Earlier
in Lesson 1, the method of resolving a vector into its
components was thoroughly discussed. During that lesson, it
was said that any vector which is directed at an angle to
the customary coordinate axis can be considered to have two
parts - each part being directed along one of the axes -
either horizontally or vertically. The parts of the single
vector are called components and describe the
influence of that single vector in that given direction. One
example that was given during Lesson 1 was the
example of Fido being pulled upon by a dog chain. If the
chain is pulled upwards and to the right, then there is a
tensional force acting upwards and rightwards upon Fido.
That single force can be resolved into two components
- one directed upwards and the other directed rightwards.
Each component describes the influence of that chain in the
given direction. The vertical component describes the upward
influence of the force upon Fido and the horizontal
component describes the rightward influence of the force
upon Fido.

The task of determining the amount of
influence of a single vector in a given direction involves
the use of trigonometric functions. The use of these
functions to determine the components of a single vector was
also discussed in Lesson 1 of this
unit. As a quick review, let's consider the use of SOH
CAH TOA to determine the components of force acting upon
Fido. Assume that the chain is exerting a 60 N force upon
Fido at an angle of 40 degrees above the horizontal. A quick
sketch of the situation reveals that to determine the
vertical component of force, the sine function can be used
and to determine the horizontal component of force, the
cosine function can be used. The solution to this problem is
shown below.

As
another example of the use of SOH CAH TOA to resolve a
single vector into its two components, consider the diagram
at the right. A 400-N force is exerted at a 60-degree angle
(a direction of 300 degrees) to
move a railroad car eastward along a railroad track. A
top view of the situation is depicted in the diagram.
The force applied to the car has both a vertical (southward)
and a horizontal component (eastward). To determine the
magnitudes of these two components, the sine and cosine
function will have to be used. The task is made more clear
by beginning with a diagram of the situation with a labeled
angle and a labeled hypotenuse. Once a triangle is
constructed, it becomes obvious that the sine function will
have to be used to determine the vertical (southward)
component and the cosine function will have to be used to
determine the horizontal (eastward) component. The
triangle and accompanying work is shown below.

Anytime
a force vector is directed at an angle to the horizontal,
the trigonometric functions can be used to determine the
components of that force vector. To assure that you
understand the use of SOH CAH TOA to determine the
components of a vector, try the following three practice
problems. To view the answers, click on the button.

An important concept is revealed by the
above three diagrams. Observe that the force is the same
magnitude in each diagram; only the angle with the
horizontal is changing. As the angle which a force makes
with the horizontal increases, the component of force in the
horizontal direction (F_{x}) decreases. The
principle makes some sense; the more that a force is
directed upwards (the angle with the horizontal increases),
the less that the force is able to exert an influence in the
horizontal direction. If you wish to drag Fido horizontally,
then you would make an effort to pull in as close to a
horizontal direction as possible; you would not pull
vertically on Fido's chain if you wish to pull him
horizontally.

One
important application of this principle is in the
recreational sport of sail boating. Sailboats encounter a
force of wind resistance due to the impact of the moving air
molecules against the sail. This force of wind resistance is
directed perpendicular to the face of the sail, and as such
is often directed at an angle to the direction of the
sailboat's motion. The actual direction of this force is
dependent upon the orientation of the sail. To determine the
influence of the wind resistance force in the direction of
motion, that force will have to be resolved into two
components - one in the direction which the sailboat is
moving and the other in a direction perpendicular to the
sailboat's motion. See diagram at right. In the diagram
below, three different sail orientations are shown. Assuming
that the wind resistance force is the same in each case,
which case would produce the greatest influence in the
direction of the sailboat's motion? That is, which case has
the greatest component of force in the direction parallel to
the boats' heading?

Many people believe that a sailboat cannot
travel "upwind." It is their perception that if the wind
blows from north to south, then there is no possible way for
a sailboat to travel from south to north. This is simply not
true. Sailboats can travel "upwind" and commonly do so by a
method known as tacking into the wind. It is true to
say that a sailboat can never travel upwind by heading its
boat directly into the wind. As seen in the diagram
at the right, if the boat heads directly into the wind, then
the wind force is directed due opposite its heading. In such
a case, there is no component of force in the direction that
the sailboat is heading. That is, there is no "propelling
force." On the other hand, if the boat heads at an angle
into the wind, then the wind force can be resolved into two
components. In the two orientations of the sailboat shown
below, the component of force in the direction parallel to
the sailboat's heading will propel the boat at an angle into
the wind. When tacking into the wind, a sailboat will
typically travel at 45-degree angles, tacking back and forth
into the wind.

Check Your
Understanding

The following problems focus on concepts discussed in
this lesson. Answer each question and then click the button
to view the answer.

1.
The diagram at the right depicts a force which makes an
angle to the horizontal. This force will have horizontal and
vertical components. Which one of the choices below best
depicts the direction of the horizontal and vertical
components of this force?

2. Three sailboats are shown below. Each sailboat
experiences the same amount of force, yet has different sail
orientations.

In which case (A, B or C) is the sailboat most likely to
tip over sideways? Explain.

3.
Consider the tow truck at the right. If the tensional force
in the cable is 1000 N and if the cable makes a 60-degree
angle with the horizontal, then what is the vertical
component of force which lifts the car off the ground?