# Module 4 -- Net Force and Vector Addition

It is very common for objects to experience multiple interactions simultaneously. When an object experiences multiple forces their effects will combine with each other, but they cannot simply be added together. The reason is that forces are not just numbers, they have direction associated with them as well. Quantities like force that have both size (called 'magnitude') and direction are called vectors.

#### Learning Goals

After completing this module you should be able to:

• Find x and y components of a vector given as a magnitude and direction, and vice-versa.
• Define the net force acting on an object in terms of the individual forces acting on it.

## Forces are Vectors

In the figure below, two people are pushing a heavy crate on a very slippery floor. One of the people pushes with a force of 40 Newtons (in the International System of Units, forces are measured in Newtons, which is abbreviated by the symbol N). The other person pushes with a force of 30 N. It is clear that if the people push in the same direction, the magnitude of the total (or net) force on the crate is 70 N. If the people push in opposite directions, the magnitude of the total force is only 10 N, since they are pushing against each other. If, however, the people push at right angles to one another, the net force is found by constructing a vector triangle. The net force is given by the sum of the force vectors (the hypotenuse of a triangle because these forces are perpendicular), and therefore has a magnitude of 50 N, which is more than either force alone, but less than the sum of their magnitudes.

## Summing Arbitrary Vectors

### Rule 1: Think Geometrically: Geometric or graphical additions of vectors

Think of vectors as arrows, and add them by putting the tail of one on the arrow of the other as is shown in the two diagrams immediately above. It does not matter which vector's tail is placed on the arrow of the other - the sum is the same (called the commutative property). This geometric addition is very helpful in sketching out preliminary predictions an problems because the direction of Fnet will always point in the same direction as the acceleration vector. You can draw any vector anywhere on the page, pointed in the correct direction and scaled appropriately, so that the length drawn on your paper corresponds to the magnitude.

1. Choose an appropriate scale for length.
2. Draw one of the vectors in the sum to scale, in the correct direction.
3. Draw the next vector in the sum with the tail of 2 on the head of the first.
4. Draw the next vector in the sum with the tail of 3 on the head of 2.
5. Keep drawing in this tail of the new on the head of the preceding vector until all the vectors in the sum are drawn.
6. Finally, draw the vector that starts at the first tail and ends on the final head, put an arrow on it and measure its length according to the scale.

The resultant vector (or net vector of the sum) is that last final vector you drew.

### Rule 2: Compute Vectors Sums or Differences Using Components

The two mathematical operations of addition and subtraction of vectors are critical to further applications of Newton's Laws of Motion. There are two very simple techniques which lend themselves to a clear physical interpretations that you must master. Recall Newton's second law is expressed as a vector sum: $\vec{F}_{net} = m\vec{a}$ . Because any vector can be written as a set of it components, using the conventional "xyz" cartesian coordinate system, Newton's second law gives rise to a system of three nearly identical scalar equations (the only difference being the identification of the three cartesian directions):

• $F_{net,x} = \sum F_{x} = ma_{x}$

and

• $F_{net,y} = \sum F_{y} = ma_{y}$

and

• $F_{net,z} = \sum F_{z} = ma_{z}$

The three component equations are all straightforward scalar equations. Since all three of these equations are identical in form, we can focus on vectors in a plane (two dimensional) to get the basic operations of vector manipulations graphically. These mathematical procedures are the same whether the situation is 1-dimensional, 2-D or 3-D.

The major steps to vector addition are: first, decomposition (determining the scalar components) and then addition of the components.

1. Sum the corresponding components, paying careful attention to the direction (sign) of each component.
2. Use the resultant mutually perpendicular component sums to form the resultant vector.
3. Use those perpendicular components to draw the resultant vector.
4. Use Pythagorean theorem to determine the magnitude of the resultant vector.
5. Use appropriate trigonometry to determine the direction of the resultant vector with respect to the arbitrary and convenient coordinate system chosen.