# rotational motion

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## Rotational motion

The motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity; also, the rotation of a particle about a fixed point in space. Rotational motion is illustrated by (1) the fixed speed of rotation of the Earth about its axis; (2) the varying speed of rotation of the flywheel of a sewing machine; (3) the rotation of a satellite about a planet; (4) the motion of an ion in a cyclotron; and (5) the motion of a pendulum. Circular motion is a rotational motion in which each particle of the rotating body moves in a circular path about an axis. Such motion is exhibited by the first and second examples. For information concerning the other examples *See* Harmonic motion, Particle accelerator, Pendulum

The speed of rotation, or angular velocity, remains constant in uniform circular motion. In this case, the angular displacement Θ experienced by the particle or rotating body in a time *t* is Θ = ω*t*, where ω is the constant angular velocity.

A special case of circular motion occurs when the rotating body moves with constant angular acceleration. If a body is moving in a circle with an angular acceleration of α radians/s^{2}, and if at a certain instant it has an angular velocity ω_{0}, then at a time *t* seconds later, the angular velocity may be expressed as ω = ω_{0} + α*t*, and the angular displacement as Θ = ω_{0}*t* + ½α*t*^{2}. *See* Acceleration, Velocity

A rotating body possesses kinetic energy of rotation which may be expressed as *T*_{rot} = ½*I*ω^{2}, where ω is the magnitude of the angular velocity of the rotating body and *I* is the moment of inertia, which is a measure of the opposition of the body to angular acceleration. The moment of inertia of a body depends on the mass of a body and the distribution of the mass relative to the axis of rotation. For example, the moment of inertia of a solid cylinder of mass *M* and radius *R* about its axis of symmetry is ½*MR*^{2}.

The action of a torque *L* is to produce an angular acceleration α according to the equation below, where *I*ω, the product

*I*ω of a rotating body, and hence its angular velocity ω, remains constant unless the rotating body is acted upon by a torque. Both

*L*and

*I*ω may be represented by vectors.

It is readily shown that the work done by the torque *L* acting through an angle Θ on a rotating body originally at rest is exactly equal to the kinetic energy of rotation. *See* Angular momentum, Moment of inertia, Rigid-body dynamics, Torque, Work