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Force Table
Lab Partners: Person 1, Person 2, Person 3, etc.
Instructor, T.A.: Your Instructor, Your TA
MM/DD/YY
ABSTRACT
This experiment was conducted to show how vectors affect one
another- in particular,
how opposing vectors can be added up to cancel each other out
to create a system in equilibrium,
which was done by hanging different masses over various
angles on a force table. As a result,
each case showed that when summed all forces added to 0.
INTRODUCTION
Vectors are extremely important in physics, as they provide a
way to show quantity that
has not only a magnitude, but a direction as well, which is
extremely important when explaining
things like motion. Although these vectors are more complex
than just a single number, they can
be manipulated by other vectors fairly easily. This makes
combining certain measurements that
could involve a multitude of vectors, as well as manipulating a
single vector as it can be added or
subtracted from itself, fairly simple.
This experiment showed the use of a force table to prove this
manipulability with vectors
by setting mass as forces on certain angles in order to cancel
each other out. This works as an
example because all three of the masses had some sort of force,
in this case being caused by
acceleration due to gravity, being applied to them in the
direction they were angled. It also
helped to demonstrate graphical methods for manipulating
vectors by means of “tip-to-tail”
measurement. This type of measurement aids in the visual
representation of vectors and gives
understanding to how a system of vectors looks when in
equilibrium, in this case a quadrilateral
formed by four vectors of different magnitude and direction. A
number of equations were used in
this experiment, and are as follows:
Instructor name.
Fx = 0Σ (1)
Fy = 0Σ (2)
Fx = Fcos( )θ (3)
Fy = Fsin( )θ (4)
g = 9.8 m/s2 (5)
F = mg (6)
Equations (1) and (2) show how F x and F y , the horizontal
and vertical components of
force F (Newtons ), when in an equilibrium-system should sum
to 0. Equations (3) and (4) show
how the force F is geometrically related to the horizontal and
vertical components, respectively,
by means of angle (degrees ). Equation (5) is a constant that
states how the acceleration due toθ
gravity, g (meters/second 2 ), is equal to 9.81. Equation (6) is a
variation of Newton’s Second Law
that shows that the force due to gravity on an object is
equivalent to g multiplied by mass m
(kilograms ).
PROCEDURE
The force table, which allows a central equilibrium to be
reached by hanging multiple
masses at different angles, was set up with 3 points to be
determined. The force table with a
3-pulley setup is seen in Figure 1. The pulleys were attached
around the circumference with a
ring and three strings that could spin freely placed in the center
of the table. The first trial
included forces in the first quadrant at 0o degrees and 200
grams and 90o degrees and 400 grams,
and one component to be measured based on these. The third
string was then pulled across the
table until the ring was centered on the middle post, and the
angle was recorded. A mass hanger
was then added to the string on this side, and weight was added
until the ring again centered
itself in the middle. At this point, all components were put in
equilibrium. This process was
repeated once more, only the 900 angle was
adjusted to 135o with the same weight of 400
grams. The third angle was measured, weight
was added, and equilibrium was reached, and
the results were recorded. Following this, a
procedure to allow for a graphically calculated
equilibrium-system was conducted. An arbitrary
quadrilateral, with a base line on the horizontal
of some graph paper, was drawn. Each side
represented a tip-to-tail vector, with the final tip
touching the tail of the horizontal vector. The angle of each
vector was found using a protractor
and the magnitude of each was found with a ruler using the
conversion 0.25” = 25 grams. These
values were translated to the force table, where 4 hangers and
pulleys were used to represent the
four vectors in the quadrilateral. Small adjustments were made
until equilibrium was established.
RESULTS AND CALCULATIONS
The following table shows angles, mass, and force for the first
setup in the experiment,
which involved starting angles that were orthogonal. Following
the table are sample calculations
demonstrating how the results were found.
Table 1: Orthogonal Starting Forces, Resultant Force
Angle (o) Mass (kg) Force (N)
0 0.200 1.96
90 0.400 3.92
246 0.450 4.41
F = mg (6)
g = 9.8 m/s2 (5)
450 g 10-3 = 0.450 kg×
0.450 kg 9.8 m/s2 = 4.41 kg = 4.41 N× m
s 2
A table for the vertical and horizontal components is shown
below, with sample
calculations following:
Table 2: Orthogonal Starting Forces, Components
Angle (o) Force (N) X-component (N) Y-component (N)
0 1.96 0.200 0.000
90 3.92 0.000 0.400
246 4.41 -0.183 -0.411
Sum 0.017 -0.011
Fx = Fcos( )θ (3)
Fx = Fcos( ) = (.450)cos(246) = -0.183 kgθ
Fy = Fsin( )θ (4)
vy = vsin( ) = (.450)sin(246) = -0.411 kgθ
Fx = 0Σ (1)
Fx = 0.200 + 0.000 + -0.183 = 0.017 0Σ ≈
Fy = 0Σ (2)
Fy = 0.000 + 0.400 + -0.411 = -0.011 0Σ ≈
The following table shows angles, mass, and force for the
second setup in the experiment,
which involved starting angles that were in the first and second
quadrants.
Table 3: First and Second Quadrant Forces, Resultant Force
Angle (o) Mass (kg) Force (N)
0 0.200 1.96
135 0.400 3.92
285 0.330 3.23
A table for the vertical and horizontal components in this
section of the experiment is
shown below.
Table 4: First and Second Quadrant Forces, Components
Angle (o) Force (N) X-component (N) Y-component (N)
0 1.96 0.200 0.000
135 3.92 -0.283 0.283
285 3.23 .085 -0.319
Sum 0.002 -0.036
The following page consists of the graphs of Force and
components for each of the two
trials. They are based on the data in Tables 1, 2, 3, and 4:
The following table shows the angles, magnitudes, and forces
for the quadrilateral system
of vectors that was drawn on the graph paper. Again, the
magnitudes are based on 0.25” = 25 g.
Table 5: Quadrilateral System, Forces
Angle (o) Mass (kg) Force (N)
0 0.25 2.4525
116.57 0.1118 1.0968
180 0.15 1.4715
243.43 0.1118 1.0968
A table for the vertical and horizontal components in this
section of the experiment is
shown below. This is the data the was calculated before being
put on the force table.
Table 6: Quadrilateral System, Components, Theoretical
Angle (o) Force (N) X-component (N) Y-component (N)
0 2.4525 2.4525 0
116.57 1.0968 -0.4905 0.9810
180 1.4715 -1.4715 0
243.43 1.0968 -0.4905 -0.9810
Sum 0 0
DISCUSSION, DATA ANALYSIS, AND CONCLUSION
In total, the data was fairly accurate in portraying systems in
equilibrium. Each case
showed fairly close results, meaning each mass system tested
ended with the ring being centered.
In the first setup, where the starting angles were orthogonal, the
resultant force was in the
quadrant opposite, which is expected. When considering the
vectors in terms of parallelograms,
the resultant of those two forces would be on a line somewhere
in between them, which when
applied to create equilibrium would be in the opposite direction,
For the case where the angles
were in different quadrants, the resultant was more difficult to
predict. In the case of creating a
resultant force that will put a two-force system into equilibrium,
it can be judged that the applied
resultant force will tend to be directed more in parallel to the
strongest of the two forces already
in the system; in other words, the angle between the resultant
vector and the strongest
pre-existing vector will be the closest to 180 degrees. This
proved true in the case of this
experiment, as the third force applied created a much more
obtuse angle with the hanger
producing a greater force out of the two forces already on the
table.
The quadrilateral equilibrium system was a way to demonstrate
differences in
tip-to-tail/graphical versus component methods of vector
computation. As the data shows, while
graphical representation can be useful for visualization,
component methods give much better,
more precise theoretical values that are better suited for further
calculation as these attributes
have much less error. They are also more easily organized and
are more efficient to use, which is
why they are preferable. Like the two previous sections of the
experiment, the quadrilateral that
was conceived demonstrated a system of equilibrium; however,
it gave more rise to sources of
error due to the discrepancies and variations between the
computed model and the force table
model.
One possible source of error that created some of these
unexpected results was that the
angles were all measured by eye. Obviously depending on the
perspective each angle can look
different, so this could have resulted in some inaccuracies when
measuring not only the
components, but the masses as the qualifications for equilibrium
would have changed. Another
source of error is that the force table had not been calibrated.
Similar to the source above, this
error may have caused an imbalance in the equilibrium, thus
producing inaccurate component
measurements. Another source of error may have been the
strings on the center ring, and the
friction in the pulleys. It was noted after the experiment that
they were not freely sliding around
as they should and were sometimes catching on certain portions
of the ring. This was also the
case with the pulleys. This means that some of the results may
be off as the angle may not have
been accurately measured after adding mass to the first two
strings.
Overall, this experiment showed how important vectors are and
how their angles and
magnitudes significantly affect their behavior and interaction
with one another. It demonstrated
not only how to solve equilibrium systems, through components
as well as graphical
representation, but how to calculate, create, and check
equilibrium systems.

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  • 1. Force Table Lab Partners: Person 1, Person 2, Person 3, etc. Instructor, T.A.: Your Instructor, Your TA MM/DD/YY ABSTRACT This experiment was conducted to show how vectors affect one another- in particular, how opposing vectors can be added up to cancel each other out to create a system in equilibrium, which was done by hanging different masses over various angles on a force table. As a result, each case showed that when summed all forces added to 0. INTRODUCTION Vectors are extremely important in physics, as they provide a way to show quantity that has not only a magnitude, but a direction as well, which is extremely important when explaining things like motion. Although these vectors are more complex
  • 2. than just a single number, they can be manipulated by other vectors fairly easily. This makes combining certain measurements that could involve a multitude of vectors, as well as manipulating a single vector as it can be added or subtracted from itself, fairly simple. This experiment showed the use of a force table to prove this manipulability with vectors by setting mass as forces on certain angles in order to cancel each other out. This works as an example because all three of the masses had some sort of force, in this case being caused by acceleration due to gravity, being applied to them in the direction they were angled. It also helped to demonstrate graphical methods for manipulating vectors by means of “tip-to-tail” measurement. This type of measurement aids in the visual representation of vectors and gives understanding to how a system of vectors looks when in equilibrium, in this case a quadrilateral formed by four vectors of different magnitude and direction. A number of equations were used in this experiment, and are as follows:
  • 3. Instructor name. Fx = 0Σ (1) Fy = 0Σ (2) Fx = Fcos( )θ (3) Fy = Fsin( )θ (4) g = 9.8 m/s2 (5) F = mg (6) Equations (1) and (2) show how F x and F y , the horizontal and vertical components of force F (Newtons ), when in an equilibrium-system should sum to 0. Equations (3) and (4) show how the force F is geometrically related to the horizontal and vertical components, respectively, by means of angle (degrees ). Equation (5) is a constant that states how the acceleration due toθ gravity, g (meters/second 2 ), is equal to 9.81. Equation (6) is a
  • 4. variation of Newton’s Second Law that shows that the force due to gravity on an object is equivalent to g multiplied by mass m (kilograms ). PROCEDURE The force table, which allows a central equilibrium to be reached by hanging multiple masses at different angles, was set up with 3 points to be determined. The force table with a 3-pulley setup is seen in Figure 1. The pulleys were attached around the circumference with a ring and three strings that could spin freely placed in the center of the table. The first trial included forces in the first quadrant at 0o degrees and 200 grams and 90o degrees and 400 grams, and one component to be measured based on these. The third string was then pulled across the table until the ring was centered on the middle post, and the angle was recorded. A mass hanger was then added to the string on this side, and weight was added until the ring again centered
  • 5. itself in the middle. At this point, all components were put in equilibrium. This process was repeated once more, only the 900 angle was adjusted to 135o with the same weight of 400 grams. The third angle was measured, weight was added, and equilibrium was reached, and the results were recorded. Following this, a procedure to allow for a graphically calculated equilibrium-system was conducted. An arbitrary quadrilateral, with a base line on the horizontal of some graph paper, was drawn. Each side represented a tip-to-tail vector, with the final tip touching the tail of the horizontal vector. The angle of each vector was found using a protractor and the magnitude of each was found with a ruler using the conversion 0.25” = 25 grams. These values were translated to the force table, where 4 hangers and pulleys were used to represent the four vectors in the quadrilateral. Small adjustments were made until equilibrium was established.
  • 6. RESULTS AND CALCULATIONS The following table shows angles, mass, and force for the first setup in the experiment, which involved starting angles that were orthogonal. Following the table are sample calculations demonstrating how the results were found. Table 1: Orthogonal Starting Forces, Resultant Force Angle (o) Mass (kg) Force (N) 0 0.200 1.96 90 0.400 3.92 246 0.450 4.41 F = mg (6) g = 9.8 m/s2 (5) 450 g 10-3 = 0.450 kg× 0.450 kg 9.8 m/s2 = 4.41 kg = 4.41 N× m s 2
  • 7. A table for the vertical and horizontal components is shown below, with sample calculations following: Table 2: Orthogonal Starting Forces, Components Angle (o) Force (N) X-component (N) Y-component (N) 0 1.96 0.200 0.000 90 3.92 0.000 0.400 246 4.41 -0.183 -0.411 Sum 0.017 -0.011 Fx = Fcos( )θ (3) Fx = Fcos( ) = (.450)cos(246) = -0.183 kgθ Fy = Fsin( )θ (4) vy = vsin( ) = (.450)sin(246) = -0.411 kgθ Fx = 0Σ (1) Fx = 0.200 + 0.000 + -0.183 = 0.017 0Σ ≈ Fy = 0Σ (2)
  • 8. Fy = 0.000 + 0.400 + -0.411 = -0.011 0Σ ≈ The following table shows angles, mass, and force for the second setup in the experiment, which involved starting angles that were in the first and second quadrants. Table 3: First and Second Quadrant Forces, Resultant Force Angle (o) Mass (kg) Force (N) 0 0.200 1.96 135 0.400 3.92 285 0.330 3.23 A table for the vertical and horizontal components in this section of the experiment is shown below. Table 4: First and Second Quadrant Forces, Components Angle (o) Force (N) X-component (N) Y-component (N) 0 1.96 0.200 0.000 135 3.92 -0.283 0.283
  • 9. 285 3.23 .085 -0.319 Sum 0.002 -0.036 The following page consists of the graphs of Force and components for each of the two trials. They are based on the data in Tables 1, 2, 3, and 4: The following table shows the angles, magnitudes, and forces for the quadrilateral system of vectors that was drawn on the graph paper. Again, the magnitudes are based on 0.25” = 25 g. Table 5: Quadrilateral System, Forces
  • 10. Angle (o) Mass (kg) Force (N) 0 0.25 2.4525 116.57 0.1118 1.0968 180 0.15 1.4715 243.43 0.1118 1.0968 A table for the vertical and horizontal components in this section of the experiment is shown below. This is the data the was calculated before being put on the force table. Table 6: Quadrilateral System, Components, Theoretical Angle (o) Force (N) X-component (N) Y-component (N) 0 2.4525 2.4525 0 116.57 1.0968 -0.4905 0.9810
  • 11. 180 1.4715 -1.4715 0 243.43 1.0968 -0.4905 -0.9810 Sum 0 0 DISCUSSION, DATA ANALYSIS, AND CONCLUSION In total, the data was fairly accurate in portraying systems in equilibrium. Each case showed fairly close results, meaning each mass system tested ended with the ring being centered. In the first setup, where the starting angles were orthogonal, the resultant force was in the quadrant opposite, which is expected. When considering the vectors in terms of parallelograms, the resultant of those two forces would be on a line somewhere in between them, which when applied to create equilibrium would be in the opposite direction, For the case where the angles were in different quadrants, the resultant was more difficult to predict. In the case of creating a resultant force that will put a two-force system into equilibrium, it can be judged that the applied resultant force will tend to be directed more in parallel to the strongest of the two forces already
  • 12. in the system; in other words, the angle between the resultant vector and the strongest pre-existing vector will be the closest to 180 degrees. This proved true in the case of this experiment, as the third force applied created a much more obtuse angle with the hanger producing a greater force out of the two forces already on the table. The quadrilateral equilibrium system was a way to demonstrate differences in tip-to-tail/graphical versus component methods of vector computation. As the data shows, while graphical representation can be useful for visualization, component methods give much better, more precise theoretical values that are better suited for further calculation as these attributes have much less error. They are also more easily organized and are more efficient to use, which is why they are preferable. Like the two previous sections of the experiment, the quadrilateral that was conceived demonstrated a system of equilibrium; however, it gave more rise to sources of error due to the discrepancies and variations between the
  • 13. computed model and the force table model. One possible source of error that created some of these unexpected results was that the angles were all measured by eye. Obviously depending on the perspective each angle can look different, so this could have resulted in some inaccuracies when measuring not only the components, but the masses as the qualifications for equilibrium would have changed. Another source of error is that the force table had not been calibrated. Similar to the source above, this error may have caused an imbalance in the equilibrium, thus producing inaccurate component measurements. Another source of error may have been the strings on the center ring, and the friction in the pulleys. It was noted after the experiment that they were not freely sliding around as they should and were sometimes catching on certain portions of the ring. This was also the case with the pulleys. This means that some of the results may be off as the angle may not have been accurately measured after adding mass to the first two strings.
  • 14. Overall, this experiment showed how important vectors are and how their angles and magnitudes significantly affect their behavior and interaction with one another. It demonstrated not only how to solve equilibrium systems, through components as well as graphical representation, but how to calculate, create, and check equilibrium systems.