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].
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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.