Our Lady of Fatima – Quezon City

College of Engineering
Civil Engineering Department

PLATE 1
EQUILIBRIUM

SUBMITTED BY

SECTION

DATE SUBMITTED

INSTRUCTOR ENGR. JAEMAR LI REYES

 PLATE 3
EQUILIBRIUM

I. OBJECTIVES

1. Demonstrate the addition of several vectors to form a resultant and equilibrant force using a force
table.
2. Illustrate and practice graphical and analytical solutions for the addition of vectors on the worded
problems.

II. MATERIALS
Force table with pulleys, ring, and string
Mass holders and slotted masses
Protractor
Compass

III. THEORY
Physical quantities which can be categorized in two ways: Physical quantities which have only magnitude
are known as scalar quantities. It is fully described by a magnitude or a numerical value. Scalar quantity
does not have directions. One of its examples are mass, speed, distance, time, energy, density, volume,
temperature, distance, work and so on.

Meanwhile, the physical quantities for which both magnitude and direction are defined distinctly are
known as vector quantities. For example, a boy is riding a bike with a velocity of 30 km/hr in a north-east
direction. Other examples are displacement, acceleration, force, momentum, weight, the velocity of light,
a gravitational field, current, and so on.

Force is a vector quantity which refers to the cause of change in the state of motion of a particle or body’.
Force is the product of mass of the particle and its acceleration.
𝐹 = 𝑚𝑎 Formula 3.1

In case of several forces with varying magnitudes and directions that act at the same point, resultant
force, is the single force, which is equivalent in its effect to the effect produced by the applied forces. It
can be found theoretically by the process of vector addition.
As seen in Figure 3.1, using the Cartesian coordinate system, a point on the graph is specified by two
scalar numbers, x and y. The coordinate x gives the distance and direction along the x-axis, and the
coordinate y gives the distance and direction along the y-axis, from the origin to get to the point in
question. The absolute value of the coordinate indicates the length along the x and y axis, while the sign
(positive or negative) refers to the direction (left or right, up or down) of motion. The vectors can be
decomposed by the use of Formula 2.1-2.4.

Figure 3.1 Cartesian Coordinate System

𝐹𝑥 = 𝐹𝑐𝑜𝑠𝜃 Formula 2.1
𝐹𝑦 = 𝐹𝑐𝑜𝑠𝜃 Formula 2.2

𝐹 = √𝐹𝑥2 + 𝐹𝑦2 Formula 2.3

𝐹
𝑡𝑎𝑛𝜃 = 𝐹𝑦 Formula 2.4
𝑥

Using the graphical vector addition process known as the polygon method, where one of the vectors is
first drawn to scale. Each successive vector force to be added is drawn with its tail starting at the head
of the preceding vector force. The resultant vector is the vector drawn from the tail of the first arrow to
the head of the last arrow, as shown in Figure 3.2. This method is prone to error and inaccuracy that may
be due to human or instrument error.

Figure 3.2 Vector Addition using Graphical Method

To eliminate error, analytical method of vector addition can also be utilized. Using the Triangle Law of
Vector Addition, two sides of the triangle represent two vectors, third side taken in opposite order
represents the vector sum of the two vectors, as seen in Figure 3.3. Meanwhile, in Parallelogram Law of
Vector Addition, two adjacent sides represent (magnitude and direction) two vectors. Diagonal through
common point represents the vector sum, as shown in Figure 3.4.

Figure 3.3 Vector Addition using Triangle Law of Vector Addition

Figure 3.4 Vector Addition using Parallelogram Law of Vector Addition

However, the Vector Addition using Analytical Method is one of the most commonly used due to its
efficient procedures. It utilizes trigonometry to express each vector in terms of its components projected
on the axes of a rectangular coordinate system decomposing vector F into |𝐹|𝑐𝑜𝑠𝜃 and |𝐹|𝑠𝑖𝑛𝜃. The
components along each axis are then added algebraically to produce the net components of the resultant
vector along each axis. Those components are at right angles, and the magnitude of the resultant can be
found from the Pythagorean theorem. Formula 3.5 to Formula 3.8 is used for calculating resultant
magnitude and angle.

Figure 3.4 Vector Addition using Analytical Method

∑ 𝐹𝑥 = 𝐹𝑥1 + 𝐹𝑥2 + 𝐹𝑥2 + ⋯ + 𝐹𝑥𝑛 Formula 3.5

∑ 𝐹𝑦 = 𝐹𝑦1 + 𝐹𝑦2 + 𝐹𝑦2 + ⋯ + 𝐹𝑦𝑛 Formula 3.6

𝐹 = √∑ 𝐹𝑥2 + ∑ 𝐹𝑦2 Formula 3.7

 ∑𝐹
𝑡𝑎𝑛𝜃 = ∑ 𝐹𝑥 Formula 3.8
𝑦

The Force Table, as seen in Figure 3.5, is a simple tool for demonstrating Newton’s First Law and the
vector nature of forces. This tool is based on the principle of “equilibrium”. 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. It is important to take note that a Resultant force is that
single force that acts alone and has the same effect in magnitude and direction as two or more forces
acting together. While Equilibrant force is a single force which will balance all other forces taken together.
It is equal in magnitude but opposite in direction to the resultant force, as shown in Figure 3.6.

Figure 3.5 Force Table

Figure 3.6 Resultant vs Equilibrant

IV. PROCEDURE
A. General Procedure on Force Table
1. Forces are applied to the small ring on the force table by strings, to which different weights can
be attached. The total force on a string is simply the weight of the masses hanging off the end
plus the weight of the hanger itself. Hence,

𝐹𝑠𝑡𝑟𝑖𝑛𝑔 = 𝑀ℎ𝑎𝑛𝑔𝑒𝑟 + 𝑀𝑤𝑒𝑖𝑔ℎ𝑡𝑠

where g = 9.81 m/s2 is the acceleration due to gravity on Earth.
 2. Note that the magnitude of the force on each string can be changed by adding or removing
different weights, and the direction of the force can be changed by moving the arms on the force
machine.
3. The pin at the center serves as a reference point for centering the ring and also prevents the ring
from falling off the table in highly unbalanced situations.
4. Using three strings instead of four is a simpler way to do this experiment, so remove one of the
four hangers. The three strings should initially be about symmetrical. Make sure that the strings
are centered properly on the ring (i.e. slide the knots so that the strings aren't twisted around the
ring). Then load the three hangers with three different amounts of weight (note: try to use pretty
heavy weights; the experiment works better then).
5. Most likely you will find that the ring will not be centered. That means that it is not in static
equilibrium - if the forces from the three strings exactly balanced, then the ring would be perfectly
centered on the table. Instead, it is being pulled to one side or the other because it is feeling a
net force; if the central pin weren't there, the ring would accelerate right off the end of the force
table in accordance with Newton's second law.
6. Move the strings until the ring is free of the pin and nearly centered. Gently tap on the force table
to temporarily eliminate friction, which will allow the ring to move more freely to its new position.
Re-adjust the positions of the strings and repeat the tapping until the ring is well- centered and
remains there. The ring is now in static equilibrium, since the vector components of all three
forces cancel one another out - there is no net force acting on the ring.
7. Record the total weight on each hanger (including the hanger itself) and the direction of each
string, which is the angle written on the force table. Be sure to number each string and its
corresponding weight and angle. Make a sketch of your force table.

V. FORMULA

∑ 𝐹𝑥 = 𝐹𝑥1 + 𝐹𝑥2 + 𝐹𝑥2 + ⋯ + 𝐹𝑥𝑛 Formula 3.5

∑ 𝐹𝑦 = 𝐹𝑦1 + 𝐹𝑦2 + 𝐹𝑦2 + ⋯ + 𝐹𝑦𝑛 Formula 3.6

𝐹 = √∑ 𝐹𝑥2 + ∑ 𝐹𝑦2 Formula 3.7

∑𝐹
𝑡𝑎𝑛𝜃 = ∑ 𝐹𝑥 Formula 3.8
𝑦

𝑚
𝐹 = 𝑚𝑎; 𝑎 = 9.81 𝑠2 Formula 3.8

VI. PROBLEMS
 Solve each Resultant and Equilibrant of the Vector Force Problems using Force Table Method,
Graphical Method, and Analytical Method.
1.

𝐹1 = 300𝑘𝑔
𝐹2 = 200𝑘𝑔
𝜃 = 30°

𝜃 = 50°

𝐹3 = 500𝑘𝑔

2.

𝐹1 = 400 𝑘𝑔

𝐹2 = 700 𝑘𝑔

𝜃 = 20°
𝐹3 = 900 𝑘𝑔

3.

𝑊 = 400 𝑘𝑔

𝜃 = 30°

300 𝑘𝑔

200 𝑘𝑔 𝐴

VII. DATA
 Problem 1
Force Magnitude Direction X Component Y Component
F1
F2
F3
∑
Resultant
𝜃 Resultant
Equilibrant
𝜃 Equilibrant
Table 3.1 Solving Resultant and Equilibrant using Analytical Method

Problem 2
Force Magnitude Direction X Component Y Component
F1
F2
F3
∑
Resultant
𝜃 Resultant
Equilibrant
𝜃 Equilibrant
Table 3.2 Solving Resultant and Equilibrant using Analytical Method

Problem 3
Force Magnitude Direction X Component Y Component
F1
F2
∑
Resultant
𝜃 Resultant
Equilibrant
𝜃 Equilibrant
Table 3.3 Solving Resultant and Equilibrant using Analytical Method

VIII. FORCE TABLE METHOD

Draw the Force Table diagram for each table and label the magnitude and direction of resultant
and equilibrant.

Problem 1

Problem 2

Problem 3
IX. GRAPHICAL METHOD
X. ANALYTICAL METHOD CALCULATION
XI. MISCELLANEOUS PROBLEMS

Using Analytical Method, compute the forces A and B in N on a separate A4 paper.

𝐵 𝐴
𝐴 𝐵 𝐵
𝐴

XII. CONCLUSION
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