Rotational

Rotational is an adjective and noun referring to rotation, the motion of something turning about an axis. In physics and engineering, rotational motion describes how bodies turn around a fixed or moving axis. It is characterized by angular velocity (ω), angular acceleration (α), torque (τ), and moment of inertia (I). For a rigid body rotating about a fixed axis, angular momentum L = I ω and the torque relates to the rate of change of angular momentum via τ = dL/dt.

Kinematics and representation: In pure rotation about a fixed axis, all points on the body move with the same angular speed. The changing orientation is described by rotation matrices or by angular parameters such as Euler angles or axis-angle representations. In two dimensions, a rotation by angle θ about the origin is given by the matrix [cosθ, -sinθ; sinθ, cosθ].

Rotational symmetry: An object has rotational symmetry if it maps onto itself under a rotation about a central point by a nonzero angle. The symmetry order is the number of distinct orientations that coincide with the original. Examples include circles (infinite order), regular polygons, and many molecules and crystals.

Mathematical structure: Rotations form the special orthogonal group SO(n). They preserve lengths and angles; rotation matrices are orthogonal with determinant 1. In three dimensions, a rotation can be described by an axis and angle, or by quaternions, which avoid some computational issues with Euler angles.

Applications: Rotational motion is central to planetary dynamics, machinery, sports, and biomechanics. In spectroscopy, rotational spectra arise from molecular rotation; the rigid rotor model yields energy levels proportional to J(J+1). Rotations are also fundamental in computer graphics, robotics, and crystallography.