# 3.1: Uniform Circular Motion and Analogy to Linear Motion

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Uniform circular motion refers to a body moving in a circular path **without angular acceleration** (circular motion is impossible without centripetal acceleration). Angular acceleration is the rate of change of angular velocity, and angular velocity is the rate of change of angular displacement. In short, any angular quantity is the same as its linear quantity, except it describes the angle between the axis of rotation and the position of object, rather than the distance-based quantities.

### Analogy between linear and angular motion

As described before, rotational motion can be understood by analogy to linear motion. This section will list the analogs and then explain why the analogy works.

For accelerated linear motion, we had 3 quantities: velocity, acceleration and displacement. These angular analogs and their symbols are listed below:

Linear/Planar quantity | Angular/Rotational quantity |
---|---|

Displacement (\(\overrightarrow{s} \)) | Angular displacement (\( \overrightarrow{\theta} \)) |

Velocity (\( \overrightarrow{v} \)) | Angular velocity (\( \overrightarrow{\omega} \)) |

Acceleration (\( \overrightarrow{a} \)) | Angular acceleration (\( \overrightarrow{\alpha} \)) |

Recall our three kinematic equations. For rotational motion, the same equation apply. The only difference is that we substitute in the angular analog of the corresponding quantities. The equations are shown below:

Linear/Planar equation | Angular/Rotational equation |
---|---|

\(\overrightarrow{s} = \overrightarrow{u} t + \tfrac{1}{2} \overrightarrow{a} t^2 \) | \(\overrightarrow{\theta} = \overrightarrow{\omega} t + \tfrac{1}{2} \overrightarrow{\alpha} t^2 \) |

\(\overrightarrow{v} = \overrightarrow{u} + \overrightarrow{a}t \) | \(\overrightarrow{\omega} = \overrightarrow{\omega_o} + \overrightarrow{\alpha}t \) |

\( v^2 - u^2 = 2as \) | \(\omega^2 - \omega_0^2 = 2\alpha \theta \) |

#### Why does this analogy work?

It might seem strange that we can swap out the values this way, but it is actually quite reasonable to do so. When we devised these equations, we didn't explicitly use any property of vectors. So, these equations just describe the relation between the magnitude of these numbers, i.e. they are just relations between quantities changing a certain way. However, note that we have used the arrow denoting a vector in the equations on both sides. This means that these equations hold for vector quantities as well.

This is because the angular quantities too form a vector similar to linear motion. This is actually a *pseudovector* and is the result of taking the cross-product(coming soon) of the radius of rotation and the angular quantity. Effectively, we can now treat rotational motion the same as linear motion.

However, it is important to note that we cannot add/subtract vectors and pseudovectors. We first have to solve the problem as a problem of linear motion at a small scale, and then calculate the pseudovector. Pseudovectors are beyond the scope of this book, but we can utilize them for our problem-solving if we keep this point in mind.