Hey, everyone. So let's check out this practice problem. We're trying to calculate the minimum amount of force necessary to produce 100 newton meters of torque using a 20 centimeter wrench. I've got my torque (τ) equal to 100 and the length of the wrench is 0.2 meters. Right? That's the length of this wrench over here. And I want to calculate the minimum force (fmin). Right? That's the minimum amount of force necessary. So how do we do that? Well, I've got forces and torques. This is really just a play on the torque equation, so let's set it up.
So I've got torque equation as τ=fr*sin(θ). Here's what's going on in this problem. If I want to figure out the least amount of force necessary to produce 100 newton meters, really another way of phrasing it is I want to find the maximum torque possible using the least amount of force, right? You want to grab the wrench and using the least amount of force, produce the most efficient torque possible. Okay? That's really what's going on here. So I've got my torque equal to 100 Newton meters. That's what I need in order to tighten this bolt. So this is fmin over here.
And then what am I going to plug in for r and then sin of theta? So really, if I want to minimize this number here, then I want these numbers to be the maximum they possibly can be. If these values are the greatest they possibly can be, then this number can be the smallest number and still get to 100 Newton meters. So, I want these two things to be max. Alright, so what do we plug in for r and sin(θ)? If you grab this wrench here, you imagine that there's a bolt right here. So, what is the distance in which you can maximize your torque? Well, it's basically the farthest you can possibly go from the axis of rotation. In other words, you want your r to be as far as possible, so you want basically to pull right here at the end of the wrench. If you grab in the middle, it's not going to turn as much, but you grab all the way at the end, that's where you get the most amount of torque. It's kind of like if you push a door. You don't want to push the door in the center; you want to push it all the way at the end to get the most amount of push.
So, this 'r' really just becomes 'l', the length of the wrench itself. So r equals l, and that's 0.2. So that's what goes inside of here. Then, how do we maximize sin(θ)? What's the angle at which we push it? Well, hopefully, you guys remember that the sine of an angle ranges between 0 and 1, and the maximum possible value is when you have sin of 90 degrees. So, what this means is that when you're at the end of the wrench here, you don't want to pull off in one direction or another. You basically want to pull off exactly at 90 degrees. That's the angle. If you pull at any other angle here, you're actually going to produce less force or less torque. So, that's what happens is you just have the sin of 90°.
So then, let's go ahead and plug everything in. So we've got fmin*0.2*1=τ, which equals 100. So, 100 divided by 0.2, that's your fmin. And if you work this out, what you're going to get is 500 Newtons. Alright, and that's the answer. Hopefully that makes sense. Let me know if you have any questions.