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.

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

Radial acceleration

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:

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

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