# Torque and Moment of Inertia

## Torque

Torque ( **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"**.

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 ( **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:

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.

## Unbalanced Torque

Just as Newton's second law of motion states that

Linear | Angular |
---|---|

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

## Angular Momentum

Just as linear momentum is mass times velocity ( **angular momentum** is moment of inertia times angular velocity.

is **angular momentum**, in. is the **moment of inertia**, in kilograms per metre squared.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).

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

## 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,

is the **rotational kinetic energy**of the mass, in joules.is the **moment of inertia**, in kilograms per metre squared.is the **angular velocity**, in radians per second.