**Where are we going with this?**The information on this page connects to standards such as:

**Angular Kinematics Overview**

*(Wow… This sounds fancy!)*

Angular kinematics is a way to describe something moving around a circle. This is a lot like geometry!

It is pretty cool, though and NASA has this…

https://www.grc.nasa.gov/www/k-12/airplane/angdva.html

## Angular Displacement

** Angular displacement**: The amount that the angle changes of from the initial position to the final position.

Angular displacement is sometimes represented by the letter phi φ. Angles are often represented by θ. And the ∆ is our good buddy to represent change of.

So we can say that φ is essentially ∆θ.

We have established that ∆ anything is where did end up compared to where it started. For instance

∆T = Tf - Ti

∆d = df - di

So,

∆θ = θf - θi

That should make sense, right? Back to the "sometimes a phi" thing:

φ = θf - θi

Keep in mind that "f" and "i" are only one way to indicate beginning and ending conditions. "0" and "1" are also seen and many sources use them for Angular displacement.

Source |

So…

φ = θf - θi

∆θ = θf - θi

is the same as

φ = θ1 - θ0

∆θ = θ1 - θ0

**Sample 1**: A thing is at 30°. It moves to 50°. What is its angular displacement? (Ans. 20°)

**Sample 2**: A thing is at 0 rad. It moves to π rad. What is its angular displacement? (Ans π rad)

## How Far Did It Move

Geometry allows for some fun stuff when the angles are in radians. Trust me on this!

In general, any angle θ measured in radians is defined by the formula

θ = s/r

where s is arc length and r is radius

that means that

∆θ rad = ∆s/r

(This is probably pretty useful!)

Of course you can convert from rad to degrees.

Multiply degrees by π/180 to get radians.

Multiply radians by 180/π to get degrees.

*But, that would be annoying! Google it, if you want.*

**Sample 3**: A thing is at a radius of 2 from a point and is moving in a circle. If it moves such that it's arc length is 4, what is its angular displacement? (Ans 2 rad)

It follows that, in a case where something is measured from a reference point (a line or angle or something), if that something moves a final angular displacement can be found (by mathing it) as…

∆θ = θf - θi

θf = θi + ∆θ

This should remind you of one of the liner displacement formulas!

df = di + ∆d

## Angular Velocity

** Angular velocity**: The amount that the angle changes in some period of time. That is to say, it is the angular displacement / elapsed time.

For instance, something with an angular velocity of 20° per second would turn 20° in 1 second, 40° in 2 seconds, 60° in 3 seconds… etc.

Other ways to describe angular velocity is to discuss revolutions per some period of time. The rate that car engines turn is reported in RPM or revolutions per minute. Revolutions per second are also normal.

Officially (SI UNITS) the units on angular velocity are radians per second (rad/s)

It is handy to know that a circle is 360° around or 2π radians.

We need a formula!

*Because of something to do with something or for some reason… Never mind.*

**The symbol for angular velocity is the Greek omega (not W or w!) ω**

∆t = tf - tior∆t = t1 - t0

and angular displacement is

∆θ = θf - θior∆θ = θ1 - θ0

ω = ∆θ/∆t

ω = (θf - θi)/(tf - ti)ω = (θ1 - θ0)/(t1 - t0)

**Sample 1**: A thing rotates 12° in 4 seconds. What is its angular velocity? (Ans. 3°/s)

**Sample 2**: A thing rotates 12 rad in 4 seconds. What is its angular velocity? (Ans. 3 rad/s)

**Sample 2**: A thing rotates 360 times in 1 minute (360 RPM). What is its angular velocity in degrees per second? (Ans 21,600°/s)

360 RPM is 60 rotations per second. Each rotation is 360°. Thus∆θ = 60 • 360°∆t = 1ω = ∆θ/∆t

ω = ∆θ/∆t

∆t•ω = ∆θ

∆θ = ω∆t

This should remind you of a linear motion formula!

∆d = v∆t

## Angular Displacement Part II

What if we took our angular displacement formula and did some math? That could be fun!

θf = θi + ∆θ

∆θ = ω∆t

Let the magic happen!

θf = θi + ω∆t

*Wanna know a secret (that really isn't a secret)? Go to the bottom of this page!*

## Angular Acceleration

** Angular acceleration**: The amount that the rate at which angular velocity changes over some period of time. That is to say, it is the change in angular velocity / elapsed time.

Since we are using a for linear acceleration let's use a Greek a for angular acceleration—α

Where

α is angular acceleration

∆ω is change in angular veloicty

∆t is elapsed time

α = ∆ω / ∆t

Let's do that math magic!

α = ∆ω / ∆t

∆ω = α•∆t

*Pssttt… It's not REALY magic… *

If something has angular velocity to start with and it gets more, then the final angular velocity is…

ωf = ωi + ∆ω

and then…

ωf = ωi + ∆ω

∆ω = α•∆t

ωf = ωi + α•∆t

*You might be thinking of the linear kinematic equation for final velocity!*

## Summary

**: The amount that the angle changes of from the initial position to the final position.**

*Angular displacement*∆θ = θf - θi

φ = θf - θi

θf = θi + ∆θ

θf = θi + ω∆t

In general, any angle θ measured in radians is defined by the formula

θ = s/r

where s is arc length and r is radius

that means that

∆θ rad = ∆s/r

**: The amount that the angle changes in some period of time. That is to say, it is the angular displacement / elapsed time.**

*Angular velocity*ω = ∆θ/∆t

∆θ = ω∆t

ωf = ωi + α•∆t

** Angular acceleration**: The amount that the rate at which angular velocity changes over some period of time. That is to say, it is the change in angular velocity / elapsed time.

Since we are using a for linear acceleration let's use a Greek a for angular acceleration—α

α = ∆ω / ∆t

∆ω = α•∆t

## Not Really A Secret!

So abut that final displacement thing…

Where *X* is an amount thing (displacement, velocity, cookies, money, pizza )and *∆t* is time and *r* is rate the thing changes…

and where the subscript *f* is final amount and *i* is initial amount…

Xf = Xi + r∆t

**EXAMPLE**: Bob has 4 things. Every minute he gets 2 more things. How many things will Bob have after 5 minutes.

Where X = things…

Find Xf where…

Xi = 4 things

r = 2 things/minute

∆t = 5 minutes

Xf = Xi + r∆t

Xf = 4 + 2•5

Xf = 4 + 10

Xf = 14

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