Uniform circular motion

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A uniform circular movement is a movement in which the path curve runs on a circle ("circular movement") and the amount of the path speed is constant ("uniform"). It is thus a form of rotation . In contrast to the uniform movement , the speed vector does not remain constant here, because its amount remains constant, but its direction changes constantly. The consideration of such fundamental motion sequences helps with the interpretation and characterization of complex processes in the field of kinematics and dynamics .

properties

Graphical analysis of the speed vector during circular motion
Graphic analysis of the acceleration vector during circular motion

A circular path is a closed trajectory in a plane with a constant distance from a center point. The distance represents the arc length and results from the angle and the radius .

A movement on the circular path can thus be described solely by the rate of change of the angle, the angular velocity . This remains constant in the case of uniform circular motion.

is constant

The amount of the speed is thus:

is constant.

Since the trajectory is closed, the movement always returns to the same point. The time interval required for this is called the period of rotation.

Vectorial consideration

As with every movement, the speed vector is tangential to the trajectory, i.e. here tangential to the circle. It is thus perpendicular to the radius vector . He points in the direction of movement.

Using the vectorial view, the acceleration required for a change in direction can also be determined without changing the amount of the speed. The acceleration is derived in the same way as when considering the velocity vector. The acceleration vector is perpendicular to the velocity vector and points to the center of the circle.

The direction of the acceleration is thus clarified, but not the amount. The small angle approximation helps here , in which the arc length between the speed vectors of the same length increasingly corresponds to the direct distance between the vector peaks. Since the change in angle of the circular motion is also reflected in the velocity vectors, the limit crossings can be equated as follows :

.

Since the acceleration vector always points towards the center of the circle, it is called centripetal acceleration and in connection with the mass the same applies to the centripetal force .

Derivation via polar coordinates

The circular motion of a particle can be efficiently represented in polar coordinates. In Cartesian coordinates is

In this case, referred to the distance between the position of the particle and the origin, which is the center of the circular motion and the angle between the line connecting the origin and location of the particle and the axis. In the case of circular motion, the radius is constant. Then the transformation in polar coordinates is:

speed

The speed is the derivation of the place. The position vector must also be differentiated in polar coordinates. Since the distance is constant, it follows

.

The derivative of the unit vector in -direction is proportional to the unit vector in -direction, da

is orthogonal to . The following applies to the speed

with the angular velocity .

acceleration

In the case of movements, the amount of speed is constant. As in the case of speed, the time derivative is reduced to the derivative of the direction vector. The acceleration of the uniform circular movement can therefore be achieved by means of

to calculate. With

follows

.

See also

literature

  • Lehmann, Schmidt: Abitur training / physics / kinematics, dynamics, energy / vocational high school / technology . 1st edition. Stark Verlagsgesellschaft, 2001, ISBN 978-3-89449-176-5 .
  • Ekbert Hering, Rolf Martin, Martin Stohrer: Physics for engineers . 8th edition. Springer, Berlin Heidelberg New York 2002, ISBN 3-540-42964-6 .

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