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## Physics library

# Relationship between angular velocity and speed

How angular velocity relates to speed. Created by Sal Khan.

## Want to join the conversation?

- when you push a heavy swinging door, why is the door harder to push open if you mistakenly push on the side closer to the hing?(as opposed to pushing on the door handle on the opposite side of the hinge)(13 votes)
- There are two way of looking at this. One has to do with the concepts of rotational motion which involve moment or moment of inertia. The other has to do with levers and work.

The easiest to explain is using the lever. You are probably familiar with a teeter-totter where you have a lever with a pivot point in the middle of the lever so that to lift 50Kg on on end takes 50kg of force on the other. If you move the pivot point to one side of the other the ration of weight(force) to balance it changes. The case of the door the pivot point it at one edge and the "weight" can be considered to be at the center of the door. The closer you are to the pivot point the harder you have to push. This comes from the amount of work that is required to move the door.

To swing the door from the closed position to fully open is the same regardless of where it is pushed, this comes from the conservation of energy. Work is defined as a force acting over a distance, if there is no movement there is no work. So if it takes X amount of work to open the door if you push on the side opposite the hinge it moves mich further than close to the hinge so since the amount of work will always be X the force will have to be greater near the hinge.

For example if a door requires 10 Nm (Newton meters the SI unit of work) to open and the edge opposite the hinge moves 2 meters when you open it it only takes a 5N force to open it but if the portion of the door near the hinge you push on moves only 0.1m then you need to use a force of 100N to open it.(33 votes)

- Is there a recommended order I should be watching these in (besides the order on the left of the screen)? I'm watching these to supplement a class I'm taking this term, so I
**am**getting it, but there seems to be very little in the way of order, as the excellent series on Cryptogaphy had.

I found an iPhone application that has a bunch of KA videos that don't seem to be on the site (Introduction to Motion 1, introduction to Motion 2, etc); The App Store had reviews that indicated that these were old videos that had been taken off the main site? I know this is getting off topic, but I'm finding this non-The-Big-Bang-Theory type of Physics much more interesting than I expected to, so I wanna learn more about it.(20 votes)- As I already answered Femke a few questions down, it would help to watch "Intro to vectors and scalars" first off, to explain the terminology. (unless you already understand it) And then Acceleration and Balanced and Unbalanced forces, working to an explanation of motion, and then Newton's Laws. After that, you can start watching the many problem solving videos :)

It's good that you're interested in physics! There's a lot more to science than just cosmology and astronomy.(13 votes)

- How does angular velocity relate to linear velocity?(10 votes)
- Linear velocity is speed in a straight line (measured in m/s) while angular velocity is the change in angle over time (measured in rad/s, which can be converted into degrees as well). Since the arclength around a circle is given by the radius*angle (l = r*theta), you can convert an angular velocity
**w**into linear velocity**v**by multiplying it by the radius**r**, so**v = rw**.(11 votes)

- why is it that angular displacement is not a vector, whereas, an infinitely small angular displacement is a vector?

please help me.(4 votes)- If you look at angular displacement as a whole,the direction keeps changing every millisecond or so ,since the path followed is curved.We can't do vector math with that.Now,imagine zooming in on the circumference of a circle.There will be a point where we have zoomed in so much that the part of the circumference we're looking at would look like a straight line.It's sorta what happens when we think roads are straight even though the surface of Earth is curved.This is what happens with infinitely small angular displacement.It can be considered a straight line,so it'll have a definite direction and will allow us to use it for vector math.

Hope that helps :)(2 votes)

- Why not just say 6.28 per seconds instead of 2pi? If there are 6.28 radians, 2pi*2pi, in a 360% circle, what is the purpose of saying 2pi? Maybe I'm just confused.(2 votes)
- 2 pi is simpler and more precise.

2*pi is not exactly 6.28.

If you think of the unit as a "pi radian" then you have 1 "pi radian" to go half way around and 2 "pi radians" to go all the way around. 1 and 2 are much nicer to deal with than 6.28.(4 votes)

- Angular velocity (w) = Radians (Angle covered) / second. At:5.53, it is written that Angular velocity = w Radians / second. Why did he add an 'w' there instead of simply writing Radians / second.(2 votes)
- That extra ω must be a mistake. Angular velocity = radians per second.

Maybe he meant that linear velocity (v) = radius (r) * angular velocity (ω).

ω = dθ/dt

v = r*ω = r*dθ/dt(3 votes)

- What is the difference between lower case omega and upper case omega?(2 votes)
- At about4:40,Sal talks about the derivative of an angle.What does that mean?(1 vote)
- A circular turn table has a block of ice placed at it's centre.The system rotates with an angular speed omega(w) about an axis passing through the centre of the table.if the ice melts on it's own without any evaporation. What is the effect on the angular speed?(1 vote)
- As the ice melts and water spreads on the turn table, moment of inertia increases and hence angular speed decreases to keep angular momentum constant.(2 votes)

- Are the following motions are same or different ? and how ?

1. Motion of tip of second hand of a clock .

2. Motion of entire second hand of a clock .(1 vote)- we can say that The tip has a unique tangential velocity but the same angular velocity as the rest of the hand

all other parts of the hand (below the tip) have less tangential velocity than the tip

every part of the hand (including tip) has the same angular velocity

make sense??(2 votes)

## Video transcript

Let's say we have some object that's moving in a circular path Let's say this is the center of the object path, the center of the circle So the object is moving in a circular path that looks something like that counterclockwise circular path--you could do that with clockwise as well I want to think about how fast it is spinning or orbiting around this center how that relates to its velocity? So let's say that this thing right over here is making five revolutions every second So in 1 second, 1 2 3 4 5. Every second it's making 5 revolutions So how could we relate that to how many radians it is doing per second? Remember radians is just one way to measure angles You could do with how degrees per second If we do it with radians, we know that each revolution is 2 pi radians If you go all the way around a circle, you have gone 2 pi radians which is really just you say you've gone 2 pi radii, whatever the radius of the circle is and that's where actually the definition of the radian comes from So if you go 5 revolutions per second and they're 2 pi per revolution then you can do a little bit of dimensional analysis. These cancel out and you get 5 times 2 pi which gets us to 5 times 2 pi gets us 10 pi radians per second And it works out the dimensional analysis and hopefully it also makes sense to you intuitively If you're doing five revolution a second, each of those revolutions is 2 pi radians so you're doing 10 pi radians per second. You're going 1 2 3 4 5, so that gives us 10, or 2 pi 2 pi 2 pi 2 pi 2 pi radians every time, you're doing it five times a second. So you're doing it 10 pi radians per second So this right here, either five revs per second or 10 pi radians per second they're both essentially measuring the same thing how fast are you orbiting around this central point? And this measure of how fast you're orbiting around a central point is called angular velocity It's called angular velocity because if you think about it this is telling us how fast is our angle changing, or speed of angle changing When you're dealing with it in two dimensions and this is typically when in a recent early physics course how we do deal with it Even though it's called the angular velocity it tends to be treated as angular speed It actually is a vector quantity and it's a little unintuitive that the vector's actually popping out of the page for this. It's actually a pseudo-vector and we'll talk more about that in the future So it is a vector quantity and the direction of the vector is dependant on which way it's spinning. So for example when it's spinning in a counterclockwise direction there is a vector, the real angular vector does pop out of the page We start thinking about operating in three dimensions And if it's going clockwise, the angular velocity vector would pop into the page The way you think about that, right-hand rule Curl your fingers of your right hand in the direction that it's spinning and then your thumb is essentially pointing in the direction that the actual vector or the pseudo-vector's gonna going We'll not think too much about that For our purposes, when we're just thinking about two-dimensional plane right over here we can really think of an angular velocity as a--the official term is a pseudo-scaler but we can include that as a scaler quantity, as long as we do specify which way it is rotating So this right over here, this 10 pi radians per second we could call this its angular velocity And this tends to be denoted by an omega a lower case omega right there Upper case omega looks like this So there's a couple of ways you could think about it You could say angular velocity is equal to change in angle over a change in time So for example, this is telling us 10 pi radians per second Or if you want to do in the calculus sense and take instantaneous angular velocity it would be the derivative of your angle with respect to time How the angle is changing with respect to time With that out of the way, I want to see if we can see how this relates to speed How does this relate to the actual speed of the object? So to get the speed of the object, we just have to think about how far is this object traveling every revolution that it's doing And what we can do right over here--let's say that this radius is r So in every revolution, it is traveling 2 pi r Let's say this is r meters. Give ourselves some units right over there So the circumference over here is going to be 2 pi r meters Let's say that the angular velocity is equal to omega radians per second And so how many revolutions is that per second? We can go backwards from what we did over here We have one revolution is equal to 2 pi radians Just to be clear, sometimes angular velocity is actually measured in revolutions per second but the SI unit is in radians per second So here I want to convert omega from radians per second into revolutions per second Radians cancel out. We are left with--we get omega over 2 pi revolutions per second We know how many meters we get for a revolution We have 2 pi r meters per revolutions So we copy and paste this So our angular velocity, if we want revolutions per second it's going to be omega over 2 pi revolutions per second Omega is in radians per second if we put it into revolutions per second omega / 2 pi revolutions per second And then let's multiply that times--we want to convert this into meters per second So how many meters do we have per revolution? Well, we're gonna travel a whole circumference per revolution, so 2 pi r meters per revolution So these two cancel out 2 pi cancels out with the 2 pi So you end up getting omega times r meters per second And just like that, we have the magnitude of the velocity or we could say the speed of the object as it goes around in a circle So what we can say is the magnitude of the velocity-- I'll specify that by v--I want to be clear. This is not vector quantity. It's not the velocity. It's the magnitude of velocity Or we can say this is the speed. It's going to be equal to omega times r So the speed is equal to the angular velocity times r I guess we could say the magnitude of the angular velocity times the radius I don't want you to be confused. I am not saying that this is a vector quantity If this was a vector, I would put an arrow right over there And if this was a vector, I would put an arrow over there then I'll be referring to the thing that's popping out of the page but here I'm talking about the magnitude of the angular velocity and so writing in words, you get speed is equal to angular velocity-- if you want to be particular, this is the magnitude of the angular velocity-- times the radius of the circle that you are going around and if you want to solve for angular velocity you divide both sides by radius and you get angular velocity Omega is equal to speed which we're using v for, divided by the radius So we can actually use this information to do other interesting things later on But hopefully this gives you a sense of how all of this stuff is related