Why is a uniform circular motion speed uneven
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 of_{}be 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 a_{T} = 0 the normal acceleration is not equal to zero. The circular movement is always an accelerated movement.
Circular movement with tangential acceleration = 0
_{}
Because of v_{T} = 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 f_{Max} = 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
_{}
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