Torques
Transcript
Now we can talk about torque and the idea of torque. So, the first thing I'll say, is that when a force is applied at the center of mass of an object, or in line with the center of mass, it will simply accelerate the object. So imagine if someone is tackling you and they hit you right on your mid line you are going to go straight backwards, probably fall on your keyster.
Similarly suppose a car is stopped at an intersection, say making a left turn or something like that and another car comes and t-bones it, hits it right at the center of mass. Well, what's gonna happen to that car is that it's just gonna slide directly back in the direction in which it was hit. And it probably, it may even fold a bit or something like that, but it's primarily gonna be linear motion.
By contrast, the force applied offline from the center of mass can also twist or spin the object in addition to moving it. Suppose someone is tackling you, and you partially step out of the way, and you get hit, say, in the shoulder, well among other things its going to spin you around. At least its going to turn you. And much in the same way, the car parked at the intersection.
A car going at a high speed hits it, right at its front, or right at its back. Well, part of what's going to happen is that parked, stopped car that gets hit at the front or the back is going to spin around. And in fact it may spin around several times depending on how hard it was hit. And so that's the idea here. We can also use an offline force to generate leverage.
This is another big idea. A lever is a simple machine that uses such a principle. And I'll assume that you have the basic idea of levers. Any upward force, in line with the box's center of mass would have to be as large as the box weight, so would have to be at least as large as capital M times g. By contrast, by using a long lever arm, we can apply a force that is far offline from the vector of the box weight, and the further we are, the less force we need to use.
You may be familiar with Archimedes' famous statement, give me a lever arm long enough and a firm place on which to stand and I will move the Earth. This idea that there's no upper limit as long as you have a lever arm long enough, you can generate any kind of force with that lever arm. The idea of force not in line with the object's center of mass but off line as if out some lever arm, leads to the more general idea of torque.
A torque is a way to express the leverage that a force has precisely because it acts at some radius from its target. The general formula for torque T, is Torque equals force times sin theta. So, lets talk bout that formula a little bit. So, here is the diagram, Force, F is simply the force, which as vector naturally has a direction.
R is the displacement vector from the center of rotation to the point where the force is applied. Often that center of rotation is the center of mass of the object. And theta is the angle between those two. So let's think about that angle. What's going on with that angle?
When that angle is 90 degrees, when the force is perpendicular to the radius arm, that produces the maximum possible torque for that particular force and that particular radius. When the angle is zero or 180 degrees. That means that the force is in line with the radius. And so there may be a linear force on the object at that point, but there's no torque.
There's no spinning force. So the sine is zero. Remember sine of 0 is 0, and also sine of 180 is 0, so that's when the torque is 0. Torques are often used in problems involving levers or anything that is balancing.
So for example, here is an example of a torque problem. We have 3 masses that are on a seesaw kind of thing, on some kind of balance, each one has a different mass And a different distance. And we're pressing down with this force f. And we want to know, how much do we have to press down in order to hold everything in balance?
We wanna keep that in balance. So how much do we have to press down at that location? 10 meters away from the fulcrum on the right side. Well, the way we're gonna do this is balance all the forces. We're gonna say that the torque counter-clockwise equals the torque clockwise.
So the counter-clockwise torque, that is, the weight of 50 Newtons times 12 meters, plus the weight of 200 Newtons times 3 meters, those are the two weights on the left side producing a counter clockwise torque. On the right side, we have the weight of 120 Newtons pressing down with 5 meters, and then we have 10 meters times F we don't know the value of F.
And so, what we get is 600 Nm + 600Nm = 600 Nm times 10 meters times F So 600 Newton meters equals 10 meters times F divide by 10 meters and we get F equals 60 Newton. So it would have to press down with 60 Newtons to hold that in balance. Here's a practice problem. Pause the video and then we'll talk about this.
Okay, so in this problem, what we have is kind of a primitive model of the lever action happening in the human forearm. And so the biceps brachii can be modeled as follows. The fulcrum is the elbow, point B is the attachment of the biceps. And this exerts an upward force 4 cm away from the fulcrum. And of course, all this would be inside the flesh of the upper arm of course.
A 20 kg weight in the hand produces a downward force. So, for the purpose of this modelling we're actually ignoring the weight of the forearm itself. We're just gonna worry about the 20 kilogram weight in the hand and this produces the downward force A. And this is 30 centimeters away from the fulcrum.
And we wanna know what force the bicep has to exert at B to balance the weight at A. So this is a torque problem. Again, we're gonna say if we're just balancing, means that the torque counter-clockwise equals the torque clockwise. The counterclockwise torque from this perspective would be the 200 Newton force at a distance of 30 centimeters.
And so here everything's given in centimeters, so I'm just gonna leave it in centimeters rather then change it to meters and get a bunch of decimals. So I have 20 Newtons times 30 centimeters and then this equals the clockwise torque which is provided by the biceps. This is 4 centimeters times F, the force from the biceps. Well, notice that I could divide by 4 centimeters, and I could make things much easier.
I'm gonna divide 200 by 4, well, that's easy, then I'll get 50 Newton. And I'll divide centimeters by centimeters just to cancel the centimeters. Well, then I've made my life much easier cuz now it's just 30 times 50 time Newtons. And so that's gonna be 1500 Newtons. And so that's the force that the biceps have to exert.
Notice that it's a much, much higher force. And of course this is the value of doing curls, for example. The bicep is exerting a force that is many times greater than the weight in the hand and so here, the answer is C. In summary, torque involves a force that acts off-line from a center as if it's along some lever arm for example.
We have the formula for torque, and of course we remember that we get the maximum torque when the force and the radius are perpendicular, and we have zero torque when the force and radius are in line, when they're parallel. And when the torques have balanced, the counter clockwise torque equals the clockwise torque.
Read full transcriptSimilarly suppose a car is stopped at an intersection, say making a left turn or something like that and another car comes and t-bones it, hits it right at the center of mass. Well, what's gonna happen to that car is that it's just gonna slide directly back in the direction in which it was hit. And it probably, it may even fold a bit or something like that, but it's primarily gonna be linear motion.
By contrast, the force applied offline from the center of mass can also twist or spin the object in addition to moving it. Suppose someone is tackling you, and you partially step out of the way, and you get hit, say, in the shoulder, well among other things its going to spin you around. At least its going to turn you. And much in the same way, the car parked at the intersection.
A car going at a high speed hits it, right at its front, or right at its back. Well, part of what's going to happen is that parked, stopped car that gets hit at the front or the back is going to spin around. And in fact it may spin around several times depending on how hard it was hit. And so that's the idea here. We can also use an offline force to generate leverage.
This is another big idea. A lever is a simple machine that uses such a principle. And I'll assume that you have the basic idea of levers. Any upward force, in line with the box's center of mass would have to be as large as the box weight, so would have to be at least as large as capital M times g. By contrast, by using a long lever arm, we can apply a force that is far offline from the vector of the box weight, and the further we are, the less force we need to use.
You may be familiar with Archimedes' famous statement, give me a lever arm long enough and a firm place on which to stand and I will move the Earth. This idea that there's no upper limit as long as you have a lever arm long enough, you can generate any kind of force with that lever arm. The idea of force not in line with the object's center of mass but off line as if out some lever arm, leads to the more general idea of torque.
A torque is a way to express the leverage that a force has precisely because it acts at some radius from its target. The general formula for torque T, is Torque equals force times sin theta. So, lets talk bout that formula a little bit. So, here is the diagram, Force, F is simply the force, which as vector naturally has a direction.
R is the displacement vector from the center of rotation to the point where the force is applied. Often that center of rotation is the center of mass of the object. And theta is the angle between those two. So let's think about that angle. What's going on with that angle?
When that angle is 90 degrees, when the force is perpendicular to the radius arm, that produces the maximum possible torque for that particular force and that particular radius. When the angle is zero or 180 degrees. That means that the force is in line with the radius. And so there may be a linear force on the object at that point, but there's no torque.
There's no spinning force. So the sine is zero. Remember sine of 0 is 0, and also sine of 180 is 0, so that's when the torque is 0. Torques are often used in problems involving levers or anything that is balancing.
So for example, here is an example of a torque problem. We have 3 masses that are on a seesaw kind of thing, on some kind of balance, each one has a different mass And a different distance. And we're pressing down with this force f. And we want to know, how much do we have to press down in order to hold everything in balance?
We wanna keep that in balance. So how much do we have to press down at that location? 10 meters away from the fulcrum on the right side. Well, the way we're gonna do this is balance all the forces. We're gonna say that the torque counter-clockwise equals the torque clockwise.
So the counter-clockwise torque, that is, the weight of 50 Newtons times 12 meters, plus the weight of 200 Newtons times 3 meters, those are the two weights on the left side producing a counter clockwise torque. On the right side, we have the weight of 120 Newtons pressing down with 5 meters, and then we have 10 meters times F we don't know the value of F.
And so, what we get is 600 Nm + 600Nm = 600 Nm times 10 meters times F So 600 Newton meters equals 10 meters times F divide by 10 meters and we get F equals 60 Newton. So it would have to press down with 60 Newtons to hold that in balance. Here's a practice problem. Pause the video and then we'll talk about this.
Okay, so in this problem, what we have is kind of a primitive model of the lever action happening in the human forearm. And so the biceps brachii can be modeled as follows. The fulcrum is the elbow, point B is the attachment of the biceps. And this exerts an upward force 4 cm away from the fulcrum. And of course, all this would be inside the flesh of the upper arm of course.
A 20 kg weight in the hand produces a downward force. So, for the purpose of this modelling we're actually ignoring the weight of the forearm itself. We're just gonna worry about the 20 kilogram weight in the hand and this produces the downward force A. And this is 30 centimeters away from the fulcrum.
And we wanna know what force the bicep has to exert at B to balance the weight at A. So this is a torque problem. Again, we're gonna say if we're just balancing, means that the torque counter-clockwise equals the torque clockwise. The counterclockwise torque from this perspective would be the 200 Newton force at a distance of 30 centimeters.
And so here everything's given in centimeters, so I'm just gonna leave it in centimeters rather then change it to meters and get a bunch of decimals. So I have 20 Newtons times 30 centimeters and then this equals the clockwise torque which is provided by the biceps. This is 4 centimeters times F, the force from the biceps. Well, notice that I could divide by 4 centimeters, and I could make things much easier.
I'm gonna divide 200 by 4, well, that's easy, then I'll get 50 Newton. And I'll divide centimeters by centimeters just to cancel the centimeters. Well, then I've made my life much easier cuz now it's just 30 times 50 time Newtons. And so that's gonna be 1500 Newtons. And so that's the force that the biceps have to exert.
Notice that it's a much, much higher force. And of course this is the value of doing curls, for example. The bicep is exerting a force that is many times greater than the weight in the hand and so here, the answer is C. In summary, torque involves a force that acts off-line from a center as if it's along some lever arm for example.
We have the formula for torque, and of course we remember that we get the maximum torque when the force and the radius are perpendicular, and we have zero torque when the force and radius are in line, when they're parallel. And when the torques have balanced, the counter clockwise torque equals the clockwise torque.
