# Uniform Circular Motion

## Introduction

A uniform circular motion occurs when a body moves with constant speed on a circular path.

## Experiment

A ball is attached with a rope ($$\ell = r = \rm 5 \,\, m$$) to a pillar and pushed so that it moves in a circle around it. Neglecting air friction and gravity, the ball moves at a constant speed on a circular path around the pillar.

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Legende
Geschwindigkeit
Beschleunigung
Winkel
Angle-time curve

The angle-time curve is a straight line passing through the origin. This shows that angle and time are proportional to each other.

The proportionality factor is a new physical quantity angular velocity $$\omega$$ of the body (see below).

$$\phi(t) = \omega \cdot t$$
Distance-time curve

The distance-time curve is a straight line passing through the origin. This shows that the distance traveled and the time are proportional to each other.

The proportionality constant is the velocity $$v$$.

$$s(t) = v \cdot t = \omega \cdot r \cdot t$$
Angular velocity - time curve

The angular velocity $$\omega$$ of the body is constant. It indicates how quickly the angle changes with time.

$$\omega = \dfrac{\Delta \phi}{\Delta t} = \rm konst.$$
Velocity-time curve

The velocity $$v$$ is constant and can be determined from the angular velocity.

$$v = \dfrac{\Delta s}{\Delta t} = \dfrac{\Delta \phi \cdot r}{\Delta t} = \omega \cdot r = \rm konst.$$

The magnitude of the velocity of a uniform circular motion is constant. However, the direction of the velocity keeps changing (see green arrow in the animation). The reason for this is the radial acceleration $$a_\rm{r}$$. It is always directed radial (towards the center of the circle).

$$a_\rm{r} = \dfrac{v^2}{r} = \omega^2 \cdot r = \rm konst.$$

## Period and frequency

The period $$T$$ is the time required by the body for one revolution around the circle. It is closely associated with the frequency $$f$$, indicating the number of revolutions, which the body does over a given timespan.

$$T = \dfrac{1}{f} \qquad \Rightarrow \qquad f = \dfrac{1}{T}$$

From these variables, also velocity and angular velocity can be calculated.

$$v = \dfrac{2 \,\, \pi \,\, r}{T} = 2 \,\, \pi \,\, r \,\, f$$ $$\omega = \dfrac{2 \,\, \pi}{T} = 2 \,\, \pi \,\, f$$

## Calculations for cicles

The relationship between radius $$r$$ and perimeter $$U$$ reads:

$$U = 2 \,\, \pi \,\, r \qquad \Rightarrow \qquad r = \dfrac{U}{2 \,\, \pi}$$

### Sources

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