Angular velocity and speed | Uniform circular motion and gravitation | AP Physics 1 | Khan Academy
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- Опубліковано 25 сер 2024
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Angular velocity and speed.
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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).
Really helpful.
Very cool!
This would’ve been helpful last year in AP Physics lol
First videos very helpful!
Summer Fun!
Fantastic
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.
haha lol, but he explained the concepts pretty great.
no he counted correct 3 seconds, counting 1 missisipi takes 1 second so he counted 3 so correct 3 seconds you noob
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...
whats missisipi?
Just a way to waste 1 sec
wish ya'll didnt use radians.. I have no idea wtf this means