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### Course: AP®︎/College Physics 1 > Unit 5

Lesson 4: Torque and angular acceleration# Torque Basics

A torque is the rotational equivalent of a force. When a torque is applied to an object it causes the object to rotate. Learn how forces create torques and how to calculate the net torque on a rotating object.
Created by David SantoPietro.

## Want to join the conversation?

- For the last example - if we set up the x and y axis with the pivot point as the origin, the two left forces would have negative distances, and the 8N force would be negative? This would still gave us the same result, but is more consistent with x and y values...(1 vote)

## Video transcript

- [Instructor] Imagine
you've got a door here with a blue door knob. Any one of these 10 newton forces will cause the door to rotate
around the hinge or the axis, or sometimes this is
called the pivot point. Any one of these forces will
cause the door to rotate. My question is, if you
could insert one of these in one of these locations, which one of these forces, if any, would cause the most angular
acceleration of this door? And you might think, "Oh,
well, 10 newtons is 10 newtons. They'll all cause the same
amount," but that's not true. It turns out we put door knobs at the end of doors for a reason. This red 10 newtons at the outside edge will cause the most angular acceleration. It'll cause this door to
speed up most rapidly. And this used to bother me. I was like, "How come this
is getting an advantage?" I think the best way to
think about it is this. Even though these forces
are all going through the same angle, so they've
gone through 20 degrees, and now they've all
gone through 30 degrees, now they've all gone
through 45, 60 and 90. Even though these forces have all gone through the same angle, they have not gone
through the same distance. Some of these forces have been exerted through
a larger distance. So just look at it. If you imagine rotating this thing, that red force, this
outside pink force here goes through a much larger distance than that inner yellow force. This force has gone through
very little distance whatsoever. And you might think, well,
why does that matter? Well, it matters because
if you remember work done. Work done is proportional
to the amount of force, but these are all 10 newtons so that doesn't really matter here. And it's also proportional
to the amount of distance through which that force is applied. And because this outside force has gone through so much more distance than these inner forces, it's done more work over the same angle. And if you do more work, you input more kinetic
energy into the door, it's gonna be moving
faster for the same angle compared to what's caused
by these other forces here. And this is why in angular mechanics you can't just think about forces, you have to think about
something called torque. The symbol for torque is this fancy T. It's the Greek letter tau. And the amount of torque
caused by a force, so you need a force to cause a torque, but it's more than just force. You have to multiply r, the distance from the access to the force by the amount of force in order to find how much
torque is being exerted by a given force. The more torque that's exerted, the more angular acceleration you'd get, the faster you'd get
something to speed up. Now you might wonder like, "Okay, I get that more
force gives me more torque, how come this is just r
and not like r squared? It seems kind of random, maybe
its like square root of r." Well, if you remember arc
lengths from back in the day, arc length is r times theta. So if I'm twice as far away from an axis, I get twice the arc length. If I get twice the arc length,
I get twice the work done. You get twice the work done, you get twice the input kinetic energy and it turns out twice the kinetic energy will give you twice the
angular acceleration. This is why everything
is just proportional to r in terms of torque, it's
not like r squared here. So for example, let's just say
the distance from the axis, because that's what matters, to this 10 newtons here was one meter. And from the access to the
purple force was two meters. And from the access to this
doorknob force was three meters. What this torque formula
means is that even though these are all 10 newtons, they'd all be exerting
different amounts of torque. I'd have to take the one
meter times 10 newtons would give me 10 newton meters. So the unit for torque
is meters times newtons, but we usually write it as newton meters. If you buy a torque wrench, you could set it in newton
meters or in foot pounds, if you're doing the US system. And then this purple force,
even though it's 10 newtons, you'd have to take two
meters times 10 newtons. This would exert a torque
of 20 newton meters and the doorknob force wins the battle because it would have three
times 10 would exert a torque of 30 newton meters. So the same size force can
exert a different amount of torque, depending on how
far away it is from the axis. So one area you have to
be careful of this torque technically is a vector. It has a direction, it could be positive or negative. If you're doing full-blown
engineering 3D physics, technically these torques would
point out of the screen here out of the page, but for
intro algebra-based physics, and for most problems, you
can usually get away with just considering
counterclockwise or clockwise as being the direction of the torque. That is to say these forces
were making this object rotate in the counterclockwise direction, so they were all at the same sign. The convention is to call
counterclockwise positive. So we'd call these all positive. If there were any forces that
tried to rotate the system clockwise, you'd call
those torques negative. You can do it either way, as
long as you're consistent. Most books pick this as
the convention though, so you should be aware of that. And then the last little
bit to be careful about, I'm drawing all these forces
nice and perpendicular to the r, and if that's the
case, you just do r times F. If your force has different components, you need to make sure
that the only component you plug in here is the
perpendicular piece. So if this had some weird angle here, you'd only want the piece
that was directly into this perpendicular lever arm
here at a perpendicular angle. We'll talk about that more later. For now, let's just try some
problems to kick the tires and get used to this formula. So imagine this example here where you've got the fancy door, you know, with a fancy hotel or restaurant that's a rotating circle and you can go in from either direction. This would be a bird's-eye view. Now, imagine you go into the hotel, you're pushing over here, you took physics, you know what to do. So you exert this 20 newtons over here. Let's say someone else comes
in from the other edge. It's all awkward and they're
trying to go in the other way and it's a stalemate. You're both pushing with
forces, but nothing's happening. And that doesn't mean
the two forces are equal. If you're in a stalemate here
in terms of angular motion, that means your torques
are equal and opposite. They're opposed, they
have the same magnitude, but they'll have opposite
directions of torque. So if you're locked in a stalemate here, that means the torque that
you exert with your 20 newtons has to be equal to the
torque from the other person. So let's try to figure out how much force would this person have to exert? It's not gonna be 20 newtons. They're pushing closer to the axis here so they're gonna have
to push with more force. How much more force? Well, we can use the formula
for torque to find it. The torques have to be equal. If there's no rotation
here, you're balanced out. If your force is 20 newtons, you're exerting a force three
meters away from the axis. That's your r, would be three meters times 20 newtons means you're
exerting 20 times three, so 60 newton meters of torque. That means the other
person has to be exerting 60 newton meters of torque,
but there r isn't two. Be careful here, you always
have to measure from the axis, the point where you're rotating about. That'd be one meter. This door's all symmetric here. So it'd be one meter times F. And if you take this 60 newton meters and you divide by one meter, you're gonna get that this force here is gonna have to be 60 newtons. So this person is gonna
have to exert more force. In fact, they pushed three
times closer to the axis, so they're gonna have to exert three times the force that you do. You have a three times advantage
here in holding this door compared to the other person. All right, let's try one more just to make sure we understand it. Let's say it's now rush hour, you know, bird's eye view here, same circular door. Three people are trying
to go through at once. It's gonna be a mad house. This time I want to know,
it's not gonna be a stalemate. This door is gonna
rotate in some direction. I want to know what the net torque is. So just like you can find net force, you can find the net torque,
but you gotta be careful. These might have different signs, you gotta add or subtract accordingly. So start over here. How much torque would
be from this 10 newtons? Well, it's exerted three
meters away from the axis, so it's r is three meters. So the torque from that
force would be three meters times 10 newtons, and
since this is directed counterclockwise, I'm just
gonna call that positive and I'll have to be
consistent with that choice. So now let's consider this eight newtons. You might think it would
have an oppositely directed sign of torque from this 10 newtons 'cause the eight is down, the 10 is up, but it's also trying to rotate this door in the counterclockwise direction. So in terms of forces, this 10 newton and eight
newton are oppositely directed, but in terms of torques,
they are the same direction. They're both causing
rotation counterclockwise. So if I called this torque
from the 10 newtons positive, I've gotta call the torque from
this eight newtons positive 'cause it's trying to exert a
torque in the same direction. So I'd have one meter
is the r for the eight times eight newtons would be the torque from the eight newtons. And then I have one more force here. This five newton is trying
to rotate clockwise. Since I called counterclockwise positive, I'm gonna have to make
this a negative torque, so minus three meters,
the r from the access to this five newtons is three meters multiplied by five newtons. And if you take 30 plus eight minus 15, you're gonna get a total of positive 23 newton meters of torque, so this is not a stalemate. There will be an amount
of angular acceleration caused by this net torque. So to recap, just like net forces can cause regular acceleration, net torques can cause
angular acceleration. If there is no net torque, that means there is no
angular acceleration. The way you find the
torque from a given force is you take r, the distance from the axis to where that force is
applied and you multiply by the amount of force, as long
as it's that amount of force that runs perpendicular to this lever arm or this r direction. Be careful that torque is a vector. We typically count
counterclockwise as positive and clockwise as negative,
but if you're consistent, you can call whichever one of these you want to be positive as long as you call
the other one negative.