Torque and Moment of Inertia

Torque

Torque ( $τ$ ) is the tendency of a force to turn an object around an axis - the angular counterpart to a linear force, hence it can be thought of as "angular force".

Formula

$τ = F r$

Variable Key

$τ$ is torque, in newton metres.
$F$ is the linear force applied, in newtons.
$r$ is the distance between the force and the centre of rotation, in metres.

Moment of Inertia

Moment of inertia ( $I$ ) is the tendency of an object to resist change in angular motion - the angular counterpart to inertia, hence it can be thought of as "angular inertia". Linearly, inertia is entirely dependent on mass, however with moment of inertia the radius is also involved as shown below:

Formula

$I = m r^{2}$

Variable Key

$I$ is the moment of inertia, in ${kg m}^{2}$ .
$m$ is the point mass of a rotating object, in kilograms.
$r$ 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.

Unbalanced Torque

Just as Newton's second law of motion states that $F = m a$ , a similar logic can be applied to an unbalanced torque, by substituting the linear properties with their angular counterparts:

Linear	Angular
$F$ Force	$τ$ Torque
$m$ Mass	$I$ Moment of Inertia
$a$ Linear Acceleration	$α$ Angular Acceleration

Formula

$τ = I α$

Variable Key

$τ$ is unbalanced torque, in newton metres.
$I$ is the moment of inertia, in ${kg m}^{2}$ .
$α$ is angular acceleration, in radians per second squared.

Moment of Inertia Proof

For the curious, this can also be used to explain why the moment of inertia is proportional to $r^{2}$ and not $r$ , which may seem unintuitive. This can be done by substituting in the equations for torque and angular acceleration, then Newton's second law. Notice how the two $r$ terms multiply together, whereas the $a$ terms cancel out:

\begin{aligned} τ & = I α \\ F r & = I \frac{a}{r} \\ m a r & = I \frac{a}{r} \\ I = m & a r \times \frac{r}{a} \\ I = m & r^{2} \end{aligned}

Angular Momentum

Just as linear momentum is mass times velocity ( $p = m v$ ), angular momentum is moment of inertia times angular velocity.

Formula

$L = I ω$

Variable Key

$L$ is angular momentum, in ${kg m}^{2} s^{- 1}$ .
$I$ is the moment of inertia, in kilograms per metre squared.
$ω$ is the angular velocity, in radians per second.

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).

Summary

In the absence of external torques, the total angular momentum of a system remains constant.

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 $L$ constant, and vice versa.

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, $E_{k} = \frac{1}{2} m v^{2}$ , we can substitute the variables for their rotational counterparts to get the following formula:

Formula

$E_{k} = \frac{1}{2} I ω^{2}$

Variable Key

$E_{k}$ is the rotational kinetic energy of the mass, in joules.
$I$ is the moment of inertia, in kilograms per metre squared.
$ω$ is the angular velocity, in radians per second.