Physics Course/Oscillation/Oscillation Side by Side

Oscillation Side by Side
When apply a force on an object of mass attach to a spring. The spring will move a distance y above and below the equilibrium point and this movement keeps on repeating itself for a period of time. The movement up and down of spring for a period of time is called Oscillation

1
The force acts on the object to pull the object down
 * F = m a

The Restoring Force of spring to push the object up can be calculated by Hook's Law
 * Fs = - k y

The oscillation stops when the two forces are equal or the net force on object is zero
 * m a = - k y
 * y = $$\frac{m a}{k}$$
 * a = - $$\frac{k}{m} y$$
 * $$t = \frac{k}{m} \frac{y}{v}$$

2
Any force acting on an object can be expressed in a differential equation
 * $$F = m \frac{d^2y}{dt^2}$$

Equilibrium is reached when F = Fs
 * $$F = m \frac{d^2y}{dt^2} = - k y$$
 * $$F = \frac{d^2y}{dt^2} + \frac{k}{m} y = 0$$
 * $$s^2 + \frac{k}{m} s = 0$$
 * s = ± j $$\sqrt{\frac{k}{m}} $$
 * s = $$e^ j\sqrt{\frac{k}{m}}t +  e^ -j\sqrt{\frac{k}{m}}t$$
 * $$y = A Sin {\frac{k}{m}}t$$