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If an object is rotating about an axis with angular velocity omega, then a point at distance r from the axis is moving at speed v = r * omega.  A mass m at this distance therefore has kinetic energy ½ m v^2 = ½ m * (r * omega)^2.

If omega is given in units of radians / second, then r * omega will be in units of distance / second, where the unit of distance is that of the quantity r.  That is, if r is in meters, then r * omega will be in meters / second; if r is in cm then r * omega will be in cm / second.

The reason for this is that a radian of angle corresponds to an arc distance on a circle, where the arc distance is equal to the radius of the circle.

The moment of inertia of a mass m located at distance r from the axis of rotation is

• I = m r^2.

An object rotating about its a axis typically has its mass distributed over a range of distances from its center, so r is different for different parts of the object and I is the sum of all the m r^2 contributions from different parts of the object.

The moment of inertia of a uniform disk of mass M and radius R rotating about an axis through its center and perpendicular to its plane is

• I =1/2 * M R^2.

The moment of inertia of a sphere of mass M and radius R rotating about an axis through its center and perpendicular to its plane is

• I = 2/5 * M R^2.

The moment of inertia of a rod of mass M and radius L rotating about an axis through its center is

• I = 1/12 * M L^2.

The moment of inertia of a rod of mass M and radius L rotating about an axis through one of its ends is

• I = 1/3 * M L^2.

The torque exerted by a force F acting at distance r from the axis of rotation is

• torque = F * r * sin(theta),

where theta is the angle between the moment-arm and the force.  The moment-arm is the vector from the axis of rotation to the point of application.

The angular acceleration of an object with moment of inertia I when subject to a torque tau, both measured with respect to the same axis of rotation, is

• alpha = tau / I (analogous to a = F / m)

The work done by a net torque tau as an object rotates through angular displacement `dTheta about its axis of rotation is

• `dW = tau * `dTheta (analogous to `dW = F * `ds).