Another, and probably by far the most expansive, place that you will see vector components is in Newton’s Second Law, a.k.a. Free Body Diagrams problems. This is where the “identify your angle” step really starts to get tricky, as the forces are usually not as straight forward and in line with the axes as in kinematics problems. This problem gives a general angle theta, and asks for the normal force, N, which is seen pointing upward directly from the incline. The picture gives us some additional information, the weight, W, is pointing directly downwards. Surprisingly, with our 5 steps, this is all the information we’ll need to solve this problem.
Step 1: Identify your coordinate axis and draw it on the paper. Unlike the kinematics problem, this time we’re going to align our coordinate axis a little differently, not just the up and sideways coordinate axis. The alignment we will choose will point the x-axis parallel with the incline, and the y-axis parallel with our eventual normal force. This makes sure that our Normal force will be equal to some trigonometric function times the Weight, and not have to break the Normal force itself into components.
Step 2: Identify the vector and draw it on the paper. Contrary to what the problem asks, for right now we’re after the weight, W. So, as shown above in red, the interest becomes finding the y-component of the weight. This begs the next piece of the puzzle, the angle.
Step 3: Identify the angle and draw it on the paper. Because the interest is in the y-component of the weight, the angle we’re going to need is the one shown above, tucked between the weight and the w-component. Unlucky for us, it’s not the one given in the problem, at least exactly. This is actually the same angle theta, but how is this possible? This is not the same angle theta given in the problem! These are the geometric gymnastics I was hemming and hawing about in the kinematics problem, and why I was so happy they gave us an easy angle back then. On the next page we will look into exactly what is going on with this angle and how exactly this manages to be the same angle.
I think the caption says it all. On to step 4.
Step 4: Use sine & cosine relationships for component definition. Here, given the weight, W as the hypotenuse, with this angle we can say that the y-component, will be identified as the adjacent side of a triangle. We will then be able to define Wy as the weight W times the cosine of the angle theta.
Not really needed for our purposes here, but the Normal force, N, opposes the y-component of the weight. This is where the phrase “equal and opposite” comes in. The weight W is given as the mass of the object times gravity, g. Thus, our normal force is equal to mg*cos(ϴ). If you were bogged down with some of the frills in the problem, i.e. what was exactly going on with the normal force, or the weight, or whether or not you would know how to do this problem without reading the steps, don’t worry. The idea is to see that these same steps can be applied to a variety of different problems, and that the procedure here is roughly the same. This internal skeleton will be the foundation for a much bigger framework as we introduce other physical concepts.
