18. (Problem Solving)

Force Tables

A tool we will use frequently to solve force problems is a force table. We fill one out using the following steps:

Make a list of all the forces acting on the object. List the type of force, followed by the source of the force. For example, "normal, table" or "static friction, floor". (For weight you can just write "weight".) You might want to do this at the same time you are creating a force diagram.
Add two columns to the right of the list, and label them "x" and "y". (You only need one if it's a 1D problem.)
Draw a basis, specifying the $+ x$ and $+ y$ directions.
For each force, if the force points along the $x$ -axis, then fill in the $x$ column with the magnitude of the force, and give it a sign (+ or -) depending on whether it points in the $+ x$ or $- x$ direction. If the force does not point along the $x$ -axis at all, then put a dash in that column. Do the opposite for the $y$ column.
If a force points at an angle, break the force up into components and fill each column with the appropriate component.
If you don't know the magnitude of a force, give it a variable name.
If the object is in equilibrium, then when you've completed the table, each column must add up to zero, and so you can write two equations (one for each column).

Suppose a 60N box is being pulled on by a rope at a 40° angle with a tension force of 30N.

We define a basis so that $x$ is the horizontal direction and $y$ is the vertical direction.
We start with weight. Weight is vertical, and points in the $- y$ direction, so we write "-60" in the $y$ column, and put a dash in the $x$ column.
Next we see that the table is touching the box, and so there is almost certainly a normal force from the table. The normal force is vertical (perpendicular to the table's surface), and points in the $+ y$ direction. I'm not given the magnitude of this force, so I call it $N$ .
The rope is exerting a tension force. It's not horizontal or vertical, so we need to split it into components . The vector points upward and to the right, both positive directions, so I explicitly write a + in front of both.
There must be some other force cancelling on the horizontal component of the tension, so there must be static friction from the table, which points in the - $x$ direction. I'm not given its magnitude, so I call it $S$ . (You may be tempted to say "But I know what it is! It's $- 30 \cos 40^{\circ}$ !" But that's getting ahead of yourself. Only fill in values you are given, for the forces you are given them.

Now that I have all the forces, I add up each column and set it equal to zero: $+ 30 \cos 40^{\circ} - S = 0$ $- 60 + N + 30 \sin 40^{\circ} = 0$

You will only fill in both columns if the force points at an angle: that is, not entirely along the x or y axes. Otherwise one column will be blank.
Always include a sign in front of each number, either + or -. This will help remind you to pay attention to the direction of each force.
It's a red flag if you write the same thing in both columns (like "20" and "20" or "N" and "N"): while this can happen, it's a bit of a coincidence and should make you look twice.
When writing a vector in components, each column will have the following pieces: a) a sign (+ or -); b) the magnitude of the force (give it a name like T if you don't know what it is); c) either "sin" or "cos" (one in each column); and d) the angle.
If you have NO idea what a force is, magnitude or direction, you can assign a different variable to each column. For instance, a completely unknown tension might have $T_{x}$ in the $x$ column and $T_{y}$ in the $y$ column.
You should do as little math as possible when filling in the table; save the solving for afterwards.
This is a matter of taste, but I usually don't include the units in the table-- see how I wrote "60" rather than "60N"-- to avoid confusing the N for a variable. Do include the N in the final answer, however.