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PHYS2125 Physics Laboratory I

1
Pre-lab
Lab 3: Vector Analysis of Forces
Name : _______________________
NetID : _______________________ Grade:_____________/30
Section : ______________________
**You are encouraged to have discussions with TA and other students, but you are required to
do calculations and answer questions individually and independently.
Question: [30]
(1) Fill in the table below [10].
Vectors Magnitude?direction method Component method
? 1 @
^ ^
??
______@______ ^ ^
?
______@______ ^ ^
??
______@______ ^ ^
?
______@______ ^ ^
?
______@______ ^ ^
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
2
??
??
(2) Draw the forces ? , ? ,
? , and ? on the
right figure [4].
(3)
i. Draw free-body diagram for mass with 4 vectors , , , and . (Pay
attention to the given x-direction.) [4]
If there is no friction, the mass will (A) move up, (B) move down, or (C) stay at rest on
the inclined plane.
Answer:_______________ (A or B or C)[1]
ii. Draw the free-body diagram for mass with the same 4 vectors. [4]
If there is no friction, the mass will (A) move up, (B) move down, or (C) stay at rest on
the inclined plane.
Answer:_______________ (A or B or C)[1]
Draw here:
??
??
??
Draw here:
??
??
??
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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iii. Please show that
(A)

[3]
(b)

[3]

PHYS2125 Physics Laboratory I

1
Lab Manual
Lab 3: Vector Analysis of Forces
? Before the lab, read the theory in Sections 1-3 and answer questions on Pre-lab.
Submit your Pre-lab at the beginning of the lab.
? During the lab, read Section 4 and follow the procedure to do the experiment.
You will record data sets, perform analyses, answer questions, and have Check
Boxes checked on Report Sheets. Submit your Report Sheets before you leave
the lab.
1. Introduction
A vector is a quantity that has both a magnitude and a direction. This is different from a scalar
quantity, which is just a single number. In this lab we will learn to accurately describe and manipulate
vectors, including resolving them into components and adding them together. We will
add vectors graphically, in terms of components, and also experimentally using the force table
apparatus.
2. Key Concepts
? Vectors and vector addition
? Free body diagrams
? Forces
? Equilibrium
3. Theory
3.1 Vectors
When completely describing a physical quantity, sometimes it is necessary to assign it a magnitude
and direction rather than using only a magnitude. Quantities needing only a single value
are scalars, and quantities requiring a magnitude and a direction are vectors. Vectors are usually
written with an arrow on top, like ? , while scalars are without the arrow, like .
To identify and use a vector, first establish a coordinate system, or set of axes, that is appropriate
for the system you are trying to describe. This will give you a frame of reference from
which to make your measurements. An example is labeling North on a map. In this lab manual,
we will use the – and -axes. All angles, as denoted by , are measured counterclockwise from
the -axis (see Figure 1). Remember, some problems are made easier by a proper choice of axes.
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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Vectors can be described in two ways: the
magnitude?direction method or the component
method. The magnitude?direction
method describes the vector with a given
magnitude and direction, but one needs to be
careful to include enough information in giving
the direction. For example, one has to
mention clearly what the direction is with respect
to. This is very intuitive to graph but
mathematically more difficult to use. The component form of a vector breaks up the magnitude
and direction into how much along the -axis and how much along the -axis (see Figure 1).
Vectors in component form look like
? ^ ^
Here ^and ^are unit vectors in the and directions, respectively. We will use this notation
in the manual.
To find the components of vectors, you need to use the trig identities

Which equation one needs to use depends on the coordinate system. For the angles used in
Figure 1, for example, if ? describes a force of 5 N (Newton) @ 30 degrees, can be computed
from cosine and from sine,

? ^ ^
If the component form is given, the angle can be solved for by

and the magnitude
of the vector is solved by | ? | v

( )

.
3.2 Adding Vectors
Adding two vectors results in a third called the resultant vector. Now that a vector is broken
up into its components, adding multiple vectors is simply adding up all of the components to
find the component of the resultant vector and adding all of the components to find the
component of the resultant vector. For example, if ? ^ ^and ?? ^ ^then
? ?? ^ ^ ^ ^. Notice that you need to be careful to take the sign of
the components into account when adding vectors.
To graphically find the resultant vector, use the ?tail to tip? method. Once a coordinate axis is
drawn, the ?tail? of the first vector begins at the origin. Measure the appropriate angle using a
protractor, and draw the length of the vector to match the vector’s magnitude. The ending point
??
??
??
???
???? ?? ??
???? ?? ??
??? ??????^ ??????^
Figure 1: The relation between the components
of a vector and the angle ?? measured counterclockwise
from the ??-axis.
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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of the first vector is known as the ?tip,? and an arrow is drawn at this end. From the ?tip? of the
first vector, start the ?tail? of the next vector. To measure the angle for the second vector, hold
the protractor parallel to the original axis. Be sure to use the same scale as the first vector in
drawing the length of the second vector. Continue this way until all vectors to be added have
been drawn. The resultant vector begins at the origin and ends at the tip of the last vector added.
To experimentally find the resultant vector, it is easiest to first find the equilibrant vector,
and the resultant is then equal in magnitude but in opposite direction to the equilibrant. Opposite
direction for component form amounts to changing the sign of each component. When an
angle and magnitude are given, the opposite direction is given by adding
to the angle. The
equilibrant vector is the vector that balances out the system. It brings the system to equilibrium
and therefore keeps the system stationary.
For more information on vectors, please see the sections on vector components and vector
addition in your textbook. There are also some great figures to help you understand the techniques.
When you add two or more vectors, it can be very helpful to make a table for organizing the
data. An example is given in Table 1. Another thing that is sometimes helpful in picturing these
vectors and their components is to draw them all on the same coordinate system with their tails
all at the origin.
Vector -component -component Magnitude Angle
? v

( )

?? v

( )

? v

( )

?
? ?? ?

v

( )

Table 1: Table that is helpful in organizing data to add multiple vectors.
3.3 Free Body Diagrams
It is often beneficial in a problem to be able to draw a picture. A free body diagram is a representation
of all the vectors of the same type that are acting on the system. In this lab, these vectors
will be forces. If the free body is stationary, that means the body is in equilibrium; the sum
of all the forces acting on the body is zero. For simplicity, assume all objects are a single point at
the center. In a free body diagram, all the ?tails? of the vectors acting on the body are located at
this center point. Be sure to label each vector and that all vectors represent the same type of
quantity. A simple check can be done with unit analysis. If all the vectors must add, they must
have all the same units. Like the graphical method of finding vectors, each vector length must
represent the magnitude of the vector, and must be labeled with its angle if it is not on an axis.
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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??
??
???
????
???
Figure 4: Vectors on the x-y plane.
Figure 2 shows the free body diagram of a golf ball at impact. The ball starts to move because
the sum of the forces from the club, the tee, and the Earth is not zero. For more information,
please see the section in your textbook on free body diagrams.
Question (write down the answers in Pre-lab)
(1) As shown in Figure 3, six vectors with unit length (
) form a regular hexagon. Please describe
the vectors using the magnitude?direction method
and the component method by filling in the table. (Note that
the angles in the magnitude?direction method are measured
counterclockwise from the -axis).
(2) Consider the vector forces in Figure 4. Please draw forces ? ,
? , ? , and ? , such that
i. ? has twice the strength and is opposite to ? .
ii. ? is the resultant force of ? and ? .
iii. ? is the resultant force of ?? and ? .
iv. ? is the equilibrant force of ? , ?? , and ? .
(3) Figure 5 is an inclined-plane system that will be studied
in the first part of this experiment. As labelled in
the figure, the ^ ^ direction is parallel (perpendicular)
to the inclined plane, and the gravitational
acceleration is toward the ground. If the hanging
mass is too small, the block mass on the inclined
plane slides down. When the hanging mass is
gradually increased to a lower-bound value such
that the block mass just stops sliding down, the
forces on satisfy the equilibrium conditions:
{
^
^

Here is a friction force and is a normal force. Both of
them are acted by the inclined plane. If we keep
increasing the hanging mass to a higher-bound value
??
??? ??
????
???
????
???
???

Figure 3: Six vectors forming
a regular hexagon.
Gravitational force:
Earth on Ball
Normal force: Tee on Ball
Normal force:
Club on Ball
Figure 2: The cartoon (left)
and free body diagram
(right) of a golf ball at impact.
??
??
??
??^
Figure 5: Inclined-plane system.
??
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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such that the block mass is just about to slide up (but not moving yet), the forces on satisfy
the equilibrium conditions (here we assume that the magnitudes of and remain the
same):
{
^
^

i. Use 4 vectors , , , and to draw the free-body diagram for mass in the case
corresponding to Eq. . How does mass move if there is no friction ( )?
ii. Use the same 4 vectors draw the free-body diagram in the case corresponding to Eq. .
How does mass move if there is no friction ( )?
iii. Combing the ^-direction conditions in Eqs. and , please show that

We will experimentally examine Eq. in the lab. Note that to verify Eq. we do not
need to know the details about gravitational acceleration or friction in the system.
4. Experiment
The first part of the experiment studies force vectors acting on a metal roller on an inclined
plane. The system looks similar to Figure 5. Since the roller will move only along the inclined
plane, its motion and the forces that cause it to move are one-dimensional. The forces are provided
by a string (attached to a mass hanger) pulling at the end of the inclined plane as well as
by the ^ component of the roller’s weight. By adjusting the mass on the hanger, you will be determining
experimentally what force is needed to bring the roller to equilibrium and verifying
the relation of Eq. you have shown in Pre-lab. In the second part of the experiment, you will
be balancing forces on a ring, but this time in two dimensions using the force table. You will analyze
the equilibrium of forces by decomposing each force into ^ and ^ components.
4.1 Equipment
? Triple beam balance
? Set of masses
? Mass hangers (3)
? Short red cable
? Inclined plane
? Metal roller
? Force table with ring, pulleys (3), and
black cables
4.2 Procedure
Forces in 1D
1. Measure the masses of the metal roller and the hanger. Record the measurements on Report
Sheets. Note that for only one measurement, the error is the measurement error.
2. Adjust the angle of the inclined plane to , connect the red cable to the roller and
hanger, and put them on the inclined plane, as in Figure 6. Make sure that the cable is parallel
to the inclined plane when the roller is pulled by the cable (you could drag the hanger
a little bit to stretch the cable).
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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3. Gradually add masses on the hanger until the
roller can maintain equilibrium at the middle of
inclined plane. Because the smallest mass is 5g,
you need to find mass
such that when
is
added on the hanger, the roller is at rest, and
when 5g is taken off, i.e. on the
hanger, the roller starts to move down. Record
on Table R1 on Report Sheets.
4. Keep adding masses until the roller starts to move up. You need to find mass
such that
when
is added on the hanger, the roller is at rest, and when additional 5g is put on, i.e.
on the hanger, the roller starts to move up. Record on Table R1.
5. Repeat 3 and 4 for different inclined angles and .
Check point 1
Ask your TA to check your data recorded and get their initials on Check Box 1 on Report Sheets.
Forces in 2D
1. Measure the mass of three hangers used with the force table. Record this measurement on
Report Sheets. In all the work below, you will need to use the total mass (mass of the
hanger plus the added mass) in your calculations.
2. Place the ring around the center peg. Make sure that all three cords are parallel to the
force table?you might need to adjust the heights of the pulleys to achieve this.
3. Place the first two pulleys at the angles given for hanger 1 and hanger 2 in Trial 1 of the table
on Table R2 on Report Sheets, and add the specified masses to the hangers at each angle.
4. Place the third pulley where you would guess the equilibrant vector to be, and hang a trial
mass. (Note that you can put the cords of more than one hanger over the same pulley if
you need to.) Adjust both the angle and mass of hanger 3 until the ring appears centered
around the peg.
5. Test the equilibrium position by giving the system a small pull along one of the vectors. If
the ring oscillates slightly but remains centered, you have found the correct equilibrant
force. That is, you have found the force that balances out the net effect of the first two.
6. Record the mass added on hanger 3 and the angle of hanger 3 on Table R2.
7. Repeat 3-6 with different configurations to complete Table R2.
Check point 2
Ask your TA to check your data recorded and get their initials on Check Box 2 on Report Sheets.
Figure 6: Roller on the inclined plane.
PHYS2125 Physics Laboratory I
The University of Texas at Dallas
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4.3 Analyses
I. Forces in 1D (write on Report Sheets)
1) Analyze the measurements by examining Eq. (6).
2) Make your conclusion, answer questions, and discuss possible sources of errors.
II. Forces in 2D (write on Report Sheets)
1) Compute each force and the resultant force the force table.
2) Compare the experimental relation between the forces with the theory and make a conclusion.

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