Rotation around a fixed axis

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The simplest three-dimensional case of rotation is rotation of a body about a fixed axis of rotation: each point of the body moves in a plane perpendicular to the axis, carrying out a circular motion, with the circle centered at the intersection of the plane and the axis.

The axis can be within the body, in which case the body is said to rotate upon itself, or spin, or outside it.

In the case of a rigid body the angle of rotation is a function of time which is the same for every point of the body.

An object may allow rotation with respect to an attached other object by means of one or more hinges (e.g. a door, scissors, a hinge joint).

Contents

[edit] Speed of rotation, angular acceleration, and torque

The speed of rotation is given by the angular frequency (rad/s) or frequency / rotational speed / revolutions per minute (turns/s, turns/min), or period (seconds, days, etc.).

With one direction of rotation considered positive, the sign of the angular frequency indicates the direction of rotation.

The time-rate of change of angular frequency is the scalar version of angular acceleration (rad/s²). This change is caused by the scalar version of the torque, which can have a positive or negative value in accordance with the convention of positive and negative angular frequency. The ratio of torque and angular acceleration (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia.

The energy required for / released during rotation is the torque times the rotation angle, the energy stored in a rotating object is one half of the moment of inertia times the square of the angular frequency. The power required for angular acceleration is the torque times the angular frequency.

[edit] Vectors

According to the right-hand rule, moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like most screws.

The angular velocity vector also describes the direction of the axis of rotation. In the case of a fixed axis this direction is along that axis and the rotation process is described by a scalar, the angular frequency, as a function of time.

Similarly the torque vector also describes around which axis it tends to cause rotation, or in other words, the direction in which it tends to change the angular velocity vector. To maintain rotation around the fixed axis the total force has to be zero and the total torque vector has to be along the axis, so that it only changes the magnitude and not the direction of the angular velocity vector. In the case of a hinge, only the component of the torque vector along the axis has effect on the rotation, other forces and torques are compensated by the structure.

[edit] Centripetal force

In the case of a spinning object, internal tensile stress provides the centripetal force that keeps the object together.

A rigid body model neglects the accompanying strain.

If the body is not rigid this strain constitutes a change of shape. This is also expressed as changing shape due to the "centrifugal force".

Celestial bodies rotating about each other often have elliptic orbits. The special case of a circular orbits is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion. The centripetal force is provided by gravity, see also two-body problem. This usually also applies for a spinning celestial body, so it need not be solid to keep together, unless the angular speed is too high in relation to its density. For example, a spinning celestial body of water must take at least 3 hours and 18 minutes to rotate, regardless of size, or the water will separate. If the density of the fluid is higher the time can be less. See orbital period. See also oblate for oblateness due to rotation, in particular of a fluid celestial body.

[edit] Constant angular speed

The simplest case of rotation around a fixed axis is that of constant angular speed. The total torque is zero: in e.g. the case of the rotation of the Earth around its axis there is very little friction, in the case of e.g. a fan the motor applies a torque to compensate for friction. The angle of rotation is a linear function of time, which modulo 360° is a periodic function.

An example of this is the two-body problem with circular orbits.

[edit] See also

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