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You slip a wrench over a bolt. Taking the origin at the bolt, the other end of the wrench is at $x=18 \mathrm{cm}, y=5.5 \mathrm{cm} .$ You apply a force $\vec{F}=88 \hat{\imath}-23 \hat{\jmath} \mathrm{N}$ to the end of the wrench. What's the torque on the bolt?

$-52 \hat{k}$

Physics 101 Mechanics

Chapter 11

Rotational Vectors and Angular Momentum

Moment, Impulse, and Collisions

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Cornell University

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Sheffield

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All right. So this problem we're talking about, uh, taking a wrench to a bolt and twisting the bull. And so we want to get the torque, given the forests and the position. So let's go ahead and get the givens down and I'll discuss the approach. The problem. Um, so the wrench and is at X equals 18. Why is 5.5? I think I will go ahead and convert that two meters and the torque force are the forests is eat in the eye hot minus, um, 23 in the J hot new ends you, but so I'll go ahead and draw this out. I'm here is the bolt. And then here is Here's the end. So, um, so this is where the force is applied, and then the forces Basically, it's been the positive X direction and then the negative y direction. So mostly in the positive x a little bit in the positive. Why? So here's our forest factor, and here's our our vector. And then, er vector. We go from here to here because it always goes from the axis of rotation to where the force is applied. And, um, so now I want to get the Twerk. And so I mean, we could do it. Um, because there's a couple ways you could do it, Um, could do about it Kind of drawing out triangles and finding the angle between these two. And, um, I think that, but I think the easiest way is to use the the formula torque is our across off and then just compute the cross product of the vectors. I think if we don't, it would be a mistake to not use this because it will make the problems seem a lot more complicated. Um, so basically, we wanted to do is take the determinant of this following three by three matrix. Hopefully, there's something in the book about how to do this, but you can probably learn from my example. You haven't seen this already. So, um so basically, we're gonna We're making a matrix of three by three. And for here, we're gonna put the are vector. So the are vector. Um, so the R vector is up into the right. So it is. It's exposition is 0.18 Why is point over I five? That's gonna look a little messy. I know. 55 And then there's nothing in the K direction. And then, um, the force is gonna b e t eight and then negative 23 then nothing in the y direction. And I just need to compute the determinant of this matrix. I'm gonna go through it quickly if you don't understand that Google finding determinant of a three by three matrix. So basically, um, looking here, this sub matrix right here on defining the determinant that this time zero this time zero is nothing that I look at those matrix with 0.180 88 0 and find the determinant of that kind of pretending like this stuff isn't here. And then again, I get 0.18 Time 00 Time t eight for the K though this I evaluate this sub matrix and I don't got a zero determinant. So I did get k hot. Um, and then I want I'm going to do 0.18 times negative 23. And then, um, oops. Minus my nose 0.55 times 88. And that I'm gonna go ahead and put that into a calculator. But before I do it Let's just do a little check. So this thing, this negative and this is negative. So it's gonna be negative. Is that, um does that make sense? That's a direction that the angular momentum will change. So I would say, um, so if this is X and this is why for a right handed coordinate system, see is out of the page of Positive Z is out of the page. And so this thing is going to rotate clockwise, and so the angular momentum change is gonna be into the page. And so that is negative. So So that does check out. Oh, so now what I want to do is just plug this into a copy later. So 00.18 times 23 plus negative or sorry plus 0.55 times. Mediate. Yeah, I got 8.98 I get a big point Any, um, in the K hat direction. I just didn't wanna type in all the negatives into the calculator. And I'm just gonna double attract that in the back of the bug. They degrees. I like to look over my work if it, um, doesn't agree. Yeah, gives nine point House. I guess it is more correct. Only have 26 pigs. I think, sometimes like the answers and mastering physics or whatever. We'll have, like just always three sig figs, regardless of what's give another problem or something. And so, even though it's not really correct, you have to get used to, like, just keep an eye on the sig figs. But it's really only incorrect to have 266 here creep, so

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