**Concept:** Net Torque & Sign of Torque

Hey guys! In this video we're going to talk about what happens when you have multiple forces producing multiple torques on the same object. In other words, trying to get the object to spin in different directions. Let's check it out. The first thing we need to talk about is the sign of torque. You might remember that with different forces, if you have a force going in different directions like the positive x-axis and the negative x-axis, we use signs to indicate direction. Same thing happens with torques, but the direction depends on which way you're trying to spin something. If you're trying to spin something in the clockwise direction, which is direction of the clock, it's negative and counterclockwise is positive. There's two ways that you can remember this. The reason the counterclockwise is positive is because it follows the unit circle. The unit circle spins this way. It starts at zero degrees over here and it grows in that direction. Another way you can think about this is that the clock, the direction of the clock, is backwards. Clock is backwards. Those are two ways you can remember the direction. This is a standard direction for rotation so you should remember that. What happens if you have multiple torques? If you have multiple torques, we can calculate the net torque. The net torque is the resultant torque. It's one torque that represents a combination of all of them. The net torque is simply the addition of the individual torques. If you have two torques, it would look like simply T1 + T2. The big difference between forces and torques is that forces are vectors, they have a directional _. It can be pointing in different directions. A torque is a result of a force and a torque is a twist that's either clockwise or counterclockwise. That's not really a direction in the same way that a force can point in an infinite number of angles. This is just really two options. It's either going to the right or going to the left or just going clockwise or going counterclockwise. It's not a direction in terms of _, so there is no theta because they are scalars, not vectors. They just have an orientation that could be either positive or negative. We're going to use simple additions and not vector additions to find torque. Remember for forces, if you push this way with a 3 and in this way with a 4, the net force is not seven. The net force is five because this is vector addition. With torques, if you have a torque of 3 and a torque of 4, the answer is always 7 because it's simple addition, not vector addition. Forces are vectors; torques are scalars. Let's do a quick example here. I have two forces act on the same door. The door is 3 meters long. It says F1 acts on the center of the door, so this is the middle right here so I'm going to say that this is 1.5, half of the length and this is 1.5, the other half of the length. It says F2 is directed at 30 degrees above the x-axis, so it looks like this. We want to know what is the net torque and we also want to use signs Ð positive or negative Ð to indicate whether they're clockwise or counterclockwise the direction, the orientation of the torques, the direction of the spin I should say. What we're going to do is we're going to do T1 + T2. There are two forces therefore there could be as many as two torques. I want to remind you that I force may give you a torque. If you have two forces, you could have two torques, you could have one, you could have zero but you can't have three. The maximum amount of torques you can have is the number of forces you have too. WeÕre gonna do that. IÕm going to expand each one of these guys. The definition of a torque is Frsin_, so this is going to be F1R1sin_1. IÕm going to leave a little space here for us to indicate whether that torque is positive or negative, and then IÕm going to do the same thing here, F2R2sin_2. Remember, to solve torque problems, those three steps are you have to draw your r vector, you have to figure out your _, and then finally you plug it into the torque equation. Let's do that first. Let's draw the r vector for each one of these guys. What's R1 and whatÕs R2? R is a vector, an arrow, from the axis of rotation to the point where the force is applied. R1 is from the axis over here to the point where the force is applied. R2 is from here all the way to the end. This is R2 over here. R1 is 1.5, and R2 is 3 meters. We got those two guys figured out. How do we know which way it moves? We're going to use the R vector to figure out whether these individual torques are positive or negative. I want you to think of here's the door which is my R vector is happening along the door and I want you to think of what would happen if you pull on the door or on your arm in these directions. F1 is pointing down this way right here, F1 pointing down. What you have, imagine if you're pulling on your arm like this. I can pull my thumb down like this, this would spin in this direction. F1 is trying to get this thing to have a torque in this direction. IÕm going to call this T1. Let's put the bar back here, the door. What if you're pulling this way, F2? Then you can think about it this way or just pull a finger at an angle. If you pull it this way, it's going to try to cause it to spin like this. Even though it's not 90 degrees, it's at an angle, it's still going to try to do it this way and that's because it's basically if you are above this line here, you're going to do that. If you are any tiny bit below this line, youÕre going to cause it to do it this way. You're going to cause rotation to go that way. What you do is you extend your r vector and if you're on top of it, you're going to like that. If you're pulling sort of below it, you're going to go like this. T2 is in this direction. What signs are these? This is going in a clockwise direction, T1, so T1 is negative. T2 is going on a counterclockwise in the direction of the unit circle, so it is going to be positive. This guy is negative, this guy is positive. Let's put those signs here. T1 is negative, T2 is positive. The rest is what we've been doing so far, which is plugging in these numbers. Negative, the forces are both 50. The distance for R1 right here, 1.5 sin_. The angle between r1, r1 is the blue line in this one so r1 and F1, the angle here is 90 degrees. This guy is positive so 50. R2 is the entire length of the door so it's 3 and sin of, I have to make sure this is the correct angle. Let's be careful here. I'm going to redraw r, then I'm going to redraw F. Remember, one of the things you can do, you want them to be either all joining at the same point or exiting the same point. You basically want to make this easier for you to see. There's a few ways you could do this. I think the easiest one is just to move this r vector up like this. Then you can see that this is in fact the angle you want, which is the angle that's given. In this case, the angle given to you was the angle you're supposed to use. But remember, a lot of times the angle youÕre given is not the one you're supposed to use. We have to be careful here, just worked out that way so that's nice. Sin90 is 1, and sin30 is 0.5. This is going to be -75 + 75. This is interesting. They both added up to 0. What does that mean? If they both added up to 0, it means that they perfectly cancel each other out. You actually have no rotation at all. The net torque here, the sum of all torques which is the same thing as net torque, is 0 and what that means, we'll talk more about that later, is that you have rotational equilibrium. The two torques are cancelling so the object actually doesn't spin at all. We'll talk about that a little bit later. But that's it for this one. Let me know if you have any questions and let's keep going.

**Problem:** A 2-m long bar is free to rotate about an axis located 0.7 m from one of its ends. Two forces act on the bar, F_{1} = 100 N and F_{2} = 200 N, and both make 30° with the bar. Find the Net Torque on the bar. Use +/– to indicate direction.