# Torque and Equilibrium

by Jared Rovny

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00:02 We've discuss now the equations of motion just how things move on their own and then also how forces work and some important forces.

00:09 And how to deal with those forces using Newton's second law.

00:12 We are now going to move to equilibrium.

00:14 We will discuss Torque. How things move if they are rotating and then also the idea of center of gravity briefly and how that's related to the center of mass.

00:25 In this image you see here, we have two objects balanced around what we are calling a torque.

00:30 What that is' Is actually the pivot point where these two objects might make it rotate to the left or to the right.

00:37 An equilibrium in this case would be both of these objects where the same mass and the same distance from this center point.

00:44 These two centers of gravity as we call them or because these object aren't of zero size they are not just single little points.

00:50 They actually have some size to them.

00:52 So we have to ask ourselves, where are the objects acting? Where are these forces on this horizontal bar that you see? As we call that the center of this object gravity which is related as we'll see as soon to the center of mass of these objects.

01:08 But now let's discuss the torque itself.

01:10 What happen if something is actually rotating like it looks like these two masses could cause the bar to rotate around the center point? If I have some object that is fixed to the point but could rotate, if I apply the force to it like this gate you see here.

01:27 If we apply a force we can imagine that this entire bar where it undergo some sort of a rotational motion.

01:34 Now, if we are going to describe the sort of motion, we have to come up with a new way of thinking about motion and coordinate systems.

01:43 So far we've just dealt with things that are very linear, they move in straight line, whether that's one dimension or two dimensions.

01:48 So now we have to think of how we can describe a system like this that is rotating instead of just moving in straight lines.

01:55 The first place we start just as in the translational motion, we pick an origin is to pick an origin for rotational motion, we'll call this the pivot point because this is the point around which objects will pivot or rotate.

02:10 You might notice that if I applied the force to a gate like this one, it would depend on the rotational power I have around this pivot point on how far away from the pivot point I'm pushing with my force.

02:22 You can imagine this very easily with the example of say a door, you wouldn't want to put the handle of a door near the pivot point of the door.

02:29 You would always put the handle as far away from the pivot point as you can.

02:32 This is because you get the greatest torque, if you apply the force at the greatest distance from the pivot point.

02:39 We define the torque therefore as not just the force that you've applied but also relative to the distance away from the pivot point at which you've applied it.

02:48 In other words, if you apply force you get more torque the further away from the pivot point, you'll apply the force. But there is one caveat to this, you can look at this picture and imagine if I apply the force in this direction I probably wouldn't get any torque we don't expect this object to rotate.

03:02 Whenever asked, you that, that's exactly what you would say.

03:05 This object shouldn't rotate or on the pivot point.

03:07 No more than if a push a door in towards its pivot point, it would open or close and so we have to somehow edit our torque equation from just saying force times distance to something else to express this fact that the torque depends strongly on what direction the force is that I'm applying.

03:25 So the full torque equation instead looks like this, the torque equals the distance away from a pivot point times the force I apply but then times the sine of theta, and the sine of theta is just a way of telling us that if the force and the distance away from the pivot point are perpendicular, then we get the full torque, where as if I'm applying the force inwards or towards my radius, towards the distance away from my pivot point.

03:49 We have a reduced torque.

03:50 A few important things about torque, the units of torque as you can see just by looking at the equation are a force times a distance since the sine of theta doesn't have any units to it, it's just a number.

04:02 And so the unit of torque are Newton's times meters, that's a force times a distance.

04:09 Reiterating what we have about the directions of torque, notice what the sine of theta does and why it works for us in a way that it does.

04:16 If I have two forces, a torque and the, I'm sorry of a force and a distance that are in the same direction like if you took a door and pull it directly away from its pivot point, we get a torque of zero and that's because the angle between the distance and the force is zero they're pointing in exactly the same direction and so the sine of the angle, sine of zero is just zero.

04:37 And the same argument holds if the angle between these two is 180 degrees, in which case they're pointing in exactly opposite directions.

04:44 This would be like push a door towards its pivot point and of course the door won't open or close in that case either, and so again you get no torque.

04:51 On the other hand if you apply force perpendicular to the distance from the pivot point to where you're applying the force just like you would when pulling the handle of the door.

04:59 If they are perfectly perpendicular you have an angle of 90 degrees.

05:03 And the sine of 90 is one. Meaning that our torque equation simplifies to our original torque equation.

05:08 Torque equals force times the distance.

05:12 One simple way to solve some problems, sometimes this is easier when you are looking at the diagram and trying to figure out how to solve for the torque is to think about the distance R and the sine of theta as sort of a single unit.

05:26 R times the sine of theta, we call this quantity R times the sine of theta the lever arm.

05:31 Basically, what we're doing is saying instead of considering this entire distance R, let's consider the shorter distance that goes directly from my pivot point to the line of force without going off at a long, long angle.

05:43 So what you would do to find the lever arm is extend the line of force as we've done with this red dotted line here and then draw a line directly from your pivot point to the line of force.

05:53 So that they are perpendicular to each other.

05:56 If you look at this diagram, and consider the triangle that we have formed here with the radius, the line of force and the lever arm.

06:02 You can see that the length of the lever arm would be given simply by R times the sine of theta, using our trigonometry.

06:08 And so this lever arm L is R times the sine of theta and we can rewrite our torque equation as simply torque equals force times the lever arm.

06:18 And we'll see soon how sometimes the lever arm is much easier to find than thinking about the more abstract concept of where is my force and where is my distance and what is the angle between them.

The lecture Torque and Equilibrium by Jared Rovny is from the course Equilibrium.

### Included Quiz Questions

1. The distance between the center of rotation and the point which the force is applied
2. The perpendicular distance between the center of rotation and the line of force
3. The distance from the center of mass of the object being rotated to the center of rotation
4. The length of the object being rotated
5. The distance from the center of rotation to the line perpendicular to the line of force
1. When the angle between the force line and the line connecting the pivot point to the point which the force is applied is 90 degrees
2. When the angle between the force line and the line connecting the pivot point to the point which the force is applied is 0 degrees
3. When the angle between the force line and the line connecting the pivot point to the point which the force is applied is 45 degrees
4. Since torque only depends on the distance r and force magnitude F it is always the maximum
5. When the force is applied parallel to the line connecting the pivot point to the point which the force is applied
1. The lever arm is 1 meter
2. The lever arm is 2 meters
3. The lever arm is 0.5 meters
4. The lever arm is 1.5 meters
5. The lever arm is √3 meters
1. This increases the distance r between the pivot point and where force is applied to the door, increasing the torque
2. This minimizes the distance r between the pivot point and where force is applied to the door, increasing the torque
3. The lever arm and force can then be parallel, providing the most torque
4. The angle between the distance and force is then perpendicular, maximizing the torque
5. Since after pulling the doorknob it would be easier to pass through the door if the doorknob is placed farther away from the pivot point

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