8.3 Angular Velocity

MIT 8.01 Classical Mechanics, Fall 2016
Instructor: Dr. Michelle Tomasik

License: Creative Commons BY-NC-SA

[simon]

Weird, so many people using that glass plane are left-handed...
It actually is remarkable that she thought about doing the right-hand rule with her left hand to correctly demonstrate it with this mirrored lecture technology.

[Joseph-exsf]

Thank you for these lectures. They are incredibly helpful!

[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).


"Radians
One way to measure an angle is in radians. A full circle has 2𝜋 radians.
This week, we will use radians to measure the angles, so all angles will have units of radians, angular velocity will have units of radians/s, and angular acceleration will have units of radians/s^2.
If we multiply these by a distance, such as r, the units will be m, m/s, or m/s^2".

My guess is that actually the angular speed ω is not measured in rad/s but in (rad/rad)/s = 1/s, and the angular acceleration is not measured in rad/s^2 but in (rad/rad)/s^2 = 1/s^2.

If we say that the measure β of the angle is θ radians, we mean β = θ rad, and θ is the number of radians (it does not have the unit "rad").
For emphasis we can say that θ is measured in rad/rad = 1, since θ = β / (1 rad) and θ is a dimensionless variable.

This means that you use the equation s = θ • r, without taking into account that in it the variable θ is dimensionless.

What I consider a mistake, is present in the literature, it is not only in those web pages.

[seanhunter]

Funny to mention the observers on opposite sides of the plane seeing clockwise or counterclockwise rotation when we are on the opposite side of a glass lightboard and she has just said clockwise and counterclockwise the opposite way round to how they would appear on her side (so that when the film is reversed the writing is not in mirror writing for us).

[Ben-jkhd]

I understand the math but the introduction of angular velocity as a concept seems groundbreaking... It would have been nice to devote more time to the motivation behind angular velocity and what it represents - what do we gain (other than being able to account for two different perspectives looking at a plane). Thank you for the great instruction.

[mustafaumutozdemir]

this was the trippiest one for me :D she probably had to use her left hand to do the right hand rule

[julienscardigli]

wow, that time, she had to say "into the board" while for her it was out of the board, right?

[BoZhaoengineering]

I like the SE version very much because it is based on mathematics which makes physics ideas very clear.

[changtillend]

Thank you so much... This is Very Very Helpful Who need this...

[insearchofpeace]

If an object undergoing circular motion has linear velocity in theta hat direction and an angular velocity in the perpendicular direction to linear velocity direction, then wouldn't that object go into a spiral?

[bhoxzivanlangnamanpfhoe]

what is the difference between postdoc and phd

[catsonair]

Why is the K hat dropped when she equates it to velocity, wouldn't be ( r * W_z * K^ * theta^) ?

[giovatronic]

My teachers have always taught me this definition of angular velocity and i think it works if we want to know the turnning sense, but does it really works? i mean, that's all right but it doesn't look like if it had a real mathematical argument behind. looks arbitrary, usseful but not formal.
somebody knows something deepier about it??
(im sorry if i didn't write very well, i don't speak english)