# Second order DE model

1. Aug 8, 2008

### bishy

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

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

$$m^2+16$$

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

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

2. Aug 8, 2008

### HallsofIvy

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