# 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 = ${\displaystyle {\frac {ma}{k}}}$
a = - ${\displaystyle {\frac {k}{m}}y}$
${\displaystyle t={\frac {k}{m}}{\frac {y}{v}}}$

### 2

Any force acting on an object can be expressed in a differential equation

${\displaystyle F=m{\frac {d^{2}y}{dt^{2}}}}$

Equilibrium is reached when

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