Trigonometry - Working with linear and angular speed

[vladusa]

IMPORTANT NOTE: Radians are unitless, so you can multiply them with anything!

[noahroland]

Thank you so much. been trying to figure this out after a few hours, this video helped me a ton.

[sharond]

I was so confused about this topic and watched this video the morning of my test and took notes.I ended up getting a 17.5/26 but then she curved the test since most people misread the word problems so 17.5/22 good enough for me 😬

[JoséAntonioBottino]

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, the number of degrees), then
s = (180° / 360°) • 2 • 𝜋 • r

The units "degrees" cancel out and the result is
s = (1 / 2) • 2 • 𝜋 • r
s = 𝜋 • r

that is, half of the circumference 2 • 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, the number of radians), 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
s = 𝜋 • r

that is, half of the circumference 2 • 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 [rad/rad = 1].

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
rad • m = m

since, according to them, the radian is a dimensionless unit. This solves the problem of units for
them and, as it has served them for a long time, they see no need to change it. But the truth is
that the solution is simpler, what they have to take into account is the meaning of the variables
that appear in the formulas, i.e. θ is just the number of radians without the unit rad.

Mathematics and Physics textbooks state that
s = θ • r

and then
θ = s / r

It seems that this formula led to the error of believing that
1 rad = 1 m/m = 1

and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality
θ = 1 m/m = 1

and knowing θ = 1, the angle measures β = 1 rad.

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).

[andy_]

at 3:20, where did that "Radian Per second" come from if u didnt have it next to the 120???

[SolorioJuan]

how did you go from cm per sec to rad per sec??? How did cm become rad??

[Deus]

Where I get confused are the units. I'll be given 2 very basic units such as degrees per min and the radius, and be expected to find revolutions per hour. That makes figuring out whether I have angular speed or linear speed to be a pain.

[tokkik]

How do you know when to use linear or angular speed??

[emilycorona]

With 25800 pi miles over 63360 hrs did you divide ? Multiply ? To get aprox 12.79247 mph ????
You didn’t explain the step & it lost me

[BEJunki]

Why did you drop the radians off 14(radians) per second at 2:05? Don't you need to convert that?

[Nylrechen]

Can you please make a tutorial vedio about how to solve a Relationships between the linear and angular measures of a central angle in a unit center with word problems and solution please 🥺🥺🥺🙏🙏

[krystalyan]

A pi is supposed to be timed for the first question.