Rotational motion occurs when an extended object changes its orientation about an axis or when its particles follow circular or otherwise curved paths associated with rotation. In idealised rigid body rotation, every point of the body has the same angular displacement, angular velocity and angular acceleration at a given instant, although different points can have different linear speeds and accelerations because their distances from the axis differ. Rotational motion can occur about a fixed axis, a moving axis or a combination of translation and rotation.
Angular displacement describes the change in orientation and is commonly measured in radians. Angular velocity describes the rate of change of angular displacement, while angular acceleration describes the rate of change of angular velocity. For rotation about a fixed axis, tangential speed is proportional to angular speed and radial distance. A point farther from the axis therefore travels faster than a point closer to the axis even though both share the same angular velocity. This distinction between angular and linear quantities is fundamental to analysing rotating systems.
Rotational motion has direct analogues with translational mechanics. Angular displacement corresponds conceptually to displacement, angular velocity to velocity and angular acceleration to acceleration. Torque plays a role analogous to force, while moment of inertia plays a role analogous to mass. Rotational kinetic energy is related to moment of inertia and angular velocity. These analogies are useful for developing equations of motion, but they are not exact substitutions in every physical situation because rotational dynamics has vector properties, axis dependence and geometric constraints that have no simple one dimensional equivalent.
The moment of inertia measures resistance to angular acceleration and depends on both total mass and its distribution relative to the rotation axis. Concentrating mass farther from the axis increases moment of inertia. Different geometric objects therefore have different moments of inertia even when they have the same total mass. For composite bodies, the total moment of inertia can be obtained by summing contributions from individual components, while the parallel axis theorem allows a known moment of inertia about one parallel axis to be related to another axis displaced from it.
Rotational motion can be uniform or nonuniform. Uniform rotational motion has constant angular velocity and therefore zero angular acceleration, although points on the rotating object can still experience centripetal acceleration because their velocity directions continuously change. Nonuniform rotation involves changing angular velocity and therefore angular acceleration. Tangential acceleration is associated with angular acceleration, while radial or centripetal acceleration is associated with angular velocity and distance from the axis. The total acceleration of a point can therefore contain both tangential and radial components.
Real mechanical systems frequently combine translational and rotational motion. A rolling wheel, for example, can translate across a surface while rotating about its centre. Depending on whether the wheel rolls without slipping, the translational and angular velocities are linked by the wheel radius. More generally, rigid body motion can be decomposed into translation of a reference point and rotation about that point. The choice of reference frame and reference point can substantially simplify the mathematical description of a complex system.
Rotational motion is fundamental to engines, turbines, electric motors, generators, flywheels, gears, wheels, propellers, centrifuges and many biological mechanisms. Rotating machinery must account for inertia, torque, angular speed, vibration, balancing and material stresses. Imbalances can produce periodic forces and vibrations because the centre of mass does not lie on the intended rotation axis. Engineering analysis therefore considers both ideal kinematics and real effects such as friction, deformation, bearing loads and energy losses.
Rotational motion also appears throughout astronomy and microscopic physics. Planets rotate about their axes while orbiting stars, stars rotate and accretion systems contain large amounts of rotating material. At smaller scales, molecular rotation and microscopic angular momentum require models appropriate to the physical system. A common misconception is that every particle in a rotating body moves with the same linear velocity. In fact, linear speed generally depends on distance from the axis, whereas angular velocity is shared by points in ideal rigid body rotation.
