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

F_s = - k y

The oscillation stops when

F_s = - F_a

m a = - k y

y = ${\frac {ma}{k}}$

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

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

Equilibrium is reached when

F_s = - F_a

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

$F={\frac {d^{2}y}{dt^{2}}}+{\frac {k}{m}}y=0$

$s^{2}+{\frac {k}{m}}s=0$

s = ± j ${\sqrt {\frac {k}{m}}}t$

s = ± j $\omega t$

s = $e^{j}\omega t+e^{-}j\omega t$

$y=ASin\omega t$