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### Course: Mechanics (Essentials) - Class 11th > Unit 9

Lesson 1: Why do trains stay on track?# Radius comparison from velocity and angular velocity: Worked example

Predicting which spinning disc has a larger radius from angular velocity and the linear velocity of a point on the edge.

## Want to join the conversation?

- So Basically (Circumference Equation):

Arc length = Radius × Angle in radians

We know that arc length can be in the unit of meters because when we multiply by a radius of 4 meters, we get the linear distance of the arc length.

Now just change displacement for angular velocity:

Arc length = Radius × (Angle in radians / seconds)

That will mean that we will get a linear velocity.

Amazing.(2 votes) - I'm getting the idea that angular speed is equal to linear speed. So if angular speed equals linear speed because they are the same thing then:

Ang. Speed = Lin. Speed

Ang. Speed = |w|r

Lin. Speed = |v|

And from here:

|w|r = |v|

And so:

|v| = |w|r

|w| = |v|/r

r = |v|/|w|

Also: r in mathematics has a relation to pi as a ratio and determines the magnitude of some triangle formed by and angle theta.

If I remember geometry correctly:

C = 2πr

A(circle) = πr^2

So:

π = C/2r

π = A(circle)/r^2

And:

r = C/2π

Just how:

r = |v|\|w|

My point on this part is that radius involves a ratio-like relationship between the mathematics and physics side of the equation.(2 votes) - arent omega written as Ω(1 vote)
- Ω is capital or uppercase omega, whereas ω is lowercase omega(1 vote)

- But here the velocity os V not speed. How did we take V as speed?(2 votes)
- Angular speed is arc length traveled/time (so basically distance), and angular velocity is angular velocity/time. The "v" in this case is linear velocity, while the omega is angular velocity.(1 vote)

- I don't understand why the magnitude of angular velocity * radius gives the magnitude of velocity. Can someone explain?(0 votes)
- We know that the angular velocity is equal to the angular displacement/change in time. Also, the speed is equal to linear displacement/change in time, and we know linear displacement = magnitude of angular displacement*radius of the circle. After substituting, we get speed = (magnitude of angular displacement*radius)/change in time. But, magnitude of angular displacement/change in time = magnitude of angular velocity. So speed = magnitude of angular velocity*radius.

Sal proves it here: https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics-motion-in-a-plane/uniform-circular-motion-introduction/v/angular-velocity-and-speed(1 vote)

- Would have been clearer if you wrote Velocity = S/Delta T

Then wrote V = W*R(0 votes) - A bit confused as to how you can relate angular velocity and velocity to radius. Can't they both be spinning at the exact same speed making it harder to determine the radius?(0 votes)
- Why would it be difficult to determine the radius if they are moving at the same speed? They are two separate values derived from separate things. Angular velocity from the change in angel over time and velocity from the change in displacement over time.(0 votes)

## Video transcript

- [Instructor] We are
told a red disc spins with angular velocity omega,
and a point on the edge moves at velocity V. So they're giving us angular velocity and also you could view
this as linear velocity, and they are both vectors,
that's why they are bolded. A blue disc spins with
angular velocity two omega, so that's twice the angular velocity, so its angle is changing twice as fast, with a point on the edge
moving at velocity 2V, so the linear velocity is also twice, twice the linear velocity of the red disc. And they ask us which
disc has a larger radius. So pause this video and
see if you can answer that before we work through it together. So let's first, let's
just visualize things. So if this is our red disc, my
best shot at drawing a disc, this should be a perfect circle, but I can't hand-draw a perfect circle, but you get the idea. And let me draw the
radius of our red disc, right over there, I'll
call that R sub red, and we know a few things. We know that a point right
over here has a velocity V, so let me draw that. So let's say it's going
right in that direction, right at that moment, it has a velocity V, and we also know that it has
an angular velocity of omega. So our angular velocity
is omega in this case, in terms of how fast it is rotating, and then we have our blue disc. So let me draw the blue
disc, and I'm gonna draw it at an arbitrary radius,
'cause we haven't figured out how the radii relate
to each other just yet. This is, should be a
circle, but I'm trying to hand-draw it, alright. So this is the radius of the blue disc, so I will call that R
sub blue, R sub blue, and the velocity of an analogous point right over here is 2V. So I should make that
vector twice as tall. So this is 2V right over here, and it has an angular
velocity of two omega, so that tells us how fast
it is actually rotating. So how do we make the statement, which disc has a larger radius? Or how do we decide that? Well the key realization
is the relationship between the magnitude of angular velocity and the magnitude of velocity. A couple ways to think about it is, the magnitude of angular velocity, notice I didn't put an arrow
on top, so I'm just talking about the magnitude of
our angular velocity, times our radius is going to be equal to the magnitude of our velocity, or is going to be equal to our speed. Or another way of thinking about it, if you divide both sides by R, the magnitude of our angular velocity is going to be equal to the
magnitude of our velocity, or our speed, over R. Or we can say that R
is equal to the speed, magnitude of velocity, over the magnitude of our angular velocity. So in this situation right over here, for this red disc, we
could say that R sub red is equal to V over omega, and over here, we could say that R sub blue is equal to, well the magnitude of the velocity, the speed is going to
be 2V over the magnitude of our angular velocity,
is going to be two omega. Well notice, the twos cancel out, so this is just going to
be V over omega again. These two things are identical. That is equal to that. So they are actually going to
have the exact same radius. So they are the same radius.