(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A 32 pound weight stretches 2 feet. Determine the amplitude and period of motion if the weight is released 1 foot above the equilibrium position with an initial velocity of 2 ft/s upward. How many complete vibrations will the weight have completed at the end of 4 pi seconds?

3. The attempt at a solution

Here is the solution I have come up with:

32 = 2k

k=16

x(0)=1 ft

x'(0) = -2 ft/s

m= 32/32 = 1 slug

[tex] \frac{dx^2}{d^2t} +16x = 0[/tex]

[tex] m^2+16[/tex]

solution to xc:

[tex] x(t)= A \cos{4x} + B\sin{4x}[/tex]

with initial conditions:

[tex] x(t) = \cos{4x} - \frac{1}{2} \sin{4x}[/tex]

therefore amplitude= [tex] \sqrt{1+\frac{1}{4}} = \frac{\sqrt{5}}{2}[/tex]

and [tex] \tan{\phi} = 3[/tex]

[tex] \phi = -1.1071 + \pi = 2.034 [/tex]

so my equation for the model is

[tex] x(t) = \frac{\sqrt{5}}{2} \sin{(4x+2.034)}[/tex]

and I know for a complete vibration to occur I have to have [tex] 2n\pi + \frac{\pi}{2}[/tex]

So [tex] 48.23 = \frac{(x-6)\pi}{2}[/tex]

x is approximatly 25 which is about 6 revolutions.

in 4 pi seconds, the system has undergone approximatly 6 revolutions and the differential equation describing the system is

[tex] x(t) = \frac{\sqrt{5}}{2} \sin{(4x+2.034)}[/tex]

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# Homework Help: Second order DE model

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