PHYSICS SEMESTER ONE
LAB 3
LAB 3: VECTOR ADDITION OF FORCES
Lab format: This lab utilizes a lab kit.
Relationship to theory: This lab corresponds to Unit 3: Kinematics in Two Dimensions & Vectors
OBJECTIVES
In this experiment, a force table (or table) will be used to verify that forces can be represented
mathematically as vectors. The objectives of this experiment are

To demonstrate the process of addition of several vectors to form a resultant vector.

To demonstrate the relationship between the resultant of several vectors and the equilibrant of those
vectors.

To illustrate and practice graphical and analytical addition of vectors.
If the measurements made in this experiment match the predictions of vector algebra, then it can be
concluded that force is a vector.
EQUIPMENT LIST
Lab Kit*
 balance (Electronic Balance)
 5 ml graduated transfer pipette
 table clamp Pulleys (4)
 plastic bottles with lids (4)
Other
 thread or dental floss
 large washer or ring
 protractor
 water
*If you do not have the Lab Kit, you can use equivalent items. Contact your instructor regarding equivalent
items to ensure that the purpose of the experiment is maintained.
INTRODUCTION
A vector is a mathematical representation of a physical quantity that has both magnitude and direction.
However, not all such quantities can be represented as vectors! It is also necessary for these quantities to
combine (add, subtract) according to the mathematical rules of vector algebra. Ordinary forces have
magnitude and direction, and they do combine to produce net forces in the same way that the vector
model predicts. So: are forces also vectors?
Stresses also have magnitude and direction, but their mathematical representation is more complicated;
they must be described in terms of a mathematical object called a tensor. Mountains have magnitude
Creative Commons Attribution 3.0 Unported License
1
PHYSICS SEMESTER ONE
LAB 3
(height) and direction (up), but have many other properties that are too complex to represent as simple
vector. It is important to note that only an experiment can determine whether or not a physical quantity
has vector properties.
Several vectors added together yield a single vector that represents the sum. That single vector is called the
resultant vector of the vectors that were added. If the vectors represent forces, the sum of the vectors is
called the resultant force.
A consequence of the vector nature of forces is that a single force can be applied in such a manner as to
balance the effect of any other collection of applied forces. To see this, remember that the collection of
applied forces is equivalent to the resultant of those forces. To balance those forces, simply find a force
equal and opposite to the resultant. Such a balancing force is called the equilibrant of the applied forces.
The process of experimentally determining the value of the resultant of several forces is complicated by the
fact that when a non-zero resultant force acts on an object it tends to be accelerated. Thus, another force
must be applied to produce equilibrium (no acceleration). That force is simply the equilibrant just
discussed. So, to measure the resultant of several applied forces, one need only discover the equilibrant.
From the equilibrant, the resultant can be found very easily.
Review the mathematical theory of vector algebra in your physics textbook. You may already be familiar
with the graphical method of addition of vectors if you have taken previous physics course(s). Also review
the analytical method of vector addition, which uses the components parallel to the axes in some
convenient coordinate system. In this experiment, you will practice using both the graphical and the
analytical methods to add vectors. You will also construct a physical representation of the vector sum.
Remember, a vector is a mathematical object that, we hope, represents the physical quality of force
(among other things).
PROCEDURE
Initial Set-Up
Draw two perpendicular lines on a piece of plain paper so that the lines cross near the centre of the paper
(use a protractor!). These will be the x and y axis for this experiment. All noted angles are measured
relative to the positive x-axis. Measure angles using the protractor with base along the x-axis and the eye
of the protractor centred on the location where the perpendicular lines cross (the origin). Secure the paper
near the centre of a table top, as shown in Figure 3.1, below.
Creative Commons Attribution 3.0 Unported License
2
PHYSICS SEMESTER ONE
LAB 3
+x
+y
Figure 3.1. Axes and protractor for measuring oblique angles (180° to 360°)2.
In each of the following exercises, compare the results of the graphical and analytical methods to the
experimental method.
The masses used for the forces in this experiment are water-filled plastic bottles. You can adjust the mass
by adding or removing water using the 5 ml pipette. Screw the lids on tight.
Vector Addition I:
Given two vectors with magnitudes F1  (0.200) g N (here g represents the gravitational field strength,
9.81 N/kg), and F2  (0.200) g N at 30 and 120 respectively, find their vector sum or resultant
F  F1  F2 by each of the following procedures:
a) Graphical. Using the parallelogram method of vector addition, draw a vector diagram to scale. Use
a scale such that the finished vector diagram fills about half a sheet of graph paper. Measure and
record the magnitude and direction of the resultant. Attach your diagrams to your report.
1. Can this vector sum also be calculated by adding them in a ‘tail-to-head’ fashion? Explain.
Creative Commons Attribution 3.0 Unported License
3
PHYSICS SEMESTER ONE
LAB 3
b) Analytical. Using the law of cosines, compute the magnitude of the resultant force. Compute the
angle of orientation of the resultant from the relationship tan    F2 / F1 . (Note that  is the
angle between F and F1 .)
2. Why does this relationship work in this particular case?
c) Experimental: Tie three threads to the washer, and secure the washer over the intersection of the
axes on the paper. Suspend a total (including the bottle) of 0.200 kg mass over a pulley at 30, and
a total of 0.200 kg at 120 (angles measured from positive x-axis). This (we hope) is a physical
representation of the vector sum.
Using the third thread with water and bottle suspended from the end, experimentally determine
the magnitude and direction of the equilibrant force that maintains the washer ring in equilibrium
centred at the intersection of the axes on the paper. Record the resultant of the two applied
forces. Remember, the resultant has the same magnitude but the opposite direction as the
equilibrant. The experimental set-up for a similar experiment is shown in Figure 3.2, below.
(When you believe that you are close, jiggle the washer slightly to be sure that the equilibrium
condition is met. Be sure that all of the strings are in such a position that they are directed along
lines that pass through the centre of the washer. This is crucial because the angles are measured
from the centre of the washer.)
+
x
Figure 3.2.
Experimental setup
(the threads and axis
labels were added
graphically).
+
y
Creative Commons Attribution 3.0 Unported License
This is similar to the
setup for Vector
Addition III2.
4
PHYSICS SEMESTER ONE
LAB 3
Vector Addition II
Repeat all of the steps as for Vector Addition 1 for F1  (0.200) g N at 20 and F2  (0.100) g N at
80 . Be careful in the analytical analysis.
3. Can you use tan    F2 / F1 in this case? Why or why not?
Vector Addition III
Repeat procedure 1 with F1  Fx  (0.200) g N (i.e. at 0 ) and F2  Fy  (0.150) g N (i.e. at 90 ). In
this case, F  Fx  Fy , where Fx and Fy are the x and y-components of F , respectively. That is, the
resultant can be resolved into these components. Use one-half of another sheet of graph paper for the
graphical method.
Vector Addition IV: Vector Resolution
Given a force F  (0.300) g N at 60 , resolve the vector into its x and y-components and find the
magnitudes of Fx and Fy by the following procedures:
a) Graphical. Draw a vector diagram to scale (on the other half sheet of graph paper used in the
previous procedure) with the component vectors. Measure and record the magnitudes of the
components.
b) Analytical. Compute and record the magnitudes of the component vectors Fx and Fy using
trigonometry.
c) Experimental Clamp pulleys along lines 240 , 90 , and 0 from the intersection of axes. Place a
total of 0.300 kg on the thread at 240 . This force is the equilibrant of F  (0.300) g N at 60 .
Add masses to the bottles at 0° and 90° until the system is in equilibrium. The 0° and 90° forces are
the x and y-components of F . Record your results.
Vector Addition V
Given the force vectors F1  (0.100) g N at 30 , F2  (0.200) g N at 90 , and F3  (0.30) g N at
225 , find and record the magnitude and direction of their resultant F  F1  F2  F3 by the following
procedures:
a) Graphical: Use the polygon or head-to-tail method.
b) Analytical: Use the component method.
c) Experimental: Use the experimental apparatus.
Creative Commons Attribution 3.0 Unported License
5
PHYSICS SEMESTER ONE
LAB 3
QUESTIONS
In addition to your calculations and results, answer all numbered questions in this lab, as well as the
questions below. Remember that in this (and all labs) error calculations are an important part of your
reporting.
4. To determine the forces on the ring due to the suspended masses, it was assumed that
g  9.81 m/s 2  9.81 N/kg . The value of g might be slightly different from that value. State clearly
what effect (if any) that would have on the percentage difference calculated for the resultants in this
comparison. Don’t assume that the discrepancy in the value of g is necessarily small.


 
that the cases of Vector Addition I, II, and III in this experiment were vector subtraction  F  F  .
5. Vector subtraction A  B is a special case of vector addition, since A  B  A  B . Suppose
1
2
5.1. What effect would this have on the directions of the resultants? (Do not calculate explicitly. Simply
state in which quadrant the resultant would be in each case.)
5.2. Would the magnitude of the resultant be different for vector subtraction than for vector addition
in each case? If so, state whether the subtractive resultant would be greater or less than the
additive resultant.
REFERENCES
This lab is adapted from references 1 and 2.
The photographs are from reference 2.
1. Laboratory Manual for PHY 100 & 101 Introductory Physics I and II and PHY 120 & 121
Principles of Physics I and II, 5th Edition 2004, prepared by Jason Diemer, North Island College
2. Laboratory Manual for PHY 060 – College Preparatory Physics II, Online Edition 2005-06-01, prepared by
Dennis Lightfoot and Ron Evans, North Island College
Creative Commons Attribution 3.0 Unported License
6
PHYSICS SEMESTER ONE
LAB 3
NANSLO Physics Core Units and Laboratory Experiments
by the North American Network of Science Labs Online,
a collaboration between WICHE, CCCS, and BCcampus
is licensed under a Creative Commons Attribution 3.0 Unported License;
based on a work at rwsl.nic.bc.ca.
Funded by a grant from EDUCAUSE through the Next Generation Learning Challenges.
Creative Commons Attribution 3.0 Unported License
7