What we're towards is the ability to ensure static equilibrium, and if we don't have this condition, we have acceleration, and we don't have good structural analysis. We were looking at the phase on how to sum all the forces.
There's a very different relationship between all the different things that are going on between one force and another force. We're working towards our ability to ensure static equilibrium. If we don't have a condition then we don't have good structural analysis. We're working towards the goal of acceleration equals to 0.
We can do the trigonometry analysis, and stuff, and going in and trigonometry analysis says it does something else, so you go back and redo them until you find a mistake. We're going to have another part that we need. In physics 1, you know from experience if we got a diving board there's all the things that are going on if we have certain forces at locations, or anything in between, and forces at different locations are good when going on. Things tend to turn when you put a force on a board.
It would be terrible to get on the diving board if the board turned much earlier. We'll look at why things bend in the next term, so that they bend enough but not too little, but not break enough. In this static class, nothing bends, and nothing has mass, so it's a massless, bendless diving board. We want to know what's going on the wall where that's not in static equilibrium.
Torque is not the term we use for this class, here we'll use the term moment. They are interchangeable. The symbol we use in physics 1 is τ or and M for moment. There shouldn't be any net tendencies anywhere.
We need to have to balance all of the moments to try to balance all of the forces. Let's say you're trying hold a tray like "I don't want to spill anything!" We'll do better when the drinks are in the middle vs the side, and moving hands up to the center of the tray. The position of the force, d is the distance from point a in which the force acts. That's what we did in physics 1.
Torque = Fd and Moment = Fd. There's a relation that must be there for something to be the calculation of the moment. The smallest distance from A to the line of action is perpendicular. The minimum distance of the point of interest and the line of action on the force is perpendicular. Twice as far away of A will be twice the moment since moment = F x d. The moment exerted by the second force is twice as much because we have a greated distance from the holder to the line of action of force.
Force is a sliding vector, and we can't change the magnitude, units, or direction. We can either pull down from the bottom or top, we can slide a force vector ANYWHERE along its line of action to do its calculation, and this will be VERY useful to us in the future. D is called the minimum perpendicular distance, between the Point OF INTEREST AKA A and the line of action of the force.
There's a direction if we have some situations. We want to make sure the wall (immovable gigantic object). The same force in different direction will have the tendancy of the board of going a specific direction, we need to make sure that the wall will help to back this up, pushing back and counterbalancing both the force and the moment.
That's the introduction to moment. However, there's other things that we need to consider. We need to get a force in the opposite direction to result in an equilibrium of 0. Not only that but the moment would be different as well. How would we figure out the moment? So MA = Fdcos(θ).
We can take the force, and just split it into the vertical component and the horizontal component. This is part of the reason why you need to make a big enough drawing. You should be working on huge sheets of paper to help solve these problems. The force going horizontally is F sin θ.
Let's compute the moment about point A. We have 2 forces. So we need to calculate the moment of both of them since they both are contributing to this. F sin θ is the other component to this. So the minimum perpendicular distance in cos is d and the minimum perpendicular distance for sin is 0.
Final equation = (F cos θ ) d + F sin θ * 0.
This is all fine and dandy, but what if we get to 3 dimensions? So we can get through specific components through specific holes, and know which one is more powerful? A moment can be clockwise and counterclockwise, where we can have a positive clockwise and negative moments, where there are negative values, and reminding you for a second when we go into the 3-dimensional aspect.
Moment itself is a vector, I didn't really need to put it's a vector since we just have to plus or minus which way a particular direction in the moment went. You got to be very careful with the vectors and how we define them. Cross products are extremely important to what we are doing. We cross with the force vector. The cross product is a method of cooperation between 2 vectors that results in a vector which gives us a value with magnitude in units and direction.
Let's take the simplest possible situation we gut and test this. In a point a, a vector F is a vector and we do ra in a second. F is the magnitude multiplied by the direction which would be -j. (Remember, we are using 3 dimensional coordinates with i, j, and k, along with other things. The position vector relates the point A to the Force F. It starts at point A and it goes to the Force F, where it goes anywhere along the line of action in the force F, and we can find d how far away 2 directions there are, for example, magnitude d in the i direction).
The moments is always some position vector cross-product with some force vector.
We should get a vector via a 3x3 matrix. Across the top of the matrix are the unit vectors i, j, and k. The middle row is the 3 components of the positional vectors of the period, which is rx, ry and rz. Now we have 3 component of Fx, Fy, and Fz. Here is the cross product equation:
X direction no x values play. Y direction no y values plan, and Z direction no z values play.
So how do we take the cross product of 2 vectors, the Force vector and the distance vector?
Here is how we do it for the specified problem:
That was the hard way to do a simple problem which is 2 directions. Right hand: put your fingers in the direction of the curl, using the right hand rule. The 3 vectors in the cross product are mutually perpendicular, and this is very difficult to ascertain in 3 dimensions.
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