Harmonic motion

$$\large \vec{r}=A\cos(\omega t)\,\hat{n}.$$ $$\large \vec{v}=-\omega A\sin(\omega t)\,\hat{n}.$$ $$\large \vec{a}=-\omega^2 A\cos(\omega t)\,\hat{n} = -\omega^2 \vec{r}.$$ $$\large \vec{F}=-m\omega^2 A\cos(\omega t)\,\hat{n}= -m\omega^2 \vec{r}.$$

More details:

Harmonic motion is any oscillation that is proportional to either $\sin(\omega t)$ or $\cos(\omega t)$. The motion of the blue ball is described by the position vector,

$$\vec{r}=A\cos(\omega t)\,\hat{n},$$

where $A$ is the amplitude of the motion, $\omega$ is the angular frequency, and $\hat{n}$ is a unit vector pointing in the direction of the oscillation. The time it takes for one period is $T=\frac{2\pi}{\omega}$. The velocity is the derivative of the of the position vector,

$$ \vec{v}=-\omega A\sin(\omega t)\,\hat{n}.$$

The acceleration is the derivative of the velocity,

$$ \vec{a}=-\omega^2 A\cos(\omega t)\,\hat{n} = -\omega^2 \vec{r}.$$

The force is the mass times the acceration, $\vec{F}=m\vec{a}$,

$$\vec{F}=-m\omega^2 A\cos(\omega t)\,\hat{n}= -m\omega^2 \vec{r}.$$

The force is proportional to the position vector, $\vec{F}=-m\omega^2\vec{r}$. This has the form of a linear spring force, $\vec{F}=-k\vec{r}$ where we make the identification $k=m\omega^2$. An object of mass $m$ attached to a linear spring with spring constant $k$ executes harmonic motion with an angular frequency $\omega =\sqrt{\frac{k}{m}}$.

Harmonic oscillations can be modelled numerically with the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/numerical_integration/harmonic.en.php">Mass-spring system</a> app.






	Physik M 513.805 (511.015)