Download VECTOR ADDITION

VECTOR ADDITION
OBJECTIVE:
To find the resultant of several vectors using graphical methods.
EQUIPMENT:
Graph paper, ruler, protractor, sharp pencil, a calculator and a force table and weights (optional).
THEORY:
As discussed earlier, we distinguish between scalars, which have only numerical value or
magnitude, and vectors, which have both magnitude and direction. Speed and distance are
examples of scalar quantities. Velocity, displacement, and acceleration are vector quantities. We
may represent a vector by an arrow whose length is proportional to the magnitude of the vector
and whose direction is determined by the direction of the vector. Thus, a vector A may be drawn
as
we will use bold-face type (A) to refer to vectors. For example, suppose that we have two
velocity vectors: V1, which is 10 km/h north, and V2, which is 20 km/h east. We represent these
velocities by the following arrows, using road map directions:
The magnitude of V2 is 20 km/h and that of V1 is 10 km/h. Thus, the length of the arrow
representing V2 is twice the length of arrow V1. We specify the direction of the vector quantity
by the direction of the arrow: V1 points north and V2 points east.
Revised Summer 2010
Vector addition:
A.
Head-to-tail method
We may combine two vectors to form a third vector,
called the resultant, by two equivalent methods. The
first of these is the head-to-tail method. To illustrate,
suppose we want to add the two vectors A and B.
We begin with either vector, in this case A, and place B
so that its tail is at the head (arrow head) of A. However,
we must preserve the direction of B.
Next, we draw an arrow from the tail of A to the head of B. This new vector, C is the
resultant. Thus, we may say
C=A+B
if we are careful to note that this is vector addition, which means paying attention to the
directions. For example, if A is 10 feet long and B is 2 feet long, C will in general not be 12
feet long. Only in the special case in which A and B are parallel will the length of C be
equal to the sum of the lengths of A and B. In general, the length of C will be less than this
sum.
In the experiment you will be given different forces. You will represent these forces by
vectors of appropriate lengths by choosing a scale. Your first vector will always start from
the origin with its tail at the origin and you draw this vector at the given angle. The tail of
the second vector will be at the arrow of the first and so on. The resultant vector will be the
line joining with the head of the last vector and the origin of coordinate system with the tail
of this resultant at the origin.
B. Parallelogram method:
In this case we will start with two vectors at a time and draw them from the origin of our
coordinate system drawn at the center of the graph paper. These two vectors will represent
the adjacent sides of a parallel-ogram which is a figure whose opposite sides are parallel and
equal. Then we will complete this figure and draw its diagonal passing through the origin
which will be the resultant of these two forces. We can continue this method for any
number of forces (vectors) and find the equilibrant.
Note:
Your instructor will give you the details of this method and demonstrate the
significance of the equilibrant.
Revised Summer 2010
PROCEDURE:
a.
You are given two forces of 2N and 4N at angles of 0 and 90 measured with respect to the
positive x-axis. Use the head-to-tail method and find their resultant.
b.
Add a third force of 6N at 120 and find the resultant of all the three forces using head-totail method.
c.
Add a fourth force of 8N at 220 and find the resultant of all the four forces using head-totail method and draw the equilibrant. Record the magnitude and direction of the equilibrant
in Table 1.
d.
Duplicate the system of forces you just added graphically using the force table. (See
Procedure for the Force Table on the next page). Compare this experimental result with the
equilibrant you obtained graphically.
e.
Calculate the percent difference between your graphical results and your experimental
results. This is done by computing the difference, dividing by the average of the graphical
results and the experimental results, and then multiplying by 100.
(Optional)
Repeat a, b and c using parallelogram method and draw the equilibrant and also compare your
results with the head-to-tail method. They should agree.
QUESTIONS:
1.
Give two examples of scalar quantities. Give two examples of vector quantities.
2.
Consider two vectors 3 meters and 4 meters long. Determine the length of the resultant if:
a. the vectors are parallel (i.e. they point in the same direction)
b. the vectors are antiparallel (i.e. they point in opposite directions)
c. the vectors are perpendicular
3.
You leave your house and walk to the drugstore, which is 60 meters to the east, and then go
to the ice cream store located 80 meters north of the drugstore. Draw a vector diagram, and
from it find how far you are from home.
4.
In problem 3 find the answer without using your ruler.
Revised Summer 2010
Procedure for the Force Table
The purpose of using the force table is to demonstrate the addition of vectors, and in particular
forces. The concept of equilibrium is also demonstrated.
1.
Tie five strings to the ring in the center of the force table. Attach pulleys at the angles used
in the graphical vector addition problem and hang masses from each one that are
proportional to the forces used in that problem. The weight hanger itself has a mass of 50
grams, so it could be used to represent the 2 N force.
2.
Using the fifth string, pull on it and adjust the angle until you have the ring in the center of
the table centered on the post in the center of the table. In this condition, the ring is in static
equilibrium. Note the angle of the string you are pulling and place a pulley there. Then
hang a weight hanger from this string and add mass until the ring is again centered. You
may need to make slight adjustments to the angle.
3.
Record the angle and magnitude of the mass in Table 1. Compare these to the equilibrant
obtained on your graphical addition of forces.
Table 1. Comparison of Graphical and Experimental Results for the Equilibrant
Graphical Result
Force Table Result
% Difference
Magnitude (N)
Direction (deg)
Revised Summer 2010