Angular velocity and speed | Uniform circular motion and gravitation | AP Physics 1 | Khan Academy

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Angular velocity and speed.
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[@user-wu2rm3bh4y]

Many people wonder why radians do not appear when we have radians * meters. Here is an attempt at an explanation:
Let s denote the length of an arc of a circle whose radius measures r.
If the arc subtends an angle measuring β = n°, we can pose a rule of three:
360° _______ 2 • 𝜋 • r
n° _______ s
Then
s = (n° / 360°) • 2 • 𝜋 • r
If β = 180° (which means that n = 180), then
s = (180° / 360°) • 2 • 𝜋 • r
The units "degrees" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r
s = 𝜋 • r
If the arc subtends an angle measuring β = θ rad, we can pose a rule of three:
2 • 𝜋 rad _______ 2 • 𝜋 • r
θ rad _______ s
Then
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
If β = 𝜋 rad (which means that θ = 𝜋), then
s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
The units "radians" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
that is, half of the circumference 2 • 𝜋 • r
s = 𝜋 • r
If we take the formula with the angles measured in radians, we can simplify
s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r
s = θ • r
where θ denotes the number of radians (it does not have the unit "rad").
θ = β / (1 rad)
and θ is a dimensionless variable, θ is in rad/rad.
However, many consider θ to denote the measure of the angle and for the example believe that
θ = 𝜋 rad
and radians * meter results in meters.
Mathematics and Physics textbooks state that
s = θ • r
and then
θ = s / r
It seems that this formula leads to the error of believing that
1 rad = 1 m/m
and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI).
In the formula
s = θ • r
the variable θ is a dimensionless variable, it is a number without units, it is the number of radians.
When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed.
My guess is that actually the angular speed ω is not measured in rad/s but in (rad/rad)/s = 1/s = s^(-1).

[@virupannamedikinal]

Really helpful.

[@TheScienceCube]

Very cool!

[@MrOhhBolts]

This would’ve been helpful last year in AP Physics lol

[@summerfun5870]

First videos very helpful!

[@chitralima2208]

Summer Fun!
Fantastic

[@Eventaulyeti]

just a minor edit: Sal, when you are counting the rotation out loud, it should start from "zero mississippi". What you counted was only two seconds long.

[@lesserknownfacts7849]

haha lol, but he explained the concepts pretty great.

[@trushalwork2984]

no he counted correct 3 seconds, counting 1 missisipi takes 1 second so he counted 3 so correct 3 seconds you noob

[@gilmoses3777]

Actually the 3-second count should go "Zero Mississippi, one, two, three". Starting with "one" and ending the movement at "three" gives us only two seconds...

[@abdullahvlog2142]

whats missisipi?

[@bhavishyasharma7834]

Just a way to waste 1 sec

[@ryanmorin6430]

wish ya'll didnt use radians.. I have no idea wtf this means