is torque, in newton metres. is the linear force applied, in newtons. is the distance between the force and the centre of rotation, in metres.
Moment of Inertia
Moment of inertia (
is the moment of inertia, in . is the point mass of a rotating object, in kilograms. is the distance between the mass and the centre of rotation, in metres.
Note how this formula assumes a point mass - a mass condensed onto an infinitely small point. In practice, this provides a good enough approximation for masses with very little variance in radius, like a small ball or a thin ring - however for more accurate results, it's necessary to multiply by a constant depending on the shape of the mass. This constant can be calculated by using calculus to find a point mass which has an equivalent moment of inertia to the real mass - in other words, by finding an average moment of inertia for all points of mass in the object.
Just as Newton's second law of motion states that
is unbalanced torque, in newton metres. is the moment of inertia, in . is angular acceleration, in radians per second squared.
For the curious, this can also be used to explain why the moment of inertia is proportional to
Just as linear momentum is mass times velocity (
Conservation of Angular Momentum
Conservation of Momentum states that in the absence of external forces, the total momentum of a system remains constant - it can be transferred between objects via collisions, however the total momentum cannot increase or decrease in the process.
This applies to angular momentum as well - it is still the same concept of momentum, just considered from a different perspective (for example, linear momentum can be thought of as angular momentum with an radius that approaches infinity).
In practice, this means that if the moment of inertia of a rotating object decreases (e.g. by a mass being pulled closer to the centre), the angular velocity must increase to keep
Rotational Kinetic Energy
Just as a linearly moving mass has kinetic energy, a rotating mass has kinetic energy as well - it just revolves around a point instead of travelling in a straight line. Using the formula we already know for kinetic energy,