How Do You Calculate the Amplitude and Period of a Weight in Harmonic Motion?

[bishy]

Homework Statement

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?

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

\frac{dx^2}{d^2t} +16x = 0

m^2+16

solution to xc:

x(t)= A \cos{4x} + B\sin{4x}

with initial conditions:

x(t) = \cos{4x} - \frac{1}{2} \sin{4x}

therefore amplitude= \sqrt{1+\frac{1}{4}} = \frac{\sqrt{5}}{2}

and \tan{\phi} = 3

\phi = -1.1071 + \pi = 2.034

so my equation for the model is

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

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

So 48.23 = \frac{(x-6)\pi}{2}
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

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

 

[HallsofIvy]

In all of these the "x" variable should be "t". Other than that, I see no error or even a question!