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Physics 101/111---Lab 5: Vectors
© 1990-2007, James J. DeHaven. Ph.D.
Vectors are ubiquitous in physics, and, in addition, you find them everywhere. Velocity,
acceleration, displacement and force are all vector quantities. In this lab we will explore various
methods for adding and resolving vectors. In the process you will have the opportunity to improve
your skill with geometry and trigonometry, and you will also work with a piece of equipment, the
force table, which allows you to visualize the addition and resolution of vectors.
Specifically, in the course of this lab you will learn to add and to resolve vectors using three
distinct methods
1. Graphical
2. Analytical
3. Experimental, using the force table
Introduction and Theory
1. Graphical Addition of Vectors
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 of vector addition. 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.
F1
F2
Resultant
Fig. 1 Resultant of two forces
2. Analytical
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 i, j 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.
3. Force Table
Vectors can be added experimentally using a force table. This is a device which allows the
detection of a condition of zero-net force to signify when force vectors are precisely balanced.
-2 Equipment
Force Table Apparatus
Weight Sets
Pulleys
Ring + String
Weight Hangers
Ruler
Protractor
Spirit Level
The force table consists of a machined metal or plastic plate mounted on a stand. The plate
has angular markings and a centering pin. Pulleys are attached to the plate and aligned with the
angular markings. Weights hung from the pulleys exert forces on a centering ring. When the forces
are balanced , the centering ring is aligned with the centering pin on the force plate. The directions
of the force vectors can be ascertained from the angular positions of the pulleys , and the
magnitudes from the values of the hanging masses.
The protractor and ruler are used in the graphical method of vector analysis.
Experimental Method
Before you begin, use the spirit level to confirm that your force table is level.
A) Adding 2 Vectors
Begin by assuming that forces are being exerted on an object by two vectors at angles of 30
degrees and 120 degrees (see figure 2), and that, further, these forces are each equivalent to the
weight of a 105 gram mass.
y
120
30
Fig. 2 Arrangement of forces to be added
x
Fig. 2 Arrangement of forces to be added
-3 1) Calculate the value of the resultant vector analytically .
2) Then use the parallelogram method to make a scale drawing of your vectors graphically (use the
protractor, ruler, and graph paper). The finished diagram should fill about 2/3 of a page and should
look roughly like the one shown in figure 3.
Resultant
F2
F1
Fig. 3 Graphic representation of added vectors
Use a ruler and protractor to measure the magnitude and angle of the resultant.
Finally, use the force table to measure the weight needed to exactly
Resultant 3)
offset the resultant of your vectors. This force is sometimes referred
to as the equilibrant. Figure 4 illustrates the idea of the equilibrant, and
the ensuing text explains in detail how to perform the experiment.
Note that the equilibrant is not the same vector as the resultant. In
fact it is the vector equal and opposite to the resultant.
F2
F1
Equilibrant
Set up your force table with 105 g weights on two pulleys at 30
degrees and 120 degrees. This is illustrated schematically in figure 5.
Using a third pulley and weights, determine the magnitude and
direction of the equilibrant force that maintains the ring in equilibrium
centered around the center pin.
Fig. 4 Equilibrant balancing
resultant
-4 Resultant= - Equilibrant
(0.105 kg)(9.8 m/s 2)
(0.105 kg)(9.8 m/s 2)
Equilibrant = ?
Fig. 5 Setting up the force table
Helpful Hints: Make sure the string knots slip readily around the center ring. Pulling the ring
straight up and then releasing it helps adjust for the friction of the pulleys as the ring vibrates up
and down.
4) Repeat steps 1,2, and 3 of this procedure to add the forces exerted by a 0.105 kg mass at 20
degrees and 0.075 kg at 80 degrees.
B) Resolution of Vectors
Given a vector of 0.155 kg (times 9.8 m/s2), at an angle of 60 degrees with respect to the xaxis, find its components
1) graphically,
2) analytically and
3) experimentally
using the force table for the experimental determination You may need to substitute a 50g weight
hangar on those large tables supplied with the 100 gram weight hangers.
-5 C) Adding three vectors
Given the vectors:
0.055 kg times 9.8 m/s2 , at an angle of 30 degrees,
0.105 kg times 9.8 m/s2 , at an angle of 90 degrees,
0.155 kg times 9.8 m/s2 , at an angle of 225 degrees,
each angle being measured from the x-axis.
Find their resultant:
1) graphically
2) analytically
3) experimentally.
You may want to perform the graphical addition of three vectors in two steps.
Data
ALL DATA MUST BE ENTERED INTO YOUR BOUND LAB NOTEBOOK. THE
NOTEBOOK CANNOT BE A RING NOTEBOOK OR A SPIRAL NOTEBOOK. PAGES
MUST NEVER BE RIPPED OUT OF YOUR NOTEBOOK.
The primary data from this experiment consists of the angles and masses needed to bring
the forces into balance in each exercise. You may also want to note down how much leeway there is
in your choice of angle and mass. In other words, how much variation, in mass and angle, can the
force table tolerate while still maintaining alignment between the centering ring and centering pin.
This should give you a good idea of the error that the experimental method introduces.
Calculations
Almost all calculations are an integral part of the laboratory. The analytical addition or
resolution of of the vectors requires you to convert them to component (i, j) format, and then
reconvert them to the angle-magnitude representation.
The graphical analysis of the vectors requires you to scale the drawing to fit your graph
paper. You will have to decide on a reasonable number of squares per kilogram of hanging mass
and then apply this ratio to all your vectors so that the length of the vector will be proportional to the
force it represents.
Things to Discuss
Does experiment confirm theory. In other words, do the graphical and experimental
methods for adding vectors confirm the analytical results?
What are the primary sources of error in the experimental method, and were the errors
primarily random or systematic? Does the resolution experiment differ substantially from the
addition experiment or is there an underlying similarity between them?
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Report:
Introduction: Write a brief introduction stating the objectives of the experiment, and a concise
summary of the methods that will be used.
Experimental: Describe the experimental apparatus and precisely what variables will be measured
and how they will be measured.
Results: Summarize the results of the experiment. Show your methods for graphical, analytical and
experimental addition and/or resolution of vectors. Attach graph paper or other drawings of vectors
as required.
Discussion: Explain the significance of your results and their connection with more general
physical principles. Where it is possible, compare your numbers with accepted values. Explain any
sources of error.