# What is a uniform circular motion in physics

## Uniform circular motion

- Circular motion and centripetal force (5:02 minutes)

### introduction

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

### attempt

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

speed

acceleration

angle

**Angle-time curve**

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

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

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

The path-time curve is a straight line that runs through the origin of the coordinates. This shows that the distance covered and the time are proportional to each other.

The proportionality factor is the path speed \ (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 an angle changes over time.

$$ \ omega = \ dfrac {\ Delta \ phi} {\ Delta t} = \ rm const. $$**Speed-time curve**

The path velocity \ (v \) is constant and can be determined from the angular velocity.

$$ v = \ dfrac {\ Delta s} {\ Delta t} = \ dfrac {\ Delta \ phi \ cdot r} {\ Delta t} = \ omega \ cdot r = \ rm const. $$### Radial acceleration

The **amount** the speed is constant with a uniform circular motion. However, that is changing **direction ** the speed constantly (see green arrow in the animation). The reason for this is the radial acceleration \ (a_ \ rm {r} \). she is always *radial* (towards the center of the circle).

### Period and frequency

The period \ (T \) is the time that the body needs for one cycle. It is closely related to the frequency \ (f \), which indicates the number of revolutions that the body makes within a period of time.

$$ T = \ dfrac {1} {f} \ qquad \ Rightarrow \ qquad f = \ dfrac {1} {T} $$Speed and angular speed can also be calculated from these variables.

$$ v = \ dfrac {2 \, \, \ pi \, \, r} {T} = 2 \, \, \ pi \, \, r \, \, f $$ $$ \ omega = \ dfrac {2 \, \, \ pi} {T} = 2 \, \, \ pi \, \, f $$### Calculations on the circle

The relationship between radius \ (r \) and circumference \ (U \) is:

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

### swell

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