Rotation (motion about a center or axis)

Key characteristics and quantities

Common measures used to describe rotation include angular displacement (an angle through which an object turns), angular velocity (rate of change of angle, often denoted ω), angular acceleration (rate of change of ω), period and frequency (time and rate of full revolutions), and moment of inertia (a scalar or tensor measure of how mass is distributed relative to the axis). These quantities are related by dynamic laws: for many rigid bodies torque and angular acceleration are related approximately by τ = Iα, analogous to F = ma for translation.

Mathematical description

In the plane, a rotation by angle θ around a center can be written in coordinates as x' = x cos θ − y sin θ, y' = x sin θ + y cos θ. Such rotations form a continuous group (SO(2)). In three dimensions rotations are commonly described by an axis and angle, by rotation matrices or by parameters such as Euler angles and quaternions. A key property of 3D rotations is that they do not generally commute: the final orientation can depend on the order of successive rotations.

Physical principles

For a rigid body rotating about a fixed axis, the angular momentum L is often expressed as L = Iω for simple symmetric cases; more generally L is a vector related to the moment of inertia tensor. In the absence of external torques the total angular momentum is conserved, which explains many everyday effects (for example, an ice skater pulling in their arms spins faster). The spin of a body refers to rotation about an internal axis, while motion around an external point is usually called orbital revolution or an orbit. Angular velocity and related kinetic concepts are discussed further at angular velocity and angular momentum.

Measurement, sign conventions and rules

Angular measures are most commonly given in radians for mathematical work and degrees for many practical contexts. The right-hand rule is the standard convention to assign a direction to angular velocity and angular momentum vectors: curl the fingers of the right hand in the direction of rotation and the extended thumb points along the vector direction.

Examples and applications

• Planetary rotation: the Earth’s daily rotation about its axis and its annual orbit around the Sun illustrate spin versus revolution; see entries relevant to the Earth and the Sun.
• Mechanical devices: wheels, gears, turbines and engines use controlled rotation to transmit power and perform work.
• Stability and control: gyroscopes and flywheels exploit angular momentum and precession for stabilization in vehicles and instruments.
• Microscale and quantum: some fundamental particles have an intrinsic quantum mechanical property called spin, a concept distinct from classical rotation but related in name.

History and further reading

The distinction between a body’s own spin and its motion about another object was important in the development of astronomy and mechanics. For introductions and accessible treatments of rotational kinematics and dynamics consult basic texts and surveys linked at further reading and related resources.