# What is circular motion and its examples

## Trajectory

Circular motion

 Cartesian coordinates Polar coordinates radius Track length

### Path (tangential) speed

 Cartesian coordinates Polar coordinates

### Angular velocity

The vector of the angular velocity is directed parallel to the axis of rotation (axial vector) and perpendicular to the plane of the path. It is then perpendicular to the radius vector and perpendicular to the vector of the tangential velocity. Its direction can be clearly described by the direction of movement of a screw with a right-hand thread. The right thumb rule applies analogously.

The angular velocity w (angular difference per unit of time) is also called the angular frequency. You can from the frequency f (revolutions per unit of time) by means ofbe calculated.

The relationship between angular velocity, path velocity and radius vector is described by the Euler equation:

### Euler equation

#### Discussion of the Euler equation

The acceleration is obtained from the first derivative of the speed with respect to time:

Inserting the owler relationship for results for the total or linear acceleration:

The individual sizes have the following meaning:

 Angular acceleration Tangential orAzimuthal acceleration Normal, radial orCentripetal acceleration Total acceleration

The equations above apply under the assumption that a mass point is at rest in the rotating frame of reference.

The following graphic shows the individual components of the acceleration:

The amount of the total acceleration results from this

Also for aT = 0 the normal acceleration is not equal to zero. The circular movement is therefore always an accelerated movement.

Circular movement with tangential acceleration = 0

Because of vT = const. And w (t) = w = const. follows:

The following applies to a full cycle:

and thus

or with the frequency

also

The circular motion with constant path velocity v = wR is an accelerated motion. In order to maintain the circular motion, a force directed towards the center must be applied - the centripetal force. The following applies to the amount of centripetal force (see above):

Since the direction of the force changes continuously, the circular motion is an unevenly accelerated motion.

The circular motion with constant

Tangential acceleration

If one speaks of the uniformly accelerated circular movement, this means a circular movement with constant angular acceleration. As already discussed above, the uniform circular motion is already accelerated unevenly.

Out

follows with

the angular velocity

Furthermore you get with

The radial speed is equal to zero, the radial acceleration is equal to the centripetal acceleration.

### Example hammer thrower

A mass of m = 7.2 kg is accelerated uniformly on a radius of R = 2m and released at the angle j = 45 ° (maximum range) to the vertical. The maximum frequency of rotation of fMax = 2s-1 is reached after n = 3 revolutions.

·       Orbital speed and throwing distance

The maximum current path speed is reached after 3 revolutions at the end of the acceleration phase:

The achievable throw is thus

·       Normal acceleration and centripetal force

##### The maximum radial acceleration follows from

The maximum normal force component to be used to keep the mass on the circular path is thus

·       Angular, path and total acceleration

Requirement:Þ

If one eliminates t from the last two equations, it follows:

At the end of the acceleration phase you get with

an angular acceleration of

This results in a tangential acceleration of

This results in an overall acceleration of