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A Lab Report Format for Mapua PHY10L Course under Professor Ryan Cabrera
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Experiment 1: Resolution of Vectors
Alejo, J.T.a, Caliwag, A.a, Carlos, J.a, Casenas, J.a
Cabrera, R.M.b
(a) Group 1B, PHY10L (A5), A.Y. 2013-2014:Q1, Department of Physics,
Mapua Institute of Technology – Intramuros, Manila
(b) Faculty, Department of Physics,
Mapua Institute of Technology – Intramuros, Manila
Abstract: The scope of the experiment is within the analysis of vectors both theoretically
and experimentally and how they are compared to one another. Also, the First Condition
of Equilibrium is included in the range of the experiment by which it serves as a guide in
determining the relationships of given vectors and the resultant vector. The theoretical
aspect of the experiment is about the resolution of the Resultant Vector through the use of
Component Method. The experimental part is done through the use of Polygon (Graphical)
Method and through the use of Force Table. The goal of the experiment is to know which
of the method is the accurate, efficient, and convenient with regards to the findings of the
result. In Polygon Method, percentage errors on values of R ranges from 0.95% to 3.08%
and angle ranges from 0.31% to 0.32%. In Component Method, percentage errors on values
of R ranges from 0.50% to 1.91% and angle ranges from 0.02% to 0.05%.
1. Introduction
Physical quantities are integral in the study of Physics. These quantities are distinguished
based on their magnitude and direction. When a quantity contains only magnitude such as
mass, distance, and time, it is considered as a Scalar Quantity. Most of the time, these
quantities, together with their magnitude, are not enough to solve a problem in Physics.
When a quantity contains both magnitude and direction, the quantity is considered as a
Vector Quantity. Example of Vector Quantity are displacement, velocity, acceleration, and
force.
Vectors can be added and/or subtracted. When the vectors are combined (whether
by addition or subtraction), the produced vector is defined as the Resultant. The direction
of the Resultant Vector can be reversed and when the direction of the Resultant Vector is
opposed to the original, it is now called as Equilibrant. Nevertheless, although Resultant
and Equilibrant are of the opposite direction, they both have the same magnitude [3].
In finding the resultant, various method are used. The experiment consists of two
parts; the Polygon Method (a Graphical approach) and Component Method (an Analytical
approach). However, an experimental approach is introduced as an alternative way of
finding the resultant in the experiment and it is through the use of the Force Table.
In the Force Table Method, finding the resultant is obtained through trial and error
by which different strings and mass hangers are manipulated in order for the ring to be in
centered. A new concept is introduced in finding the resultant in relation on as to why the
ring is needed to be centered. The concept is based on the First Condition of Equilibrium
– which defines as “vector sum of all the forces acting on a body” [1].
2. Theoretical and Conceptual Framework
Vectors are important in resolution of physical quantities such as displacement, velocity,
and force. By resolution, it means decomposing a vector into component in the respective
axes. In the experiment, resolution of vectors is found by three methods namely the
Polygon Method, Component Method, and Experimental (Force Table) Method.
2a. Polygon Method
The Polygon Method (otherwise known as Head-to-Tail Method) is the method of
finding the resultant vector graphically. A protractor, a ruler, and a drawing instrument are
used to draw and as well as measure the magnitude and direction of given vectors and also
the resultant vector [6].
Fig. 1 shows the Polygon Method
Figure 1. Polygon Method [6]
The vectors F1, F2, and F3 are drawn and the resultant is traced from the initial
point (starting point of F1) up to the terminal point of F3. The resultant is geometrically
drawn – the tail lies on the initial point and the arrow-head lies on the terminal point.
After the sketching of the vectors, the magnitude of the resultant is measured using
a ruler and its direction/angle is measured using a protractor.
2b. Component Method
In the component method, finding the resultant is solved using an analytical
approach. Concepts of Summation of Components, Finding the Magnitude using
Pythagorean Theorem, and Finding the Angle of the Resultant using Inverse Tangent
Function are used to find the resultant vector.
2bi. Summations of Components
A vector is composed of components. In a 2-dimensional perspective, a vector
contains x and y components. The z-component of a vector is added in a 3-dimensional
perspective. In the experiment, we only focus on vectors in 2-dimensional plane [4].
The following figure shows the graphical analysis of the components of a vector in
a plane.
Figure 2. Analysis of Components [4]
Based from the figure, the x and y components of �� vector can be derived and their equations are as follows:
𝑎𝑥 = 𝑎 cos 𝜃 (1a) 𝑎𝑦 = 𝑎 sin 𝜃 (1b)
While the method is used in finding the x and y components of a vector, it can also
be used on finding the resultant vector by summing up the components of the given vectors
[3].
The equation for the summation of components are of the following:
∑𝑅𝑥 = 𝑅1𝑥 + 𝑅2𝑥 + 𝑅3𝑥 …+ 𝑅𝑛𝑥 (2a) ∑𝑅𝑦 = 𝑅1𝑦 + 𝑅2𝑦 + 𝑅3𝑦 …+ 𝑅𝑛𝑦 (2b)
2bii. Magnitude of the Resultant
Since the resultant is a vector, it also has components. The components of the
resultant are the sum of the x and y-components of the given vectors. Its x-component is
Eq.(3) and its y-component is Eq.(4).
By using the components of the resultant, Eq. (3) and Eq. (4), its magnitude can
then be derived and its equation is based on Pythagorean Theorem [3].
𝑅 = √(∑𝑅𝑥)2 + (∑𝑅𝑦)2 (3)
2biii. Direction of the Resultant
The direction or the angle of the resultant can be derived based on the concept of
right triangle wherein the Tangent function of the angle is equal to the ratio of its y and x-
component. Again, we will use Eq. (2a) and Eq. (2b) as the components of the resultant
[3].
tan 𝜃 = (∑𝑅𝑦
∑𝑅𝑥) (4)
To get, the angle, we transform Eq. (4) into:
𝜃 = tan−1 ( ∑𝑅𝑦
∑𝑅𝑥) (5)
Take note that the angle might vary because of the location of the x and y-
components in different quadrants.
Q1 Q2 Q3 Q4
x-component + - - +
y-component + + - -
Table 1. Components in Quadrants
2c. Force Table – Experimental Method
The Force Table is a circular instrument composed of 4 strings each with a certain
mass and center ring by which all of the strings are attached on the center. The concepts
behind the experiment is to (1) demonstrate the First Condition of Equilibrium; that if the
ring is centered, all the forces that are acting on it are zero and (2) the missing resultant is
the equilibrant of sum of the other vectors [2].
The theories and concepts in this section will all be useful in the experiment –
finding the resultant/missing vector in a given problem by using different methods. Further,
in the end of the experiment, we will learn as to which of the method is the accurate,
efficient, or practical on finding the resultant vector.
3. Materials and Methods
The materials used in this experiment “Resolution of Vectors” are the following:
1 pc. Force table - is a common physics laboratory 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.
4 pcs. Super pulley with clamp - makes set-up and alignment easy.
4 pcs. Mass Hanger – use to contain the different weights.
1 set Slotted Mass - are used in student lab classes, to teach physics and other
sciences. The slots allow them to be placed on weight hangers, which are
lightweight platforms attached to a thin rod with a hook at the top. Various
masses are added to the hanger to create the desired amount of total mass
(standard masses are 500, 200, 100, 50, 20, 10, 5, 2, and 1 gram), then the
combination is hung by the hooked end from a string or other support point
1 pc. Protractor - An instrument for measuring angles, typically in the form of a flat semicircle marked with degrees along the curved edge.
Figure 3. Setting-up of Force Table Experiment
We were oriented that we should take utmost care on the super pulleys to avoid
damages. We were advised to ask the instructor for ideal masses of the hanger to be used.
Methodology
Figure 3. Methodology of the Experiment
Fig. 3 shows the step by step process on how the experiment was done. We should
always remember to follow these steps carefully to obtain accurate results. The first step is
to set up the force table and assemble the four pulleys for the system. Secondly, attach a
hanger at the end each string and suspend a mass on each hanger. Next that we did was the
adjustment of the angle of the strings until the ring is at the center. Then, pull the ring
slightly to one side and observe if the ring returns to the center. Once the balance or the
equilibrium is obtained, record the mass of each string and its angles respectively. Lastly,
determine the resultant force by component method and the polygon method. Repeat steps
for another trial.
4. Results and Discussion
In this experiment, our task is to determine the resultant of the three vectors using two
different methods, polygon and component. We are to compare the results of the two
methods used. We have also computed the percent error based on our data. With this, we
have been able to come up with sets of data gathered from our experiment.
Preparation of Materials Setting Up the Equipment
Assemble the 4 Pulleys Attach a hanger at the end of each
string.
Adjust the angle of the strings Record the mass on each string
Determine the resultant of the three
vectors. Perform another trial by repeating
the procedures.
Table 2. Actual Values
Table 3. Trial 1 of the Experiment
Table 4. Trial 2 of the Experiment
Based from the results, the percentage error for the R of Polygon Method ranges
from 0.95% to 3.08% while its angle ranges from 0.31% to 0.32%. Meanwhile, the
percentage error for the R of Component Method ranges from 0.50% to 1.91% and its angle
ranges from 0.02% to 0.05%. We can notice that the ranges in errors of R and angle of
Component Method is lower than the ranges in errors of R and angle of Polygon Method.
We can say that finding the resultant of a vector is more accurate when we are using the
Component Method.
Actual Values Trial 1 Trial 2
F1 30g 40g
F2 40g 60g
F3 45g 85g
F4 65g 105g
θ1 45o 45o
θ2 115o 115o
θ3 190o 190o
θ4 309o 45o
Trial 1
Actual
R=F4= 65g
Actual θ =309o
COMPUTED
VALUES
Polygon
Method
% error
(polygon
method)
Component
Method
% error
(component
method)
R 63g 3.08% 63.76g 1.91%
θ 310o 0.32% 308.86o 0.05%
Trial 2
Actual
R=F4= 105g
Actual θ =320o
COMPUTE
D VALUES
Polygon
Method
% error
(polygon
method)
Component
Method
% error
(component
method)
R 104g 0.95% 105.53g 0.50%
θ 319o 0.31% 319.95o 0.02%
5. Conclusion and Recommendation
We can now conclude that the methods of finding resultant vectors from given vectors have
different characteristics. The Component Method is the more accurate method that is used
in finding resultant vectors because of its lower percentage errors. Meanwhile, the Polygon
Method is the more efficient because doing it requires less effort. You just use the ruler
and protractor to measure the magnitude and the direction of the vector. Among the three,
the Force Table (Experimental) Method is the most practical since you do a trial and error
and you are actually measure and test the directions and masses in the table. Further, the
First Condition of Equilibrium is not to be neglected because of its principle that is constant
to every vectors – that the sum of vectors in a system is equal to zero. If not for this
principle, the resolution of vectors will be difficult and almost impossible.
We recommend that for those who will do this experiment that be careful on all the
methods. Lessen the mistake in sketching the vectors in Polygon Method, avoid careless
computations in Component Method, and handle the Force Table with care for if these
precautions are not followed, errors will surely rise.
References
[1] Andrews University. (n.d.). Applied Physics Experiment 3: Vector Addition of Forces.
Retrieved from
http://www.andrews.edu/phys/courses/p131/manual/experiment3.html
[2] Davis, D. (2002). First Condition of Equilibrium. Retrieved from Eastern Illinois
University: http://www.ux1.eiu.edu/~cfadd/1150/08Statics/first.html
[3] Mapua Institute of Technology - Department of Physics. (n.d.). Laboratory Manual ,
General Physics 1. Experiment101 RESOLUTION OF FORCES.
[4] Resnick, H. &. (2011). Fundamentals of Physics 9th Edition.
[5] The Physics Classroom. (n.d.). Vectors: Motion and Forces in Two Dimensions -
Lesson 3. Retrieved from The Physics Classroom:
http://www.physicsclassroom.com/Class/vectors/U3l3a.cfm
[6] The University of Oklahoma. (n.d.). The Head-to-Tail Method. Retrieved from
http://www.nhn.ou.edu/walkup/demonstrations/WebTutorials/HeadToTailMethod
.htm
Appendix A: Application
Whenever there is direction and magnitude, there is vector. Even from the
distance that we travel every day, from the signals that our laptops receive from Wi-Fi,
the electric current that we utilize in our everyday necessities – all of these are
applications of vectors. Almost everything that has Physics has vector – and it would not
be a surprise because Vector is a special language of Physics.
Application 1. Typhoons both have magnitude and direction. Meteorologists use vectors in order to trace the path of a typhoon. (Image courtesy of PAGASA)
Application 2.Even non-Physics aspects have vectors. Economists use the application of vectors to analyze economic growth of a certain place. (Image courtesy of PHILSTAR)
Appendix B: Answers to Guide Questions
1. Why is it important for the ring to be at the center? Since the mass hangers have equal
masses, can you disregard them in the experiment? Why?
In regards with the mass of the hangers, if we ignore their masses, we will acquire
erroneous result. Suppose that in F1+F2+F3 = 0, if we change the mass in a given force,
the equilibrium will be affected and it will not be zero anymore.
2. When a pull is applied on the ring and then released, why does it sometimes fail to return
to the center?
When you pull the string, you apply external force which disturbs the equilibrium.
In our experiment, there are only four concurrent forces and the sum of these must equal
to zero. If ever an external force is applied, a total of five forces is currently acting on the
system therefore the equilibrium will not be equal to zero anymore.
3. What is the significance of the resultant𝐹1 , 𝐹2
, 𝐹3 to the remaining force 𝐹4
? What
generalization can you make regarding their relationships?
The resultant𝐹1 , 𝐹2
, 𝐹3 must be equal to 𝐹4
in terms of the magnitude but they differ
in direction. Therefore, 𝐹4 is the equilibrant of 𝐹1
, 𝐹2 , 𝐹3
.
4. If the order of adding vectors is changed (i.e from 𝐹1 + 𝐹2
+ 𝐹3 to 𝐹2
, 𝐹1 , 𝐹3
) will the resultant be different? Why?
No, there will be no difference because addition of vectors follows associative law
which states that vector can be added in any order. The resultant will be the same.
5. Which method of the resultant is more a) efficient, b) accurate, c) practical or convenient
to use? Defend your answer.
a. Efficient – Polygon Method
Efficiency means to work with less effort. Tracing the vectors then measuring them
by ruler and protractor is less work. You only draw and measure. That’s it and it is very
simple.
b. Accurate – Component Method
The Component Method is the most accurate because you can calculate up to 3 – 4
decimal places accurately. Also in the experiment, it shows less percentage error than the
Polygon Method.
c. Practical – Force Table (Experimental) Method
It is the most practical because of the reason of it is designed for actual use. You
actually measure and test the directions and masses to get a resultant. You are practicing
in a trial and error way so therefore it is more practical to use.
Appendix C: Answers to Problem Sets
1. Given the following concurrent forces:
F1=5N, North; F2=7N, 30° N of W; F3=10N, 75° W of S
Determine a) F1+F2 b)F2-F1 c) F3+F1-F2
a.) F1 + F2
R = √∑𝑭𝑥𝟐+ ∑𝑭𝑦
𝟐 = √(−6.06)2 + (8.5)2 = 10.44
Φ = tan−1 ∑𝑭𝑦
∑𝑭𝑥 = tan−1 |
𝟖.𝟓
−6.06| = 54.51° (quadrant II)
Ө = 180° - 54.51° = 125.49°
b.) F2 - F3
R = √∑𝑭𝑥𝟐+ ∑𝑭𝑦
𝟐 = √(0.70)2 + (5.31)2 = 5.36N
Φ = tan−1 ∑𝑭𝑦
∑𝑭𝑥 = tan−1 |
5.31
0.70| = 82.49° (quadrant I)
Ө = 82.49°
x - component y - component
F1 5cos90° = 0 5sin90° = 5
F2 7cos150° = -6.06 7sin150° = 3.5
-6.06 8.5
x - component y - component
F2 7cos150° = -6.06 7sin150° = 3.5
-(F3) 7cos195° = -6.76 7sin195° = -1.81
0.70 5.31
c.) F3 + F1 - F2
R = √∑𝑭𝑥𝟐+ ∑𝑭𝑦
𝟐 = √(−0.69)2 + (−0.31)2 = 0.76N
Φ = tan−1 ∑𝑭𝑦
∑𝑭𝑥 = tan−1 |
−0.31
−0.69| = 24.19° (quadrant III)
Ө = 180° + 24.19° = 204.19°
2. Given the following concurrent forces A, B, and C, determine the resultant.
A = 3ǐ + 2ǰ + 4 ǩ
B = -2ǐ + 6ǰ - 7 ǩ
C = 5ǐ - 4ǰ + 9 ǩ
R = (3 -2 +5)ǐ + (2 + 6 – 4)ǰ + (4 – 7 + 9) ǩ = 6ǐ + 4ǰ + 6 ǩ
3. Given the following concurrent forces:
F1 = 10 N at 37° N of W
F2 = 15 N, north
F3 = 14 N toward the negative z-axis
F4 = (-8ǐ + 12ǰ + 4 ǩ)N
R = √∑𝑭𝑥𝟐+ ∑𝑭𝑦
𝟐+ ∑𝑭𝑧
𝟐 = √(−15.99)2 + (33.02)2 + (18)2 = 40.87N
x - component y - component
F3 7cos195° = -6.76 7sin195° = -1.81
F1 5cos90° = 0 5sin90° = 5
-(F2) 7cos150° = -6.06 7sin150° = 3.5
-0.69 -0.31
x - component y - component z - component
F1 10cos143° = -7.99 10sin143° = 6.02 0
F2 15cos90° = 0 15sin90° = 15 0
F3 0 0 14
F4 -8 12 4
-15.99 33.02 18