Statics: 2-D Couples

We were working on the article article on Moment, which is the more developed idea. Almost every single force in torque was perpendicular. Now we've changed to moment, and it's the tendancy in physics to use torque with the symbol τ or the symbol Γ but in engineering we tend to do moment, where the work doesn't make too much sense but at least the letter does. x means cross product and M0 = r0 f F.

We need to draw the vector r0, then we need to represent it in vector form so we can finally put it into the cross product. Here the vector goes from O to f. 

Maybe we can find the line of action, that is 45 degrees. 

Maybe you like that point because the geometry is easier, etc. Anything as long as you go to the point to the line of action of the force, it will calculate the right moment, since when you do the cross product, it all comes out to that the magnitude is F x d. We just need to refer to a vector r to the line of action. Add any vector to the line of action. 

Just be careful. Let's first draw a bracket, and there's a force on that bracket, and the force comes out 40 degrees on the horizontal. Let's say there's a nice right angle bracket of some kind, and there's a force out at a specific point, and has a magnitude of 760 lbs. And we have the bracket embedded in the ceiling, so we have to design the attachment on the ceiling such that this force won't rip the bracket out of the ceiling. There's worrying about twisting and breaking, and we want to know how much moment a force exerts at a point. 

We want how much moment the force exerts about the top point, since this is one point that is the farthest away from the force, so the moment increases with the distance. We got to make sense things are welded well enough, otherwise things can fail easily. Let's put some dimensions on this. The orange will be the same thing. To calculate the moment, we need the vectors r0 and F0. 

We got a force, we have to come up with a vector that represents it. 

Calculating the moment will be a 3x3 matrix.

r0 goes out 10 inches in the x direction so we can say r0 = (10i - 12j) inches.

Fx = 582 i + 489 j. For 2D vector in xy space it always equal to 0. We design things for factor of 2 for safety considerations. so 11874 in lbs k vector or 6,000 for safety considerations counterclockwise (this is the moment). For 3 dimensional problems, it's the same thing but k doesn't equal to 0 this time. 

There are 2 things going into moment vectors.

We can replace a force by F0 => M0 to F0M0. We have a system with only a force acting on it. We can have the force on top and we can also calculate the moment, and those 2 systems are equivalent.

Let's see we have an object with a lot of forces on it, we can add up all the forces, and replace them with a single equivalent force and we got the same problem. In the class sum is 0 since we don't want the things to accelerate.  

We can reduce the equation to one force moment. So Mtotal of all vectors is ΣM.