Statics: Equilibrium in 3 Dimensions, Part 1

 Let's go at the previous problem: 

Now let's use equations to solve it. 

We'll draw diagrams here. The bottom you have to determine the mass of the pulley and the mass of the weight pulling down on it as well. We're trying to solve for the tension of the line as a function of the mass being lifted. 

Nothing else comes in regards to the tools we got besides the fact that we got 6 equations and 6 unknowns. 

Never forget the holy 2 equations of statics: 

AB is held by 2 cores attached to the colored walls. I want to find the tension of the 2 lines and the reaction force. Do determine the forces, again, draw a free body diagram. 

This is the ball and socket joint. Same concept, it's just this one is 3 dimensional. Here is the joint: 

A has 3 components Ax, Ay, and Az. 

We assume there's a uniform slender bar and stuff. 

The moment about point A into BC is going to be Rab x Tbc + Rab x Tba + (1/2Rab x W) = 0. and the sum of the moments - to 0).  where W represents the weight of the actual bar. 

Here is the list of equations that we get. 

And the final equation we can put into the equilibirum. We have 2 unknowns and 1 equation in this manner. 

TBC is 66.7 N and TBD is 167N. Let's move on to the next problem. 

Let's say the backwall of a garage is 12 meters high. There is a point 1.2 meters out in front and 2 meters up from the xy plane. A cable runs from a lot of places back to the garage and I am hired to pull parallel to the X axis with some forces, and the utility of this is to successfully pull the garage open. The mass being pulled is 200kg. We want to find the magnitude of the pulling forces, the forces of the cables going to points B and C.