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## Homework Statement

A spring with a spring constant k of 100 pounds per foot is loaded with 1-pound weight and brought to equilibrium. It is then stretched an additional 1 inch and released. Find the equation of motion, the amplitude, and the period. Neglect friction.

Find the displacement function x(t)

## Homework Equations

Fg = mg

Fs = -kx

Fnet = ma

## The Attempt at a Solution

Fnet = Fg + Fs

ma = mg + (-kx)

mx'' = mg + (-kx)

mx'' + kx = mg

Converting Units:

m = 1 kg = 2.2 lbs

g = -9.8 m/s^2 = -32.1 ft/s^2

k = 100 lbs/ft

Solving Second Order Non-Homogeneous DE:

mx'' + kx = mg

2.2x'' + 100x = -70.6

Particular Solution:

Xp = A

X'p = 0

X''p = 0

2.2*0 + 100*A = -70.6

A = -7.06

2.2x'' + 100x = 0

Roots = 0 ± 6.742i

x = -7.06 + C1*cos(6.742t) + C2*sin(6.742t)

Using Conditions:

x = -1 inch or -0.083 feet when t = 0

x' = 0 ft/s when t = 0

-0.083 = -7.06 + C1*cos(0) + C2*sin(0)

-0.083 = -7.06 + C1

C1 = 6.976

x' = -6.742*C1*sin(6.742t) + 6.742*C2*cos(6.742t)

0 = -6.742*C1*sin(0) + 6.742*C2*cos(0)

0 = 6.742*C2

C2 = 0

Final Equation:

x = -7.06 + 6.976*cos(6.742t)

Is this right?