Pure translation is movement of the center of mass of an object without any rotation. This is caused by a force acting through the position of the center of mass.
Pure rotation is circular movement about the center of mass while the center of mass remains stationary. This is caused by a force couple, where two equal and opposite forces act apart on the object (often an applied force combined with a reaction force from an axle or pivot).
A single force that does not pass through the center of mass will impart both translation and rotation.
Rotational units
Angular displacment \theta is a measure of change of rotation, measured in \rad.
Angular velocity \omega is a measure of rate of change of rotation, measured in \rad \s^{-1}.
The angular velocity \omega and the translational velocity v at a point distance r from the pivot are related by:
Angular acceleration \alpha is a measure of second degree rate of change of rotation, measured in \rad \s^{-2}.
The angular velocity \alpha and the translational velocity a at a point distance r from the pivot are related by:
Equations of rotational motion
These equations apply only to rotational motion with uniform angular acceleration. \omega_i is initial angular velocity, \omega_f final angular velocity, and \theta total displacement.
- \omega_f = \omega_i + \alpha t
- \omega_f^2 = \omega_i^2 + 2\alpha\theta
- \theta = \left( \frac{\omega_f + \omega_i}{2} \right) t
- \theta = \omega_i t + \frac{1}{2}\alpha t^2
These are exact analogues of the equations of linear motion.
Torque
The turning effect of a force is called torque (\tau). Torque is proportional to force applied and distance from the pivot, and is measured in newton-meters (equivalent to joules).
When a force is applied to a wheel, a reaction force is produced by the axle. Since this reaction force acts through the center of rotation, it does not produce any torque, so the turning effect is calculated only from the applied force and the distance from the pivot.
Torque is directly proportional to angular acceleration: the greater the torque, the greater the acceleration. The constant I is called the rotational inertia.
Torque in a beam is caused only by force applied perpendicular to the beam. If the force is not perpendicular, you must first decompose the force into perpendicular and parallel components.
Rotational inertia
Rotational interia is measured in \kg \m^2, and depends on the mass of the object and the distance of the mass from the center of rotation. It’s analogous to mass m in the force equation, because an object with a larger mass will be harder to set spinning.
If opposing masses were placed on a spinning disk, the rotational inertia would be greater the further towards the edge the masses were placed, even though the center of mass remains the same.