Basic Rotation

A more specific definition will soon be given, but as you have already experienced with early experiments involving people rotating in lab chairs, the angular position theta of a rotating object is measured relative to the positive x axis, in the counterclockwise direction.

The angular velocity omega of a rotating object is the rate of change of its angular position.

The angular acceleration alpha of a rotating object is the rate of change of its angular velocity.

These definitions are formally identical, except for the word ‘angular’, to the definitions velocity and acceleration in terms of position and time.  All reasoning used to analyze angular motion applies to rotational motion.

Rotational displacement is measured in radians.  One radian of angular displacement around a circle corresponds to an arc length equal to the radius of the circle.  So the arc length corresponding to angular displacement `dTheta, around a circle of radius r, is `ds = r * `dTheta.  Note that a complete circle corresponds to an arc of 2 pi r, so that the angular displacement is 2 pi radians.

An object with mass, which rotates about a fixed axis, has a moment of inertia, which is denoted I and is equal to the sum of the m * r^2 contributions of all particles which constitute the object. 

Moments of inertia of some common object include:

·         A hoop of mass M and radius R, with all mass concentrated at the rim of the hoop.  The moment of inertia of a hoop is M R^2.

·         A uniform disk or cylinder of mass M and moment of inertia R rotating about its central axis.  The moment of inertia of this disk or cylinder is ½ M R^2.

·         A uniform sphere of mass M and radius R rotating about an axis through its center.  The moment of inertia of a sphere is 2/5 M R^2.

·         A uniform rod of length L and mass M rotating about an axis through its center.  The moment of inertia of a rod rotating about an axis through its center is 1/12 M L^2.

·         A uniform rod of length L and mass M rotating about an axis through one of its ends.  The moment of inertia of a rod rotating about an axis through one of its ends is 1/3 M L^2.

The torque exerted by a force which is applied along a line not passing through the axis of rotation of an object is

tau = r * F,

where r is the distance from the axis to the line of application of the force.

If a torque tau is applied to an object whose moment of inertia is I the its angular acceleration is alpha = tau / I.  This is analogous to, and follows from, Newton’s Second Law a = F / m.

The kinetic energy of an object whose moment of inertia is tau and whose angular velocity is omega is KE = ½ I * omega^2.

The angular momentum of an object whose moment of inertia is tau and whose angular velocity is omega is angular momentum = I * omega.  The total angular momentum of a closed system is constant.