion

if the mass is initially released from the equilibrium position with a downward velocity of 6 ft/s. (Use

for the acceleration due to gravity.)

x(t)= ???ft

[edit]

PLEASE Show all your work! I want to learn how to do this problem, not just know the answer.

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Explanation / Answer

mg = k1 * x1, x1 = 3in= 0.25 ft

k1 = mg/x1 = 4 * 20 * 32 = 2560 lb/ft

mg = k2 * x2, x2 = 9in= 0.75 ft

k2 = mg/x2 = (4/3) * 20 * 32 = 2560/3 lb/ft

Effective k = 4 k1 * k2/(k1 + k2) = 2560(4 * 1/3 / (1 + 1/3)) = 2560 lb/ft

frequency of oscillation (f)= sqrt(k/m) = sqrt(2560 / 20) = sqrt(128) = 8sqrt(2)

x(t) is a simple harmonic motion of form:
x(t) = Acos( ft)+Bsin(ft)

velocity v(t) = x'(t) = f(Bcos(ft)-Asin(ft))

v(0) = 6

=> 6 = f * B

=> B = 6/f

x(0) = 0

=> 0 = A

therefore x = Bcos(ft) where f = 8sqrt(2), B = 6/f

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