FVCC Physics Laboratory
Vector Addition
2011
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Objective
This laboratory will investigate how to add multiple vectors both algebraically and graphically. We will
confirm our additions by using a force table to sum the vectors and test the resultant.
Objective
Objective
Objective
Objective
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Be able to express vectors in polar or cartesian coordinates
Add vectors graphically
Add vectors algebraically
Test the addition on a force table
Theory
A vector describes the magnitude and direction of some quantity such as velocity, or force. Understanding vectors and how they combine is prerequisite knowledge for nearly all aspects of physics. While
vectors can be expressed in many ways, they always need at least 2 numbers to quantify them. First
they must have a magnitude and second, they must have a direction. In this lab we will work with 2
dimensional vectors which can be easily expressed in rectangular (Cartesian) coordinates or polar coordinates. Figure 1A shows a vector expressed in rectangular components and fig.1B illustrates the vector
in polar coordinates.
Converting a vector from polar coordinates to Cartesian coordinates is done with trigonometry.
~ between these coordinates systems. To illustrate, consider
Referencing fig.1, we can convert the vector A
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Vector Addition
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Figure 1: Two possible coordinates systems used to express vectors
putting some numbers to the vectors. Imagine that we are given the vector with Cartesian components
~ = 3x̂ + 2ŷ. To convert this vector to polar coordinates, find the magnitude with Pythagrean
given by A
theorem and find the angle with an arc tangent.
√
√
R = 32 + 22 = 13
(1)
2
= 33.4 deg
(2)
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To convert a vector in polar back to Cartesian coordinates, use trigonometry. For this example, consider
~ = 5r̂+30 deg θ̂. ( This may also be written as: A
~ = 5∠30 deg.)
a vector given in polar coordinates as: A
Then the conversion is:
Ax = 5 cos 30 = 4.3
(3)
θ = arctan
Ay = 5 sin 30 = 2.5
(4)
Vectors can be added algebraically or graphically. To add them algebraically, it is best to express the
vectors in Cartesian coordinates and then add or subtract the components. Computer spreadsheets are
particularly convenient for this. To add vectors graphically, place the tail of the second vector on the
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Vector Addition
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Figure 2: Tip to Tail vector addition
tip of the first vector. The resultant vector is the connection of the tail of the first vector with the tip
of the second vector. This is illustrated in fig.2.
The goal of this laboratory is to add several vectors together with different methods and show that
all methods give the same result. We will use three methods which are: 1) graphical, 2) algebraic, 3)by
a force table. The force table is not really a method to add vectors, but rather a visual confirmation of
vector addition. Figure 3 shows the table with several vectors loaded on it.
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3.1
Experiment
Equipment list
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2
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Engineering graphing paper
Force Table
Disk weights and weight hangers
3-beam balance
Protractor
Ruler
Triangle
The idea is to add 3 vectors so we need to specify 3 actual vectors to add. The vectors we will add
should be:
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Vector Addition
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Figure 3: Force table
~ = 0.921x̂ + 0.335ŷ[N ]
A
~ = −0.980x̂ + 1.69ŷ[N ]
B
~ = 1.13x̂ + 0.950ŷ[N ]
C
(5)
~ = A
~+B
~ +C
~
R
(8)
(6)
(7)
~ is the resultant vector, or the vector which is equal to the sum of A,
~ B
~ and C.
~ These vectors
Here R
are the vectors you are to use in the exercises to follow. They are the approximate value you should
try to get. Since you will test the vector addition with a force table, the vectors you really use will be
slightly different than those listed. To convert from force into mass, use g = 9.80 sm2 .
3.2
1
Procedure
~ B
~ and C,
~ create three real vectors that will be
Using the target values given for vectors A,
masses hanging from the force table at specified angles. Measure the real values by measuring the masses for each of the vectors. The mass should include the hanger. Here is a
suggested data table. You can also put your data directly into a spreadsheet.
Physics 2 Labs
Vector Addition
Vector Mag [N]
~
A
~
B
~
C
Mag [g]
Angle x-component
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y-component
Table 1. Vector data
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The force table should have a center pin so that weight hanging unbalanced does not crash
to the table. For now, leave this pin in place.
~ B
~ and C.
~
Load each of the determined weights on the force table, creating the vectors A,
~ vector. Express the R
~ vector in both coordinate
Add these vectors algebraically to find the R
systems.
Using a protractor, ruler and graph paper, add the vectors graphically 2 times on two different
~ +B
~ +C
~ = R.
~ Then add them graphically in a different
sheets of graph paper. First, add A
~ +B
~ +A
~ = R.
~
order: C
~ vectors on your graph paper and report the graphical determination of R.
~
Measure the R
You can measure the polar form with a ruler and a protractor. The rectangular components
can be measured with a ruler alone, but it may be useful to use a right triangle to help you
break each vector into its components.
~ is the vector which is the equivalent vector to the 3 vectors on your force table. That is,
R
~ B,
~ C
~ and replace them with R.
~ Find the equilibriant vector, which is
you could get rid of A,
~
~
the vector opposite to R. Well call this the Ro . This is the vector which when loaded onto
~o in both rectangular and
the force table will balance the 3 vectors already there. Express R
polar coordinates. Here is a suggested results table.
Method
Algebraic
Graphic 1
Graphic 2
Equalibriant
Result
~
R
~
R
~
R
~o
R
Mag [N]
Table 2. Results and calculation table
Mag [g]
Angle x-component
y-component
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Vector Addition
~o on to the force table, and pull the pin! If everything stays in place, your correct.
Load R
Using significant figures, estimate the uncertainty in your results. An example of presenting
your results and uncertainties: R = 10.0 ± 0.1[N ].
Questions
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What was the best method of vector addition and what was worst? Why?
Compare the graphic method and the algebraic method. How far apart are they in
magnitude and angle?
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