Torque, moments and angular momentum
Conservation of angular momentum Angular momentum is constant when there is no net torque.
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- We learned in the videos on torque that torque equals--
- let me actually draw a picture, so you remember what
- we're talking about with torque.
- So let's say-- so that's the arm, let's say right here is
- what its pivot is.
- That's its pivot.
- And let's say that I'm putting a force right there, and it's
- perpendicular to this arm.
- And let's say the length of this arm is r.
- And this force is f.
- So we know that the torque is this force times
- the distance, right?
- And then we also know that-- what is force?
- Well, that is equal to mass times
- acceleration times distance.
- So torque is equal to mass times
- acceleration times distance.
- And then so what is acceleration?
- Well that's equal to mass times change in velocity over
- change in time, times distance, right?
- So we learned all that from our torque chapter, and you
- might want to review it, just to get an intuition of what
- torque is good for.
- But in general, if something isn't spinning, you apply
- torque, and you'll get it spinning.
- Or if something is spinning already, if you apply torque
- in the direction that it's spinning, it'll spin faster.
- Or if you go in the opposite direction, it'll spin slower.
- And what I'm showing you here, is that what happens if you
- apply no torque?
- Well if you apply no torque, then we know that this
- quantity is 0.
- Or another way to think about it-- actually why did I write
- d, it could be d, but I shouldn't have called this r,
- it should be d.
- So if this is 0-- if we are applying no torque, right--
- what do we know?
- We know that the change in velocity over change in time,
- times this distance, won't change-- that
- this quantity is 0.
- So we know that the velocity times the distance is going to
- be a constant.
- And that comes from what I just talked about.
- It falls out of Newton's laws, but it applies to spinning.
- An object that's not spinning will tend to not stay
- spinning, and an object that is spinning will
- tend to stay spinning.
- So in this case, if this object at this point right
- here-- so we're in a case where there's no torque, so
- this force is 0-- there's no force applying here.
- And whatever this object's velocity was-- its tangential
- velocity-- it's going to stay at that velocity, right?
- It's just going to keep spinning at that velocity.
- If I apply more torque it'll go even faster, if I apply
- less torque, it'll slow down a little bit.
- But we know that this velocity times
- the distance is constant.
- And actually, I don't know why I took this m out-- we know
- that the mass times the velocity times the distance,
- is constant.
- Right?
- So what does that tell us?
- Well, we learned in the angular velocity video-- my
- mind's a little slow today-- that what?
- Angular velocity is equal to velocity divided by the
- radius, and this case, the radius is this distance, so we
- could also write it as velocity over distance.
- And when I talk about radius, it's just the radius of the
- circle that you're spinning around in, right?
- This could be the circle, up here.
- So this d, that's the same thing as the radius.
- I'm just switching letters to confuse you.
- But let's see if we can write something, if we can change
- this expression to include angular velocity.
- You'll see where I'm going in a second.
- So let's solve for v.
- So let's multiply both sides of this, times d.
- So you get dw is equal to v.
- Right, I just took this d, put it on this side.
- So let's write that here-- m times dw-- right, I just
- replaced the v-- times d, is equal to a constant.
- Assuming that we have no net torque on the system.
- And so what does that get us?
- Well that gets us m times the angular velocity times d
- squared is equal to a constant.
- So what does this tell us?
- This tells us the mass of an object spinning-- let me
- rewrite this, because I think-- so what we know is
- that the mass of an object spinning, times how fast it is
- spinning, times the distance to the center of its
- rotation-- and actually I'm going to change that d to an
- r, I don't know why I even used d to begin with-- times
- that squared.
- That's going to be equal to a constant, assuming no net
- force-- no net torque.
- Another way of looking at that, if we just wanted to go
- to angular velocity from the get-go, we could have said
- torque is equal to mass times change in velocity, times
- change in time, times the radius to the center of where
- you're rotating around.
- And change in velocity is just the same thing as mass times
- change in angular velocity, times r.
- Change in time, and then there's another
- r, that's this r.
- Because velocity is angular velocity times r, and we're
- assuming r doesn't change, so any change in velocity is just
- going to be a change in the angular velocity.
- And then we would get the same thing that we just got here.
- If we have no net torque, we're going to have no change
- in angular velocity, so angular
- velocity will be a constant.
- So you'll get mass times angular velocity, times-- and
- then you have this r, and this r-- times r squared, is going
- to be equal to a constant.
- Where am I going with all of this?
- Well let's think about something.
- Let's say that I have some object traveling in a circle.
- I do everything for a reason, and you'll see my
- reason right now.
- Let's say that-- so let's say this is some type of
- retractable pole, and I'm an ice skater.
- And I'm not using the ice skater's body right now,
- because it'll become complicated.
- Let's say this is some type of robot arm, and that's its
- joint right there.
- And it's holding a mass out here.
- This is neat because after this concept, you'll
- understand what goes on in figure skating, and then the
- Olympics-- oh no, the Winter Olympics are
- over, aren't they?
- Anyway, let's say that this object is spinning
- around at some rate.
- It's spinning around at 10 radians a second.
- That's its angular velocity.
- And let's say, right now its rotational
- distance is 10 feet.
- So this is spinning on an ice skating rink, because you
- don't want friction and all of that.
- So what's its current angular momentum?
- So that's what this term right here is, angular momentum.
- So what's its current angular momentum?
- Well its mass times 10, times-- 10 is its-- actually,
- let's make this radius a different number, let's call
- it 8 feet, just you know what I'm doing.
- So its angular velocity is 10, and then its radius
- is 8, so times 64.
- So it equals 640 times mass.
- This is its angular momentum.
- Now what happens if this arm, for whatever reason,
- shortens-- and maybe it does something like this, the arm
- kind of bends.
- And then the mass comes here, it comes in
- closer to the center.
- I'll write that in a different color.
- The mass comes closer to the center, so now
- the radius is 4.
- But I've had no net torque on the system, all I've done is
- change how far it is from the center of rotation.
- How much faster is it going to spin now?
- Let's think about it.
- Its angular momentum won't change, this is a constant,
- which is its angular momentum.
- That won't change.
- So we now know that the mass times the angular-- the new
- angular momentum, we'll write that angular momentum 1--
- times the new distance squared-- times 16-- is also
- going to be equal to 640m.
- The angular momentum doesn't change.
- Let's cross out m from both sides, and then divide both
- sides by 16.
- We now have that w1 is equal to 16 goes into 640.
- So what happened?
- Originally, I was going around at 10 radians per second, when
- I halved the radius-- when I got half as close to the
- center of my rotation, I'm actually spinning around 4
- times as much.
- And that's because this term is a quadratic term.
- And you probably have observed this behavior before, when you
- see the ice skater skating around, and they're spinning
- with their arms wide open, and then they pull their arms in,
- and they go a lot, lot, lot, lot faster.
- And that's because their angular velocity, or the rate
- at which they're spinning, is proportional to the square of
- the radius around their axis of rotation.
- Anyway, I hope I didn't confuse you, and I'll do some
- more problems with this in the future, but I've
- just run out of time.
- See you soon.
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