In Unit 2 we
studied the use of Newton's second law and free-body
diagrams to determine the net force and acceleration of
objects. In that unit, the forces acting upon objects were
always directed in one dimension. There may have been both
horizontal and vertical forces acting upon objects; yet
there were never individual forces which were directed both
horizontally and vertically. Furthermore, when a free-body
diagram analysis was performed, the net force was either
horizontal or vertical; the net
force (and corresponding acceleration) was never both
horizontal and vertical. Now times have changed and you are
ready for situations involving forces in two dimensions. In
this unit, we will examine the affect of forces acting at
angles to the horizontal, such that the force has an
influence in two dimensions - horizontally and vertically.
For such situations, Newton's
second law applies as it always did for situations
involving one-dimensional net forces. However, to use
Newton's laws, common vector operations such as vector
addition and vector resolution will have to be applied. In
this part of Lesson 3, the rules for
adding vectors will be reviewed and applied to the
addition of force vectors.

Methods of
adding vectors were discussed earlier in Lesson 1 of
this unit. During that discussion, the head
to tail method of vector addition was introduced as a
useful method of adding vectors which are not at right
angles to each other. Now we will see how that method
applies to situations involving the addition of force
vectors.

A
force board (or force table) is a common physics lab
apparatus that has three (or more) chains or cables attached
to a center ring. The chains or cables exert forces upon the
center ring in three different directions. Typically the
experimenter adjusts the direction of the three forces,
makes measurements of the amount of force in each direction,
and determines the vector sum of three forces. Forces
perpendicular to the plane of the force board are typically
ignored in the analysis.

Suppose that a force board or a force
table is used such that there are three forces acting upon
an object. (The object is the ring in the center of the
force board or force table.) In this situation, two of the
forces are acting in two-dimensions. A top view of
these three forces could be represented by the following
diagram.

The goal of a force analysis is to
determine the net force and the corresponding acceleration.
The net force is the
vector sum of all the forces. That is, the net force is the
resultant of all the forces; it is the result of adding all
the forces together as vectors. For the situation of the
three forces on the force board, the net force is the sum of
force vectors A + B + C.

One method of determining the vector sum
of these three forces (i.e., the net force") is to employ
the method of head-to-tail addition. In this method, an
accurately drawn scaled diagram is used and each individual
vector is drawn to scale.
Where the head of one vector ends, the tail of the next
vector begins. Once all vectors are added, the resultant
(i.e., the vector sum) can be determined by drawing a vector
from the tail of the first
vector to the head of the last
vector. This procedure is shown below. The three vectors are
added using the head-to-tail method. Incidentally, the
vector sum of the three vectors is 0 Newtons - the three
vectors add up to 0 Newtons. The last vector ends where the
first vector began such that there is no resultant
vector.

The purpose of adding force vectors is to
determine the net force
acting upon an object. In the above case, the net force
(vector sum of all the forces) is 0 Newtons. This would be
expected for the situation since the object (the ring in the
center of the force table) is at rest and staying at rest.
We would say that the object is at equilibrium. Any
object upon which all the forces are balanced
(F_{net} = 0 N) is said to be at
equilibrium.

Quite obviously, the net force is not always 0
Newtons. In fact, whenever objects are accelerating, the
forces will not balance and the net force will be
nonzero. This is consistent with Newton's
first law of motion. For example consider the situation
described below.

An
Example to Test Your Understanding

A
pack of five Artic wolves are exerting five different forces
upon the carcass of a 500-kg dead polar bear. A top
view showing the magnitude and direction of each of the
five individual forces is shown in the diagram at the right.
The counterclockwise convention is used to indicate the
direction of each force vector. Remember that this is a top
view of the situation and as such does not depict the
gravitational and normal forces (since they would be
perpendicular to the plane of your computer monitor);
it can be assumed that the gravitational and normal forces
balance each other. Use a scaled vector diagram to determine
the net force acting upon the polar bear. Then compute the
acceleration of the polar bear (both magnitude and
direction). When finished, check your answer by clicking the
button and then view the solution to the problem by
analyzing the diagrams shown below.

The task of determining the vector sum of
all the forces for the polar bear problem involves
constructing an accurately drawn scaled vector diagram in
which all five forces are added head-to-tail. The following
five forces must be added.

The scaled vector diagram for this problem would look
like the following:

The above two problems (the force
table problem and the polar bear
problem) illustrate the use of the head-to-tail method
for determining the vector sum of all the forces. The
resultant in each of the above diagrams represent the net
force acting upon the object. This net force is related to
the acceleration of the object. Thus, to put the contents of
this page in perspective with other material studied in this
course, vector addition methods can be utilized to determine
the sum of all the forces acting upon an object and
subsequently the acceleration of that object. And the
acceleration of an object can be combined with kinematic
equations to determine motion information (i.e., the final
velocity, the distance traveled, etc.) for a given
object.

In addition to knowing graphical methods
of adding the forces acting upon an object, it is also
important to have a conceptual grasp of the principles of
adding forces. Let's begin by considering the addition of
two forces, both having a magnitude of 10 Newtons. Suppose
the question is posed:

10 Newtons + 10 Newtons
= ???

How would you answer such a question? Would you quickly
conclude 20 Newtons, thinking that two force vectors can be
added like any two numerical quantities? Would you pause for
a moment and think that the quantities to be added are
vectors (force vectors) and the addition of vectors follow a
different set of rules than the addition of scalars? Would
you pause for a moment, pondering the possible ways of
adding 10 Newtons and 10 Newtons and conclude "it depends
upon their direction?" In fact, 10 Newtons + 10 Newtons
could give almost any resultant, provided that it has a
magnitude between 0 Newtons and 20 Newtons. Study the
diagram below in which 10 Newtons and 10 Newtons are added
to give a variety of answers; each answer is dependent upon
the direction of the two vectors which are to be added. For
this example, the minimum magnitude for the resultant is 0
Newtons (occurring when 10 N and 10 N are in the opposite
direction); and the maximum magnitude for the resultant is
20 N (occurring when 10 N and 10 N are in the same
direction).

The above diagram shows what is
occasionally a difficult concept to believe. Many students
find it difficult to see how 10 N + 10 N could ever be equal
to 10 N. For reasons to be discussed in the next section of
this lesson, 10 N + 10 N would equal 10 N whenever the two
forces to be added are at 30 degrees to the horizontal. For
now, it ought to be sufficient to merely show a simple
vector addition diagram for the addition of the two forces
(see diagram below).

Check
Your Understanding

Answer the following questions and then view the answers
by clicking on the button.

1. Barb Dwyer recently submitted her vector addition
homework assignment. As seen below, Barb added two vectors
and drew the resultant. However, Barb Dwyer failed to label
the resultant on the diagram. For each case, which is the
resultant (A, B, or C)? Explain.

2. Consider the following five force
vectors.

Sketch the following and draw the resultant (R). Do not
draw a scaled vector diagram; merely make a sketch. Label
each vector. Clearly label the resultant (R).

A + C + D

B + E + D

3. On two different occasions during a high school soccer
game, the ball was kicked simultaneously by players on
opposing teams. In which case (Case 1 or Case 2) does the
ball undergo the greatest acceleration? Explain your
answer.

4. Billie Budten and Mia Neezhirt are having an intense
argument at the lunch table. They are adding two force
vectors together to determine the resultant force. The
magnitude of the two forces are 3 N and 4 N. Billie is
arguing that the sum of the two forces is 7 N. Mia argues
that the two forces add together to equal 5 N. Who is right?
Explain.

5. Matt Erznott entered the classroom for his physics
class. He quickly became amazed by the remains of some of
teacher's whiteboard scribblings. Evidently, the teacher had
taught his class on that day that

Explain why the equalities are indeed equalities and the
inequality must definitely be an inequality.