1. Find the equation of motion and graph of the system that has an associated DE

Question

1. Find the equation of motion and graph of the system that has an associated DE:

x''+8x'+16x=0     x(0)=0     x'(0)=1

2.Redo exercise 1 when the weight is released 1/2 ft above equilibrium with downward initial velocity of 5ft/sec.

3. A 128 lb weight stretches a spring 6.4 ft. The medium through which the weight moves offers a resistance numerically equal t0 2 times the instantaneous velocity. The weight is released 2 feet above equilibrium with a downward velocity of 5 ft/sec.

       a) Set up the system

        b) Find the equation of motion

        c) Graph it

        d) What is the maximum displacement of the mass on the spring?

          e) What is the velocity of the mass in 1 second, 5 seconds?

4. Using the system in exercise 3, find the equation of motion and graph when there is an external force, f(t)=5cos(4t), acting on the system. Discuss the change in the graph and how the external force affected the system.

Explanation / Answer

1)the solution of equation of motion is

x(t) = e^-4t * t
i.e Y = (e^-4 x) * x

now,
if we have x(0) = -1/2 ft
and velocity = x'(0) = 5 ft/sec

consider downword direction as +ve.

we have solution as :

x(t) = 0.5 * e^(-4t) * (6t-1)
i.e.
Y = 0.5 * (e^(-4x))* (6x-1)

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