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About this Lesson
 Type: Video Tutorial
 Length: 10:49
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 116 MB
 Posted: 07/01/2009
This lesson is part of the following series:
Physics (147 lessons, $198.00)
Physics: The Physics of Extended Objects (25 lessons, $35.64)
Physics: The Dynamics of Rotational Motion (4 lessons, $6.93)
This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.
Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campuswide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for NonPhysicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.
Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and realworld applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for NonNewtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.
About this Author
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Describing angular motion of some complex object is relatively straightforward. It's kinematics. You describe angle and angular velocity and angular acceleration. This is a description of what's going on when some object is spinning around. And you can write down kinematical equations if the acceleration and constant and figure out what's happening, but describing motion is not the same thing as explaining it. Think back to when we talked about kinematics of linear motion or translational motion and it was nice to do kinematics, but we really wanted to talk about ultimately was forces. What causes changes in velocity? It's a force.
So when we have some solid object that's capable of rotating, I'm very interested in not just describing this motion, but explaining it. As I watch it, I see that it was spinning with some fairly high angular velocity and it's slowing down. There's some angular acceleration. Why? That's dynamics, angular dynamics. I want the analog of F equals ma for things that are rotating.
So what could it look like? Could it just be that if you apply a force, you cause an angular acceleration? That seems reasonable. When you apply a force, it starts to spin up. And if you apply a fore, you can make it slow down. But if can't just be as simple as force is proportional to angular acceleration, because I'm applying a big force straight towards the center. It's not angularly accelerating, so this is a direct proof that force is not proportional to angular acceleration. And if I apply a force downward, it accelerates angularly, inward it doesn't. So you think, "Okay, angular acceleration is somehow related not just to force, but the direction of the force." But that's not the whole story either. Here's a down force. Let me just move that down force over here. Nothing happens, no angular acceleration. So it's not just the direction, it's where you apply the force as well. Apply a little force and you get some angular acceleration. Apply that same little force over here, same direction, same amount of force, but it's close to the middle, not much happens. It's like if you want to open a swinging door. You'd be dumb to go and push against the door right next to the hinge. You want to get as far away from the hinge as possible.
If you want twisty force  I need a word for that. If you want torque that's the physics word for twisty force, what you need is force, and it's got to be as far away from the center as possible, and you've got to have your angle  what angle do you want? If you're over here, you really want your force to be perpendicular to the radius. And it's the same everywhere. No matter where you are, the more perpendicular the force is to the radius, the more twisty force, the more torque you're going to get. If force is parallel to radius, you get no twisting at all.
So we write down torque, it's force times radius. And I'm going to argue that it's just the sin of the angle. It's the amount of the force perpendicular to the radius that you care about. I'm going to really show you geometrically that this is the right formula in just a second. This is torque and we give it the symbol , the Greek letter for t, is torque. And let me look at this situation in terms of pictures of a wrench. So it's basically the same story. Here's our pivot point. This is the bolt that we're trying to turn and we're applying a force somewhere over here. That's the red arrow. And the force is maybe at some funny angle and it's located a radius vector r away from the pivot point. What am I interested in? It's not this parallel component of force that's doing any twisting. F parallel is just stretching and sliding the wrench away from the bolt. It's this component, F perpendicular. And look at this little triangle here: F perpendicular is just F sin . So, indeed, F sin times radius is torque. So that's one way that people like to think about torque is F perpendicular times the radius magnitude. It's the component of force that's twisting things.
There's another geometrical way that's completely equivalent that I kind of like better. It takes a little bit of thinking about. Instead of looking at the parallel and perpendicular components of force, let me look at the perpendicular component of radius. What do I mean by that? Here's my pivot point, here's my force. Draw this dashed line, you can draw it in both directions. It's the line of force. It's the line that this force vector is sitting on. And go from the origin, from the pivot point, straight over towards that line, perpendicular over to that line. It's like the shortest possible distance. And define that to be r perpendicular. If you look at this little picture, this angle is , so this angle is . So, indeed, r perpendicular really is r times sin . So, once again, you can think about torque, instead of this way, as r perpendicular times s. And I happen to like that one, but you just need to stare at the pictures and convince yourself that they're all the same ways of getting at this quantity, and then decide which one you like to work with in problems the best.
Torque has units. Twisty force has units not of force. It's Newtons times meters, or meters times Newtons. And meters times Newtons? That's a joule and nobody ever says that torque has a certain amount of joules, because joule makes you immediately think energy or work, and torque is neither. It's twisty force. So you just call it Newton meters or meter Newtons.
When you're calculating torque, it's got a sign, it's got a direction associated with it. This is a, from your perspective, a clockwise torque. It's convention whether I call clockwise or counterclockwise plus or minus, but this torque has one sign and this torque has the opposite sign. So you have to look at your picture. This formula is only giving you the magnitude of the torque, and then there's this sign associated with it, which you just have to look at and think about. And there's a neat way of describing the direction of torque, because, in real life, you don't always have problems that just have a fixed axis like this. Many problems are, but sometimes you have a wheel that can spin around in space and be spinning. It can have arbitrary orientation. So let's think about the vector nature of all of these angular quantities. I mentioned earlier that when an object is spinning, it's got an angular velocity , and that also you can think of as just clockwise or counterclockwise, plus or minus. But if you're thinking about an object with arbitrary orientation, you can think of as a vector. Now, what direction does the vector point, the angular velocity vector? If a wheel is spinning like so, clockwise, there's points going down over here, points going up over here and points going sideways. The velocities of points aren't really telling me about the direction of the angular velocity. So we have a convention, it's called the righthand convention. Take your right hand, curve your fingers the way the wheel is rotating, and your thumb points in the direction of . So angular velocity of a wheel spinning this way is towards me. If I spin it the other way, the vector points towards you. The magnitude of the vector is just the usual old one, dimensional that we've been talking about.
If a wheel starts off at rest, like this one, and I start to spin it up so its angular velocity vector is towards me and increasing, what's the direction of change? It started off zero, it ended up towards me, and the change is towards me. So the direction of the angular acceleration vector would be toward me. Once you've picked this convention, all the other vectors sort of fall into line. You know exactly which way they're pointing.
What's the direction of torque? Let's think about that situation. If the angular acceleration is increasing towards me, which way is my torque? Well, I'm applying a force down and I'm thinking torque is supposed to cause angular acceleration, so I'd like my torque to be pointing towards me. And here's the trick: look at the vector r F. Remember the vector cross product, it's a defined mathematical quantity. r F has magnitude F times r times the sin of the angle between them. And so, let's define torque to be r F. Does that give me a sensible direction for the torque vector? In this case, r was this way, force was that way. Yes, indeed, thumb is pointing towards me, which is just what I was arguing is the direction it should be pointing, if it's going to make sense, if torque and angular acceleration are going to be pointing in the same direction. This is, in fact, our definition of torque and it's a vector quantity and it's nice to see it. We won't use it all that often. We'll mostly be working problems which are effectively onedimensional, clockwise or counterclockwise, plus or minus.
Rotational dynamics is what we're doing. We're talking about why does something spin up faster and faster or spin down slower and slower? What causes angular accelerations? And the physical answer is torque does. So torque is to angular acceleration as force is to acceleration. And just like F equals ma was our central formula, allowing us to understand dynamics, the why of particle motion, similarly, torque and the connection between torque and angular acceleration, Newton's Second Law for angular dynamics, is what we're going to be using to understand the why of circular motions.
The Physics of Extended Objects
The Dynamics of Rotational Motion
Torque Page [2 of 2]
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Great explaination but would have liked an example of calculations.