Abstract – The experiment made use of a two dimensional plane with a system of clamps, pulley, strings and hangers to which slotted mass are suspended to gain tension on the strings. A center ring must gain a state of equilibrium with the center post by placing the correct amount of force acting on the ring from four different directions on it by adjusting either the amount mass on the hangers or the angle of the string.

I. INTRODUCTION

A scalar quantity is one with only magnitude, or size. A scalar (for short) has only one numerical value. A vector has a magnitude as well as direction and thus can be represented with a vector. We can represent the overall displacement with two successive displacement vectors. The net displacement of these two displacement is a single displacement vector that we call vector sum or resultant. A component of a vector is the projection of the vector on the axis; x axis is its x component, and similarly the projection on the y axis is the y component. The process of finding the components of a vector is called resolving the vector. [1][2]

II. THEORY

To add vectors graphically, make a scale drawing and place the vectors “head to tail”—that is with the tail (origin) of the second vector starting from the head (end point) of the first. Then draw the resultant vector from the origin of the first vector to the end point of the second vector. Finally, measure the length (magnitude) and the direction of the resultant vector directly from the scale drawing.

In the analytical method, since vectors and its component vector form a right triangle, we may use trigonometry. Thus, for the vector magnitude A that makes an angle θ with the x axis, the component Ax, along the direction and the component Ay along the direction are given by

Ax=A cosθ , A y=A sin θ

If we know the components Ax and Ay of a vector, then we can obtain the magnitude and direction of the vector by applying trigonometry. The magnitude A of the vector is computed from the Pythagorean Theorem as

A=√ A x2+ A y

2

The angle θ between the vector A and the x axis is determined from

tanθ=A y

Ax

[3]

III. METHODOLOGY

1.) Experimental Set-up

Fig. 1. Force table with slotted mass suspended in hangers

2.) Materials: The materials needed to conduct the experiment are: a force table, four super pulleys with clamp, four mass hangers, a set of slotted mass, a protractor, and a ruler.

3.) Procedure: The force table, pulleys and hangers were set-up and assembled as shown in Fig. 1. The hangers are attached from the end of each strings, passing over the pulleys where the slotted mass are to be attached and suspended indiscriminately. The mass on the hangers or the angle of the strings are then adjusted until the ring is in the center with the mark on the force table and the center post as reference.

Test the strings by tugging on it lightly to one side for a few times to see if the ring will go back at the center of the table. If the ring fails to return at the center mark, continue adjusting the load or the angle of the strings. Otherwise, proceed to record the mass on each string and its angle as F1, F2, F3, F4, θ1, θ2, θ3, and θ4

respectively.

Experiment 101: Resolution of Forces

Caryl Jean P. FlorentinoPHY10-4/A1

Department of Multimedia Arts and Sciences, Mapúa Institute of Technologycarylflo23@gmail.com

The resultant of F1, F2, F3 is then determined with the use of the graphical and analytical method which is then compared to F4. Two trials of the experiment is performed.

IV. RESULTS

ActualValues

TRIAL 1 TRIAL 2

F1 30 g 40 g

F2 25 g 45 g

F3 35 g 35 g

F4 25 g 45 g

θ1 110° 349°

θ2 165° 113°

θ3 258° 166°

θ4 355° 286°

Fig 2. Data gathered for Trial 1 and Trial 2

TRIAL 1Actual R = F4 = 25 g Actual θ = 355°

Computed values

Polygon method

Percentage error

(polygon)

Component method

Percentage error

(component)

R 25 g 0% 25.36 g 1.44%

θ 357° 0.563% - 5.95° 0.27%

Fig 3. Results yielded using polygon method (graphical method) and component method (analytical method) for TRIAL 1

TRIAL 2Actual R = F4 = 45 g Actual θ = 286°

Computed values

Polygon method

Percentage error

(polygon)

Component method

Percentage error

(component)

R 45 g 0% 44 g 2.22%

θ 282° 1.40% - 16.19° 0.06%

Fig 4. Results yielded using polygon method (graphical method) and component method (analytical method) for TRIAL 2

V. DISCUSSION

From the data recorded, it is seen that the combined forces of F1, F2, F3 for the two trials is almost the same as the value for the actual R (F4) with percentage error of <5% . It can be said that the computation used in determining the resultant force is accurate.

For TRIAL 1, the experimental results for R has 0% in error and θ with a 0.563% error using the graphical method. The experimental results for R has a 1.44% error and θ has a 0.27% error using the analytical method. It

can be said that the graphical method is more accurate for finding out the mass displacement and the analytical method for finding the direction the mass is acting on for TRIAL 1.

For TRIAL 2, the experimental results for R has 0% in error and θ with a 1.40% error using the graphical method. The experimental results for R has 2.22% error and θ with a 0.06% error using the analytical method. . It can be said that the graphical method is more accurate for finding out the mass displacement and the analytical method for finding the direction the mass is acting on for TRIAL 1.

VI. CONLUSION

With the use of the force table, it is verified that the graphical method (polygon method) and the analytical method (component method) are both accurate for finding out a resultant vector with the low percentage error the results have yielded. However, the graphical method is more accurate for finding the displacement of a resultant and the analytical method is more accurate for finding the direction of a resultant. The result of the experiment yielded a 2% margin in error that can possibly merited to human error as the cause. It is also concluded that if the forces acting on an object equates to zero, the object achieves a state of equ

ilibrium.

REFERENCES

[1] Wilson, J., & Buffa, A. (2000). College physics (4th ed., Vol. 1). Prentice Hall.[2] Walker, J., Halliday, D., & Robert, R. (2008). Fundamentals of Physics (8th ed.). Hoboken, NJ: Wiley. [3] Jones, E., Childers, R. (2001). Motion in Two Dimensions, Resolution in Vectors. In Contemporary College Physics (Third edition ed.). McGraw-Hill.