Rotation and Revolution
Rotation is when the axis of rotation is within the object (rotation of the Earth about its axis in 24 hours). And, revolution is when the object revolves around an external axis at a distance (the Earth revolving around the Sun, moons around planets, etc.)
Rotational speed - Bahngeschwindigkeit
Motion in a circle is periodic. Once one revolution has been completed, the motion continues on and a new revolution begins.
The period it takes to complete a revolution is denoted by the symbol T.
If the speed around a circle with radius r does not change, the speed (Bahngeschwindgkeit) along the path is calculated with the circumference and the period T (Dauer T)
$${v = \frac{\Delta s}{\Delta t} = \frac{2 \pi r}{T}}$$
The factor $${ \frac{2 \pi}{T}}$$ is independent of the radius.
angular velocity - Winkelgeschwindigkeit
On a rotating disc (vinyl), objects at different radii to the centre of the circular motion travel across the same angle Δφ in the same time Δt.
$${\omega = \frac{\Delta \phi}{\Delta t} = \frac{2 \pi }{T}}$$
$${\omega = 2 \pi f}$$
$${\omega = \frac{v}{r}}$$
is called the angular velocity ω (Winkelgeschwindigkeit). If the angle Δφ is measured in radians (Bogenmaß), the angular velocity ω has units s-1.
curved motion and force
Because of inertia, bodies only change direction if a force is acting upon them. (z.B. beim Schleifstein lösen sich glühende Teile und fliegen geradlinig weg)
As $${\vec{F} = m \cdot \vec{a}}$$
the force and the acceleration have the same direction.
If a force acts upon a moving object, there are 3 cases.
- $${\vec{F} \parallel \vec{v}}$$
- $${\vec{F} \perp \vec{v}}$$
- $${\vec{F} ~\text{and}~ \vec{v}~\text{form an angle between 0 and 90 degrees.}}$$ $${\vec{F}~ \text{is divided into components}~\vec{F_{\parallel}}~ \text{and}~ \vec{F_{\perp}}}$$
The third case implies that the velocity in direction of the moving object would also increase in magnitude (this is not the case in circular motion).
When an object moves in a circular path at constant speed, it is the second case and there is always a force acting towards the centre of the circle called the centripetal force (Zentralkraft FZ) which points to the centre of the circular motion.
circular motion is accelerated motion
As the direction of the objects velocity is changing at every moment as it moves along a circular path, there must be an acceleration.
It is called centripetal acceleration
We know that
$${F_Z \sim m}$$
Measurements have shown that
$${F_Z \sim r}$$
and
$${F_Z \sim \omega^2}$$
Overall measurements have shown that
$${F_Z = m \omega^2 r = \frac{m v^2}{r}}$$
The centripetal force is perpendicular the velocity and directed towards the centre of the circular motion. The force results in an acceleration aZ towards the centre.
$${a_Z = \omega^2 r = \frac{v^2}{r}}$$
different perspectives - verschiedene Sichtweisen
- Trägheitskräfte
- Zentrifugalkräfte
coriolis force - Corioliskraft
A ball rolls with constant velocity vK from the centre of a disc (radius r), rotating at an angular velocity ω , towards an observer A standing off the disc, and reaches the edge after a time t.
To observer A it appears that the ball travelled along the radius and the displacement is r = vKt. On the other hand, to an observer B on the disc the ball still reaches the observer A but on a curved path. The path has gone off track by b = ωrt = ωvKt2. To observer B the ball has accelerated.
If we assume a constant acceleration aC, we have $${b= \frac{1}{2}a_C \cdot t^2}$$ Because $${b= \omega \cdot v_K \cdot t^2}$$ it follows that $${a_C = 2 \cdot \omega \cdot v_K}$$
$${F_C = m \cdot a_C = m \cdot 2 \cdot \omega \cdot v_K}$$
This force observed in a rotating frame of reference is called Coriolis force after Gaspard Coriolis (1792-1843). The accerleration aC is referred to as the Coriolis acceleration.
TODO - image of the coriolis force