Use tangent to get the direction… tan θ = Use pythagorean theorem to get the magnitude of the resultant force… ∑ F = Arrange the results in a table like this one. T 1Ĭompute the x and y components of each vector. These forces should form the ratio 1:1:√2. The horizontal tension and the vertical weight are the legs of a 45–45–90 triangle whose hypotenuse is the diagonal tension. We already said this, so there is no advantage to this method over the previous one. Thus each tension equals half the weight. Symmetry tells us the two short sides should have equal length. The two short sides lie on top of the long side. Sure it has three sides, but it covers no area. This is what we call a degenerate triangle. This means, it should be possible to arrange the three vectors in this practice problem into a closed figure - a triangle. When forces are in equilibrium, their sum is zero and their will be no resultant. The sum would be the resultant vector connecting the tail of the first vector to the head of the last. The graphical method for addition of vectors requires placing them head to tail. In this practice problem, the vectors are rigged so that the alternate solution is easier than the default solution. They don't work all the time, but when they do we should use them. Sometimes, however, there are clever solutions available. Understand the rules, describe them using commands a computer understands, put numbers in, get answers out. We use this brainless, brute force approach to problems all the time. Whenever you're given a pile of vectors and you need to combine them, components is the way to go - especially if you have no expectation of any special relationships among the vectors. We used component analysis since it's the default approach. Let's see if there isn't a simpler solution. Substitute back into the horizontal equation and compute T 1. Solve that for T 2, substitute values, and compute T 2. Substitute the result into the vertical equation. I also suggest working through the vertical equation first.

I like to put negative vectors on the left side of the equals sign and positive vectors on the right side.

Break it up into components and state the conditions for equilibrium in the vertical and horizontal directions. These are both good vectors - good in the sense that they are easy to deal with. Weight points down (270°) and T 1 points to the left (180°). Together they should equal the weight, which means each one is carrying half the load.

The two upward components should equal one another. Use a ruler and a protractor if you wish. As always, make a nice drawing to show what's going on. Describe this state using the language of physics - equations in particular, component analysis equations. The sign isn't going anywhere (it's not accelerating), therefore the three forces are in equilibrium. The sign always has weight ( W), which points down. Two-dimensional rigid bodies have only one degree of rotational freedom, so they can be solved using just one moment equilibrium equation, but to solve three-dimensional rigid bodies, which have six degrees of freedom, all three moment equations and all three force equations are required.For all solutions, let T 1 be the cable on the left and T 2 be the cable on the right. To analyze rigid bodies, which can rotate as well as translate, the moment equations are needed to address the additional degrees of freedom. We saw in Chapter 3 that particle equilibrium problems can be solved using the force equilibrium equation alone, because particles have, at most, three degrees of freedom and are not subject to any rotation. In many cases we do not need all six equations. Working with these scalar equations is often easier than using their vector equivalents, particularly in two-dimensional problems.

Statics equilibrium in 3d problems