Experiment 2
Sum of Vectors
Objectives
1. Use the force table to balance a point by the application of three known concurrent forces
2. Find the resultant of these forces by
a. The polygon method (geometrical method)
b. The method by components (analytical method)
c. The law of cosines (trigonometric method)
Theory
Forces are vectors. To define them we need to specify the magnitude as well as the direction. This
quality as a vector makes that the sum of the forces transforms into a different process to add numbers or
scalars. In this lab exercise we will do the vector sum of three known forces using three different methods.
In each of these methods we will represent the forces with arrows. If we want to draw the arrows it must be
done such that their lengths are proportional to the force while their orientation shows the direction.
Example 1
A book whose weight is 10 N is resting on a table. (N represents newtons that are the units of force in the
International System of Units. A newton is approximately 98 grams -force.) See Figure 2-1. Notice that
the arrow representing vector w has a length of about 2 cm and points in the direction of the weight of the
book. In this case we have chosen a scale such that 10 N corresponds to 2 cm. If we apply a horizontal
force F of 15 N to push the book to the right, this force is represented by an arrow parallel to the surface of
the table, with a length of 3 cm since each 5 N equals 1 cm in the scale chosen. See Figure 2-1. Observe
that when a letter is used to refer to the vector it is in bold. What length would the arrow have to represent
a vector of 100 N if we use the same scale of 5 N per centimeter?
Example 2
The necessary force to hold the weight of an adult person is approximately 600 N. In the scale used in
600 N
Examp le 1, the length of the arrow representing this force would be of
= 120 cm . Clearly in this
N
5
cm
case due to the size of the sheet paper we wouldn’t be able to draw an arrow with this magnitude. The
scale must be modified. Let’s assume that 1 cm would equal 100 N, and then the length of the arrow would
be only 6 cm. The person who draws the arrow into his/ her convenience and space available to draw the
arrows will determine the scale used.
Polygon Method to Add Vectors
This is a geometric method. It will be described with the following example.
Example 3
Let A = 10 N, B = 15 N and C = 5 N three vectors such that A points directly to the right, B to the top and
C, 45° below the horizontal line and to the left. Notice that when we refer only to the magnitude of the
vector, the bold is not used. We want to represent the vectors with arrows. The first step consists of
choosing a convenient scale that permits converting the newtons into centimeters. Let 2.5 N = 1 cm. With
this scale the lengths of the arrows that represent the vectors A, B and C will be: A’s length,
10 N
15 N
5N
= 4 cm , B’s length,
= 6 cm , and C’s length,
= 2 cm See Figure 2-2.
2.5 N
2.5 N
2.5 N
1 cm
1 cm
1 cm
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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Figure 2-1. Arrows represent vectors. Their length is proportional to their magnitude. We use F for the horizontal
force, and w for the book’s weight
Figure 2-2. Three forces A, B, and C represented by arrows
The sum of R = A + B + C is a new vector that we will call the resultant and is represented by R. The
resultant is obtained after joining the beginning of vector A with the end of vector C once vectors B and C
have been transferred from their original positions to new ones where the beginning of B joins the end of A
and the beginning of C the end of B. See figure 3. Measuring the length of R and converting it to newtons
find the magnitude of the resultant. In this example the length measured is of 5.2 cm that equals 13 N. To
determine the direction of R we measure with a protractor the angle between A and R. Its value is 60°. It
is evident that the use of this method requires paper, pencil, geometry instruments, and the ability to draw,
as well as to measure, and use such instruments correctly. The method is very simple but requires time and
it doesn’t have good precision. In the following section we will see how to add these same vectors in an
easier manner, more precise and faster.
Method by Components
In this analytical method we break each vector into two perpendicular components as illustrated in
example 4.
Example 4
Vectors A, B, and C of example 3 are used. The first step to add analytically; consist in defining a
coordinate system where the direction of each vector is specified. Look at Figure 2-4. Notice that with this
method all vectors are placed with their tails in the origin.
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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Figure 2-3. The same forces A, B y C, making a polygon with the resultant R
Figure 2-4 Vectors A, B and C set up on a Cartesian system of coordinates
Each vector will be broken down into their horizontal and vertical components. To do this, refer to Figure
2-5. In this figure vector F is located in a coordinate system. The magnitude of the vector is proportional
to the length of the arrow, while its orientation is determined by the angle θ that the vector forms with the
positive side of x-axis. Observe that the triangle Oab is a right triangle in the vertex a. The side Oa is the
leg of the right triangle adjacent to angle θ. The side ab is opposite to angle θ and the segment Ob is the
hypotenuse of the triangle. Segment ab is parallel to y-axis and cb parallel to x-axis. Segment Oa has been
named Fx, segment ab is Fy, and F is the arrow Ob.
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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Figure 2-5. A vector F is broken down into two mutually perpendicular components Fx and Fy
By elementary trigonometry we have that Fx = F cosθ and Fy = F sinθ, where Fx and Fy are known as the
Cartesian components of vector F. Now we will use these equations to find the components of vectors A,
B, and C. The angles that these vectors make with the positive side of x-axis are 0°, 90,°and 225°
respectively. Remember that the magnitudes of A, B and C are 10 N, 15 N and 5 N, then
Ax = 10cos 0º = 10 N, Ay = 10sin 0º =0
Bx = 15cos 90º = 0, By = 15sin 90º = 15 N
Cx = 5cos 225º = -3.54 N, Cy = 5sin 225º = -3.54 N
The components throughout each axis can be treated as scalars. Rx will be called the sum of the
components in the x-axis and Ry will be the sum of the components in the y-axis, which is
Rx = Ax + Bx + Cx = (10 + 0 – 3.54) N = 6.46 N
Ry = Ay + By + Cy = (0 + 15 – 3.54) N = 11.46 N
Figure 2-6 shows these components and the new vector R. From this figure we notice that angle θ can be
Ry
deduced from the tangent function where tan θ =
, while the magnitude R is obtained from Pythagoras
Rx
Theorem R =
R x2 + R y2 . We must also notice that the vectors A, B, C, and R can be written in the
following form: A = (10, 0), B = (0, 15), C = (-3.54, -3.54) and R = (6.46, 11.46), adding that the units
associated with these numbers are newtons, N. In this case, we say that the vectors are written in their
rectangular representation. Also one should know that we can write them like: A = 10 i, B = 15 j, C = 3.54 i - 3.54 j, and R = 6.46 i + 11.46 j, where i, and j are the unitary vectors, that is to say, with unit
magnitude, and parallels to the x-axis and y-axis respectively. In our particular case,
R y 11.46
tan θ =
=
= 1.77 where θ = tan-1 1.77 = 60.6°. On the other hand,
Rx
6. 46
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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R=
R x2 + R y2 = 6. 46 2 + 11 .46 2 = 13. 2N
Remember that by the polygon method we had found that θ = 60° and R = 13 N. It is necessary neither to
draw the vectors nor to define a scale for length with the analytical method or by components. The
drawing was done only to illustrate the method.
Figure 2-6. The vector sum of components Rx y Ry equals the resultant vector R whose direction is given by angle θ
and its magnitude, by the Pythagorean theorem
Law of Cosines Method
This is a trigonometric method, which we will show with the following example:
Example 5
Let us use vectors A and C from examples 3 and 4. Remember that A = (10 N, 0°) and C = (5 N, 90°). We
use this form to write these vectors like a pair of ordered numbers within parenthesis, which is known as
polar representation. It consists of presenting the components of the vectors in polar coordinates. We
wish to find vector D that represents the sum of A + C. See Figure 2-7. In this method we do not need to
draw the vectors. We do it once again to facilitate its presentation and explanation. With this method we
do not have to worry about scales.
Figure 2-7. Vectors A, C and D form a triangle
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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According to the polygon method D = A + C as you can see. In this case there are only three vectors, so
the polygon reduces to a triangle for which we know the lengths of sides A and C as well as the angle
between them. The law of cosine establishes that:
D2 = A2 + C2 –2AC cos 45° = 102 + 52 – 2(10)(5)(0.707) = 54.3
From which D = 7.4 N. The direction of the resultant vector D is obtained from the value of θ which we
calculate after applying once again the law of cosines.
C2 = A2 + D2 –2AD cos θ,
and
 A2 + D 2 − C 2
θ = cos−1

2 AD


 10 2 + 7.4 2 − 5 2
 = cos −1 

 2 × 10 × 7. 4



 = 28.7 0


Operations with vectors using a hand calculator, or a computer
Actually, operations with vectors are made using almost exclusively the method by components.
Generally we first have the vectors in their polar form and must add them. The procedure consists of
transforming them to the rectangular form, adding them and then again turns the result to the polar form
Example 6
We are going to add the vectors A = (10, 30 °), B = (24, 45 °) and C = (5, 150 °), expressed in polar form.
We use the following table:
Polar representation
Vector
A
B
C
R
Magnitude
10
24
5
R = R x2 + R 2y =
= 21 .32 + 24 .47 2 =
= 32.44
Rectangular representation
Direction
30 °
45 °
150 °
Ry
θ = tan −1
=
Rx
=
x-Component
Ax = 10 cos 30 °= 8.66
B x = 24 cos 45 °= 16.97
C x = 5 cos 150 °= -4.33
R x= A x + B x+ C x=
= 21.30
y-Component
Ay = 10 sin 30°= 5.00
By = 24 sin 45 °= 16.97
Cy = 5 sin 150 °= 2.50
Ry = Ay + By + C y =
= 24.47
24.47
= 48. 86 o
21.30
The order in which the operations are made is the following one: 1st Write the names of the vectors in the
first column, and in the firs three rows, 2nd Write their magnitudes in the second column, 3rd Write their
directions in the third column, 4th Convert the vectors to the rectangular representation, and write their xcomponents in the fourth column, while their y-components in the fifth column, 5th Add independently the
x-components and the y-components, and write the results in the fourth row in columns four and five, 6th
Convert the rectangular components of the resulting vector to its polar components. See fourth row of
columns firs and second. The sum of these three vectors gives by resultant vector in polar form: R =
(32,44, 48.86 °). We see that adding vectors, expressed originally in polar coordinates, is a long and
tedious process. Luckily, most of the hand calculators, nowadays common to university students, are able
to perform these operations with extreme facility. Let us see the particular case of the Texas Instruments
Ti-86 calculator. In order to add these same three vectors simply we turn on the calculator, and press the
keys yellow 2nd followed by MODE, that open a screen where the different ways are enlisted in which the
computer works, we noticed that the Normal way in the left superior corner of the screen is selected with a
flashing cursor, we used the arrows of the keyboard to lower the cursor to the fourth line and soon we move
a place to the right to select PolarC. Now we press key ENTER followed by EXIT . This returns us to the
original screen. Here we write the vectors in their polar form, between parenthesis, as: (10 ∠30)+(24
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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∠45)+(5 ∠150), now we press ENTER and we obtain (32.44 ∠48.96). This procedure is similar to which
must be followed with other calculators. Each student must review the manual of operations of his
calculator to learn how to handle vectors, since his success in this course will partly depend on this ability
In the computers of the laboratory there is a program that allows conducting operations with vectors in
graphical and algebraic form. In order to accede the program the left button of the mouse is pressed on the
following icon:
Vector tutorial.lnk
Next the program is opened, and the student can make the tutorial that is offered. We recommend that all
students become familiar with this resource to improve their understanding of the properties of vectors, and
how to work with them
Materials
Thread
Weights of different values
Weight holders
Graphic paper of 8 ½” by 11”
Instruments
Level to adjust table’s legs
Triangles of 45° - 45° and 30° - 60°
Metric ruler of 30 cm, graduated in mm
Protractor
Equipment
Table of forces with pulleys
Procedure
1.
The first activity of this exercise consist of being able to make that the center of concurrence of the
three threads that hold the weights in the table of force align with the center of the table after placing
the arbitrary weights in the three weight holders. Make sure that the total mass added to each of the
weight holders exceeds 100 g but is less than 500 g. This permits us to ignore the effect of friction in
the pulleys. To obtain the alignment you can do the following:
a.
b.
Relocate the pulleys in other positions in the perimeter of the table of force, and
Increase or decrease the mass of one of the weight holders.
You must keep in mind that the precision of the results that you obtain will be greater as the alignment
between the concurrence of the threads and the center of the table are better. The exercise will be more
beneficial if the added masses to the weight holders are not identical. Once you have obtained the best
alignment possible, we will anchor the ring where the threads are tied up, using a metallic bar that will
be provided for this purpose.
2.
The perimeter of the table of force is sub-divided into degrees. In the table there exists a line whose
position corresponds to 0°. This line will be designated as the x-axis. The line that corresponds to 90°
is y-axis. Take the graphic paper and chose an appropriate scale that permits you to draw the forces
which are the weights of the masses in the weight holders like arrows on the paper.
3.
Choose the center of the sheet of paper as the origin of the coordinate system and draw the x and yaxes.
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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4.
Read the angles that the threads form with the x-axis and draw three straight lines on the graphic paper
such that each one begins in the origin (center of the paper) and subtend an angle equal to that of the
corresponding thread. The length of each ones of these lines should be proportional to the weight it
represents. As mentioned before, the forces are the weights of the masses added to the weight holders
plus the weights of the weight holders. The weights are obtained by multiplying the masses by 9.8
m/s 2 .
Data Analysis
1. Use the polygon method to find the magnitude and direction of the resultant vector of the three forces.
Write both results in the lab report. Attach to your lab report the graphic paper where the original
vectors were drawn as well as the polygon. Use the two triangles 45°- 45° and 30°- 60° to facilitate the
location of the vectors to form the polygon.
2.
Name A, B, and C the three arrows that were drawn on the graphic paper. Use letter A for the arrow
with the smallest angle, B for the next to the smallest angle in the counter clockwise direction, and C
for the arrow with the biggest angle.
3.
Use the Law of Cosines to find the resultant of the sum of A + B. Identify this resultant with the letter
D. Write in your report the magnitude of D and the angle it forms with the x-axis.
4.
Use again the law of cosines to add D and C and name R the resultant vector. Write in your report the
magnitude and direction of R.
5.
Write down the magnitudes of vectors A, B, and C in the first column of Table 1of the report. Write
the directions of the x and y-components of each vector and write them down in columns 3 and 4 of the
table. Calculate:
Rx = Ax + Bx + C x y Ry = Ay + By + C y
And write the values in the corresponding place in the report.
6.
Calculate R
=
R x2 + R y2 and θ using tan θ =
Ry
Rx
and write their values in the report.
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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Experiment 2. Laboratory Report
Sum of Vectors
Section: _____________ Bench: ___________
Date: ____________________________________________
Students:
1.
___________________________________________________________
2.
___________________________________________________________
3.
___________________________________________________________
4.
___________________________________________________________
1.
Polygon Method
Magnitude of the resultant R = ________________________________ N
Direction of the resultant θ ° = __________________________
2.
& 3. Law of Cosines
Magnitude of vector D = _______________________ N
Direction of vector D, θ ° = ________________
3.
Magnitude of vector R = ____________________N
Direction of vector R, θ ° = _________________
5.
& 6 Method by Components
Table I
Magnitude of vector
Direction
x-Component
y-Component
A=
N
N
N
B=
N
N
N
C=
N
N
N
D=
N
Rx =
N
Ry =
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
N
9
Questions
1.
Did you obtain the same result with the three methods? Explain.
2.
If there were differences in the results, what were the reasons for that?
3.
Which would be the most convenient method if the number of vectors were large?
4.
What is the magnitude and direction of a vector which components are: Fx = 100 N, Fy = 173 N?
Show your calculations.
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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Experiment 2. Questions
Sum of Vectors
This questionnaire has some typical questions of experiment 2. All students taking the laboratory of
University Physics I course must be able to correctly answer it before trying to make the experiment.
1.
The sum of the two concurrent vectors of 6 N and 4 N has a magnitude of 10 N. The angle between
these two vectors is,
a. 180°
b. 90°
c. 60°
d. 0°
2.
Two vectors whit magnitudes of 15 cm and 20 cm have mutually perpendicular directions. The
magnitude of the sum of these two vectors is,
a. 35 cm
b. 20 cm
c. 25 cm
d. 15 cm
3.
Vectors are quantities that,
a. Are added like scalars or numbers
b. Are characterized as having magnitude and direction
c. Are used exclusively for forces
d. Have no magnitude
4.
We have chosen a scale of 1.5 cm/ 10 N to draw the forces. An arrow with a length of 7.5 cm
represents a force of
a. 200 N
b. 50 N
c. Data is missing
d. 288 N
5.
An arrow has a length of 12 cm in a scale of 24 cm/100 N. This arrow represents a force of
a. 200 N
b. 50 N
c. Data is missing
d. 288 N
6.
To add vectors with the geometric method, we
a. Do not need to draw the vectors
b. Should represent the vectors with arrows which lengths are proportional to their magnitudes
c. Do not need to define a scale
d. Know that the precision of the results is maximum
7.
The graphic method used to add vectors is also known as:
a. Analytical method
b. Cartesian method
c. Method by Components
d. Geometric method
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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8.
The method by components to add vectors is also known as
a. Geometric method
b. Trigonometric method
c. Algebraic method
d. Analytical method
9.
In the Law of Cosines Method
a. It is necessary to define the scale before adding
b. We need drawing material and equipment
c. Some trigonometry calculations must be done to find the x an y-components of the vectors
d. Only two vectors can be added each time
10. To specify analytically the direction of a vector ______________ is used
a. A scalar
b. The angle that makes the vector with the positive side of x-axis when it’s beginning coincides
with the origin of the Cartesian coordinate system.
c. Its magnitude
d. A protractor
11. The components of a vector are their:
a. Magnitude and direction
b. Polar coordinates
c. Projections on the Cartesian x and y-axes
d. Units
12. Suppose we have two vectors A (4, 0°) and B (-3, 180°) in polar coordinates. Their sum is,
a. (7, 0°)
b. (1, 180°)
c. (5, 90°)
d. (5, tan-1 ¾)
13. Let A and B be vectors which polar coordinates are (4, 0°) and (-3, 90°) Their sum is
a. (1, 45°)
b. (5, 0°)
c. (5, -37°)
d. (-1, 4)
Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by
the National Science Foundation (NSF) © All rights reserved (2006)
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