# Physics Course/Oscillation/Oscillation Up and Down

## The Oscillation up and down of a Spring

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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

Fs = - Fa
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

Fs = - Fa
$F = m \frac{d^2y}{dt^2} = - k y$
$F = \frac{d^2y}{dt^2} + \frac{k}{m} y = 0$
$s^2 + \frac{k}{m} s = 0$
s = ± j $\sqrt{\frac{k}{m}} t$
s = ± j $\omega t$
s = $e^ j\omega t + e^ -j\omega t$
$y = A Sin \omega t$