Kinematics is a mathematical branch of physics that uses equations to describe the motion of objects (specifically their *trajectories*) without referring to forces.

That is, you could simply plug in various numbers to the set of four kinematic equations to find any unknowns in those equations without needing any knowledge of the physics behind that motion, relying only on your algebra skills.

Think of “kinematics” as a combination of “kinetics” and “mathematics” – in other words, the math of motion.

Rotational kinematics is exactly this, but it specifically deals with objects moving in circular paths rather than horizontally or vertically. Like objects in the world of translational motion, these rotating objects can be described in terms of their displacement, velocity and acceleration over time, although some of the variables necessarily change to accommodate the basic differences between linear and angular motion.

It is actually very useful to learn the basics about linear motion and rotational motion at the same time, or at least be introduced to the relevant variables and equations. This is not to overwhelm you, but instead is meant to underscore the parallels.

Of course, it’s important to remember when learning about these “types” of motion in space that translation and rotation are far from mutually exclusive. In fact, most moving objects in the real world display a combination of both types of motion, with one of them often not being evident at first glance.

## Examples of Linear and Projectile Motion

Because “velocity” typically means “linear velocity” and “acceleration” implies “linear acceleration” unless otherwise specified, it’s appropriate to review a few simple examples of basic motion.

Linear motion literally means motion confined to a single line, often assigned the variable “x.” Projectile motion problems involve both x- and y-dimensions, and gravity is the only external force (note that these problems are described as occurring in a three-dimensional world, e.g., “A cannonball is fired…”).

Note that mass *m* does not enter kinematics equations of any sort, because gravity’s effect on the motion of objects is independent of their mass, and quantities such as momentum, inertia and energy are not part of any equations of motion.

## A Quick Note on Radians and Degrees

Because rotational motion involves studying circular paths (in non-uniform as well as uniform circular motion) rather than using meters to describe the displacement of an object, you use radians or degrees instead.

The radian is, on the surface, an awkward unit, translating to 57.3 degrees. But one trip around a circle (360 degrees) is defined as 2π radians, and for reasons you’re about to see, this proves convenient when problem solving in some cases.

- The relationship
**π rad = 180 degrees** can be used to easily convert between both units of measure.

There may be problems that include the number of revolutions per unit time (rpm or rps). Remember that each revolution is 2π radians or 360 degrees.

## Rotational Kinematics vs. Translational Kinematics Measurements

Translational kinematics measurements, or units, all have rotational analogs. For example, instead of linear velocity, which describes, for example, how far a ball rolls in a straight line over a given time interval, the ball's *rotational* or *angular velocity* describes the rate of rotation of that ball (how much it rotates in radians or degrees per second).

The main thing to keep in mind here is that every translational unit has a rotational analogue. Learning to mathematically and conceptually relate the “partnered” ones takes a little practice, but for the most part it’s a matter of simple substitution.

Linear velocity *v* specifies both the magnitude and direction of a particle’s translation; angular velocity *ω* (the Greek letter omega) represents its singular velocity, which is just how fast the object is rotating in radians per second. Similarly, the rate of change of *ω*, the angular acceleration, is given by *α* (alpha) in rad/s^{2}.

The values of *ω* and *α* are the same for any point on a solid object whether they’re measured 0.1 m from the axis of rotation or 1,000 meters away, because it is only how fast the angle *θ* changes that matters.

There are, however, tangential (and thus linear) velocities and accelerations present in most situations where rotational quantities are seen. Tangential quantities are computed by multiplying angular quantities by *r*, the distance from the axis of rotation: **v _{t}**

_{}=

**ωr** and

**α**

** = **

_{t}**α**

**r.**

## Rotational Kinematics vs. Translational Kinematics Equations

Now that the measurement analogies between rotational and linear motion have been squared away using the introduction of new angular terms, these can be used to rewrite the four classic translational kinematics equations in terms of rotational kinematics, just with somewhat different variables (the letters in equations representing unknown quantities).

There are four fundamental equations as well as four basic variables in play in kinematics: position (*x*, *y* or *θ*), velocity (*v* or *ω*), acceleration (*a* or *α*) and time *t*. Which equation you choose depends on which quantities are unknown to start.

**- [insert a table of linear/translational kinematics equations aligned with their rotational analogs]**

For example, say you are told that a machine arm swept through an angular displacement of 3π/4 radians with an initial angular velocity *ω _{0}* of 0 rad/s and a final angular velocity

*ω* of π rad/s. How long did this motion take?

While every translational equation has a rotational analogue, the reverse isn’t quite true because of centripetal acceleration, which is a consequence of the tangential velocity *v _{t}* and points toward the axis of rotation. Even if there is no change in the speed of a particle orbiting a center of mass, this represents acceleration because the direction of the velocity vector is always changing.

## Examples of Rotational Kinematics Mathematics

1. A thin rod, classified as a rigid body with a length of 3 m, rotates around an axis about one end. It accelerates uniformly from rest to 3π rad/s^{2} over a period of 10 s.

a) What are the average angular velocity and angular acceleration during this time?

As with linear velocity, just divide (**ω _{0}+**

**ω**) by 2 to get average angular velocity: (0 + 3π s

^{-1})/2 =

**1.5**

**π**

**s**.

^{-1}- Radians are a dimensionless unit, so in kinematics equations, angular velocity is expressed as s
^{-1}.

The average acceleration is given by **ω=ω _{0}+ αt**, or

**α** = (3π s

^{-1}/10 s) =

**0.3π s**.

^{-2}b) How many complete revolutions does the rod make?

Since the average velocity is 1.5π s^{-1} and the rod spins for 10 seconds, it moves through a total of 15π radians. Since one revolution is 2π radians, this means (15π/2π) = 7.5 revolutions (**seven complete revolutions**) in this problem.

c) What is the tangential velocity of the end of the rod at time t = 10 s?

Since *v _{t}* =

**ωr**, and

**ω** at time t = 10 is 3π s

^{-1},

**v** = (3π s

_{t}^{-1})(3 m) =

**9π m/s.**

## The Moment of Inertia

*I* is defined as the moment of inertia (also called *second moment of area*) in rotational motion, and it is analogous to mass for computational purposes. It thus appears where mass would appear in the world of linear motion, perhaps most importantly in calculating angular momentum *L*. This is the product of *I* and * ω,* and is a vector with direction the same as

****

*ω*.**I = mr ^{2} for a point particle**, but otherwise it depends on the shape of the object doing the rotating as well as the axis of rotation. See the Resources for a handy list of values of

*I* for common shapes.

Mass is different because the quantity in rotational kinematics to which it relates, moment of inertia, itself actually *contains* mass as a component.