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. Well, times have changed and now
you are ready for situations involving forces in two
dimensions. In this unit, we will examine the effect 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) strings or cables
attached to a center ring. The strings 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. (NOTE: This is the method
used in the "Vectors Are a Snap" Lab.)
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, each
of the three 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 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 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) was at rest
and staying at rest. We would say that the object was at
equilibrium. Any object upon which all the forces
are balanced (Fnet = 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 depressing mouse on the "pop-up
menu" 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
The following five questions correspond to the
questions on the "Addition of Forces" handout found in
your unit packet. You can view the answers to these
questions be depressing your mouse on the "pop-up
menu."
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).
3. On two different occasions during a GBS 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 Mods 5-8
physics class. He quickly became amazed by the remains of
some of Mr. Henderson's whiteboard scribblings.
Evidently, Mr. Henderson had taught his class on that day
that
Explain why the equalities are indeed equalities and
the inequality must definitely be an inequality.
