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

 

Unit of measurement: rad.s-1

 

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 or

Azimuthal acceleration

Normal, radial or

Centripetal 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