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