8.2 Circular Motion: Position and Velocity Vectors
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- Опубліковано 1 чер 2017
- MIT 8.01 Classical Mechanics, Fall 2016
View the complete course: ocw.mit.edu/8-01F16
Instructor: Dr. Peter Dourmashkin
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
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You don't know how much you've saved my educational life. God richly bless you.
wow..what a lecture...i salute you dr...you have talent in delivering education...thank you ...
Thank you so much!! I got it. Thanks for your cooperation.
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
in 6mins all my doubts have been erased, thank you
enjoyble explanation. thank you
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.
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).
On the web page ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-3-circular-motion/8-2-circular-motion-position-and-velocity-vectors/, you say:
"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, since θ = β / (1 rad) and θ is a dimensionless variable.
On the web page ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/mit8_01scs22_chapter6.pdf, you justify the result of equation 6.2.14 (p. 5) by saying that "the length of the chord approaches the arc length".
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.
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Very interesting demonstrantion
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.
It means that the value of theta at time t. It's because he has already mentioned that theta changes with time and isn't a constant.
Great loved it
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Beautiful
Wonderful ful and amazing explanation make more vedioes
Does θ(t) has properties of a vector?
Damn ,the lecture was great
Tysm❤️🔥
Thank you sir
Is there a video that covers the same topic but lets me find the velocity vectors in 3 dimensions?
In spherical or cylidrical?
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?
171.887
Arc length = r * angle
In 1:33 why not theta hat in cos ?
Thank you so much , saved a lot of hours of my jee preparation
Do you need to study polar coordinates for JEE?
@@dimlighty Better if you study.
same here mate, 2026 aspirant here, just a question, did u ever use polar co ordinates in rotational/ circular or used the normal approach taught in coaching?
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?
Yes same confusion with me. I have commented before. I'm agree what you said. Actually you put question mark but I'm sure it is.
Theta(t) is the angular displacement over time t. Yes!!
No, For decomposition of a vector you need only the magnitude of that theta.
i think he is writing on mirror
his teaching is super and fast
Is angular displacement a vector quantity ?
Yes, there's a reason we write r cap.
Super
What is he drawing on? And is he writing everything backwards so it appears normal to us?
He is writing on a big piece of glass. You can find out information about how lightboards work here: lightboard.info/. The technology used in the MIT Lightboard interactively flips the image and adds (composites) auxiliary images and video inputs into the captured video in real time, so that the instructor can immediately review the footage.
My god
You spin me right 'round, baby, right 'round
Like a record, baby, right 'round, 'round, 'round
Let's take a moment to appreciate this guy is writing backwards on glass...!
i think they flipped him after recording him
@@starryepidemic2532Oh!!! That's gruesome 😮😮😮 MIT SUCKS
Jigsaw
U went kind of strapped, is more simple than this
But what r vector represents?
it's the position vector, describes the position of a point in relation to the origin