8.2 Circular Motion: Position and Velocity Vectors

MIT 8.01 Classical Mechanics, Fall 2016
Instructor: Dr. Peter Dourmashkin

License: Creative Commons BY-NC-SA

[paulproofmath]

You don't know how much you've saved my educational life. God richly bless you.

[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.

[mohfa]

wow..what a lecture...i salute you dr...you have talent in delivering education...thank you ...

[ajaikumar]

It take 20 min for my teacher to explain this... Unfortunately I don't understand them... But ur explanation is sharp and crisp and very good to understand... Thank u sir

[SM_Int.M.S]

Needs more explanation in the part: r hat(t)= cos theta (t) (i hat)+...
Why cos or sin (theta t)??
Did you mean time t is with theta? Or just multiplied with? Theta (t) is angular displacement over time t.

[sarthakmonga]

Thank you so much, saved a lot of hours of my jee preparation

[changtillend]

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

[abcddd]

2:54 is that equation supposed to be r̂(t) = cos( θ(t) )*î + sin( θ(t) )*ĵ, where θ(t) is a function which gives the angle at any given time and r̂(t) is the r̂ vector at any given time?

[marksmod]

You spin me right 'round, baby, right 'round
Like a record, baby, right 'round, 'round, 'round

[SM_Int.M.S]

Thank you so much!! I got it. Thanks for your cooperation.

[s.u.]

Is there a video that covers the same topic but lets me find the velocity vectors in 3 dimensions?

[mihrilo]

Let's take a moment to appreciate this guy is writing backwards on glass...!

[anuradha]

i think he is writing on mirror
his teaching is super and fast

[Enterprises-zh]

Wonderful ful and amazing explanation make more vedioes

[younisalmughaizwi]

In case of circular motion, at what value of angle (in degree) the distance travelled is equal to the thrice of radius of the circle?

[koustubhjain]

Is angular displacement a vector quantity ?

[dirtybigmike]

What is he drawing on? And is he writing everything backwards so it appears normal to us?

[ruffnck]

U went kind of strapped, is more simple than this