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Force Vectors: Graphical and Analytical MethodsTITLE

I. Abstract

This report aims to study vectors and to determine the resultant and equilibrant of any number of forces

both by different methods. The methods used were graphical or polygon, analytical or componentand experimental

methods. The experiment was done in two parts: in the first part, the given vectors were (1) 50 g at 100 and (2) 70 g

at 45; while in the second part, the given vectors were (1) 80 g at 170, (2) 60 g at 245 and (3) 100 g at 310. All

methods were used in each part and the data collected were tabled and the formulas used were shown in this report.

In addition, properties of these quantities such as associativity and commutatively of the addition operation were

explored.

II. Introduction

All measurable quantities can be classified as either a scalar or a vector. Physical quantities that can be

completely specified by magnitude only are called scalars. Examples of scalar quantities are the number of students

in a class, the mass of an object, or the speed of an object, to name a few. Some physical quantities have both

magnitude and direction, these are called vectors. Examples of vector quantity include spatial displacement,

velocity, force, and acceleration. The statement "a car is traveling at 60 mph" tells us how fast the car is traveling but

not the direction in which it is traveling. In this case, we know the speed of the car to be 60 mph. On the other hand,

the statement "a car traveling at 60 mph due east" gives us not only the speed of the car but also the direction. In this

case the velocity of the car is 60 mph due east and this is a vector quantity. Scalar quantities can be added together

algebraically taking account of only the signs of the quantities. Vectors, on the other hand, may be added in several

different ways: (1) graphically, by drawing a scale diagram and measuring the magnitude and direction of the

resultant, (2) analytically, by calculation using trigonometry, (3) and by experimental method.

In graphical method, when two forces act upon an object, their combined effect can be determined by

adding the vectors which represent the forces. One method of performing this addition is known as the graphical

method. In this method, arrows are drawn in the direction of the forces. The lengths of the arrows are proportional to

the magnitudes of the vectors. The resultant is formed by constructing a parallelogram with the two components

serving as sides as shown in figure 1.

To add two vectors, slide the second vector so that its tail is at the head of the first vector. The sum of the

two vectors is a vector drawn from the tail of the first vector to the head of the second vector.To find the difference

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of two vectors, we can take the negative of the second vector and add it to the first vector following the steps

described above for addition.

In the analytical method, vectors are added by finding the components of each vector projected along the

axes of some suitable coordinate system. In other words, the vectors are written in the usual x, y format, and added.

The resultant is then found and expressed in terms of its magnitude and direction by using Pythagoras theorem and

the appropriate trigonometric functions.Addition or subtraction of vectors involves breaking up the vectors into its

components and then performing the addition or subtraction to the x and y components separately. The following are

the formulas used in analytical method to solve for resultants magnitude and direction:

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Lastly, in experimental method, we apparatus use is the force table. The method of rectangular components

enables the resultant of a set of coincident vectors of various magnitudes and directions to be determined using

simple trigonometry and without the use of the sine and cosine formulae. Force table is a simple tool for

demonstratingNewtons First Law and the vector nature of forces. This tool is based on the principle ofequilibrium. An object is said to be in equilibrium when there is no net force acting on it. An object with no net

force acting on it has no acceleration. By using simple weights, pulleys and strings placed around a circular table,

several forces can be applied to an object located in the center of the table in such a way that the forces exactly

cancel each other, leaving the object in equilibrium. (The object will appear to be at rest.) We will use the force table

and Newtons First Law to study the components of the force vector. The table consists of a circular top supported

by a heavy tripod base. There is a small peg located at the center of the top and the perimeter of the table is

graduated in degrees. Forces are applied to a small ring by means of strings connected over pulleys to weight

hangers. By varying the total mass on each string as well as the direction at which each string acts, one can adjust

the equilibrium position of the ring so that its center is the peg. This equilibrium configuration is the only one where

the angles measured along the edge signify the correct direction of each string.

Using any of these methods, a set of vectors (such as forces acting at a point) may be reduced to a singlevector. If the resultant is zero, the vectors are described as being in equilibrium. If it is not zero, a vector which is

equal in magnitude but opposite in direction to the resultant is called the equilibrant such that the set of vectors and

its equilibrant are in equilibrium.

III. Experimental Set-up

a. Apparatus: Force Table, Set of Weights, Weight Holders,

Protractor, Ruler, Graphing Paper

b. Sketch:

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c. Procedure:

IV. Observation

V. Data and Result

VI. Data Analysis

1. In this experiment, when equilibrium has been established with the ring in the center table, you will find

that the ring shifted horizontally a considerable distance away from the center of the force table and still be

in equilibrium. How do you account for this fact?

Answer:

Shifting of rings shows inequality of force, even if the other force is great, it will still be in state of

equilibrium if the opposing force is equal to the other force.

2. How do you find the vector sum and vector difference of the vector quantities?

Answer:

Vector sum is obtained by Parallelogram Rule of addition which is a partial case of general

Polygon Rule used for adding several vectors, while vector difference is obtained by Triangle Method of

subtraction.

VII. Conclusion

Problems:

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1. A spelunker is surveying a cave. He follows a passage 100 meters straight east, then 50 meters

on a direction 30 west of north, then 150 meters at 45 west of south. After a fourth unknown

displacement, he finds himself back where he started. Determine the magnitude and direction of the fourth

displacement.

Answer:

F1= 100m E

F2= 50m at 30 NW

F3= 150m at 45 SW

F1x = 100cos0 = 100 F1y = 100sin0 = 0

F2x = 50cos30 = -43.30 F2y = 50sin30 = 25

F3x = 150cos45 = -106.07 F3y = 150sin45 = -106.07

-49.37 -81.07

94.92 meters at 58.66 NE

2. A sailboat sails 2.0 km. East, then 4.0 km. Southeast, then and an additional distance in an

unknown direction. Its final position is 6.0 km directly east of the starting point. Find the magnitude and

direction of the third leg of the journey.

Answer:

F1= 2.0km EF2= 4.0km SE

F3= ?

Ffinal = 6.0km E

F1x = +2.0km = 2.0 F1y = 0 = 0

F2x = +4.0cos45 = 2.83 F2y = -4.0sin45 = -2.83

Ffinal= +6.0 km = 6.0 Ffinal= 0 = 0

F1x + F2x + F3x = Ffinalx F1y + F2y + F3y = Ffinaly

2.0 + 2.83 + F3x = 6.0 0 + (-2.83) + F3y = 0

F3x = 1.17 F3y = 2.83

3.06km at 67.54 NE

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