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Physics 161
Static Equilibrium and Rotational Balance
Introduction
In Part I of this lab, you will observe static equilibrium for a
meter stick suspended horizontally. In Part II, you will observe
the rotational balance of a cylinder on an incline. You will vary
the mass hanging from the side of the cylinder for different
angles.
Reference
Young and Freedman, University Physics, 12th Edition: Chapter
11, section 3
Theory
Part I: When forces act on an extended body, rotations about
axes on the body can result as well as translational motion from
unbalanced forces. Static equilibrium occurs when the net force
and the net torque are both equal to zero. We will examine a
special case where forces are only acting in the vertical
direction and can therefore be summed simply without breaking
them into components:
(1)
Torques may be calculated about the axis of your choosing:
(2)
where torque is specified by the equation:
(3)
where d is the lever arm (or moment arm) for the force. The
lever arm is the perpendicular distance from the line of force to
the axis about which you are calculating the torque.
Normally, up is "+" and down is "-" for forces. For torques, it is
convenient to define clockwise as "-" and counterclockwise as
"+". Whatever you decide to do, be consistent with your signs
and make sure you understand what a "+" or "-" value for your
force or torque means directionally.
Part II: Any round object when placed on an incline has
tendency of rotating towards the bottom of an incline. If the
downward force that causes the object to accelerate down the
slope is canceled by another force, the object will remain
stationary on the incline. Figure 1 shows the configuration of
the setup. In order to have the rubber cylinder in static
equilibrium we should satisfy the following conditions:
(4)
Figure 1: Experimental setup for Part II
The condition that the net force along the x-axis (which is
conveniently taken along the incline) must be zero yields the
relationship. (Prove this!)
Without static friction the cylinder would slide down the
incline; the presence of friction causes a torque in clockwise
(negative) direction. In order to have static equilibrium we need
to balance that torque with a torque in counterclockwise
direction. This is achieved by hanging the appropriate mass m.
Applying the last condition to the center of the cylinder will
result in:
where r, the radius of the small cylinder (PVC fitting), is the
moment arm for the mass m and R, the radius of the rubber
cylinder, is the moment arm for the frictional force which
accounts for M and m. Combining this equation with the
equation for Ffr from above will result in:
(5)
(6)
By adjusting the mass m, we can observe how the equilibrium
can be achieved.
Procedure
Part I: Static Equilibrium
Figure 2: Diagram of Torque Experiment Setup
1. Weigh the meter stick you use, including the metal hangers.
2. Attach the force sensor cords to the Interface box as you have
done in previous labs. Set up the hardware in Pasco by adding
two force sensors for channels A and B.
3. For today's lab you do NOT need to graph the force sensors
over time, instead, drag the Digits icon in the "Displays" area.
This will give you a digital readout of the force sensor data at
the given time. Create two Digit displays, one for each force
sensor on the meter stick. Choose “Force Channel A (N)” and
“Force Channel B (N)” in the two Digit displays.
4. Remove the meter stick from force sensor hooks, click on
Record button, and tare each force sensor without weight to
establish zero. Then hang a known mass on each force sensor to
verify that it is reading correctly. Measure the mass on the
scale first. If you need to increase the precision of the display,
hover the mouse over the increase digits icon shown in the
figure below. This will increase the number of significant
figures.
5. Keep Record button on. Set up the meter stick and force
sensors as shown in Figure 2. The meter stick will be suspended
from a beam via the two force sensors. (These will also be used
to determine the upward vertical forces at these positions.) The
force sensors must be attached vertically anywhere that yields
equilibrium using the metal hangers.
6. Attach three masses (100g, 200g and 500g according to the
table in step 9) to the meter stick using metal hangers. Neglect
the masses of the metal hangers in your torque calculations.
7. Balance your system by moving the three weights and
watching the Digits displays. All forces must be vertical to
avoid difficulties, so make sure the meter-stick is level and be
sure the force sensors are pulling straight upward. Adjust the
location of the masses so that the force meters read almost
identical forces.
8. Record the position and mass on the meter-stick for each
mass.
9. Using the following data sheet to record the results, calculate
the sum of the masses responsible for the positive forces and the
sum of those responsible for the negative forces. (Forces 5 and
6 are the force meters.) Check to see if the sums are equal.
Mass(kg)
Force #
Force (N)
Position (m)
100 = 0.1
F1
980 = .98
0.149
200 = 0.2
F2
1960 = 1.96
0.211
500 = 0.5
F3
4900 = 4.9
0.685
143.04 = .14304
F4
1401.792 = 1.401
0.5
F5
-4.60
0.091
F6
-4.50
0.944
10. Using the zero position (x = 0 m) of the meter stick as the
axis of rotation and counterclockwise torques as positive,
determine the sum of the torques acting in both directions and
record them on the data sheet. Check for equality between
positive and negative sums within the calculated uncertainties.
Now imagine the lever arm is located at the axis point in the
middle of the meter stick (x = 0.50 m) and recalculate torques.
Check for equality between positive and negative results, within
calculated uncertainties.
Torque Calculation Table
Lever Arm (m)
Axis at x=0m
Torque (N-m)
Axis at x = 0m
Lever Arm (m)
Axis at x=0.5m
Torque (N-m)
Axis at =0.5m
F1
0.149
0.14602
0.351
0.34398
F2
0.211
0.41356
0.289
0.56644
F3
0.685
3.3565
0.185
-0.9065
F4
0.5
0.7005
0.5
0.7005
F5
0.091
-0.4186
0.409
-1.8814
F6
0.944
-4.248
0.444
1.998
Sum of Positive Torques
4.61658
3.60892
Sum of Negative Torques
-4.6666
-2.7879
%diff between positive and negative torques
5.002%
82.102%
Sum of all Torques
-0.05002
0.82102
Part II: Rotational Balance
The experiment uses a mass hanging on a string over a cylinder
to exert a constant torque on the system resulting in a rotational
balance of the system.
1. Measure the outer radius of the cylinder and the radius of the
PVC fitting around which the string is wrapped.
2. Set up the apparatus as shown in Figure 3 and make sure the
mass attached to the string is not touching either the board or
the side of the cylinder. Set the angle of incline to 10o.
3. Place the string around the PVC fitting and make sure that it
goes around the fitting at least for two turns.
4. Place enough mass at the end of the string so the cylinder
does not roll down the incline. Record this mass as mmin
Figure 3
5. Start adding more mass to the mass found in the previous step
until the cylinder starts rolling up the incline. Record this mass
as mmax.
6. Find the average of mmin and mmax. Let σm= ( mmax -
mmin )/2. Place all these values in the following table. Repeat
the experiment for θ= 15 and 20 degrees.
Run
θºtheory
M (kg)
mmin
mmax
maverage
σm
θºexperiment
σθ
1
10
0.25697
0.1 kg
0.25 kg
0.175
0.075
0.02878
2
15
0.25697
0.15 kg
0.25 kg
0.2
0.05
0.031088
3
20
0.25697
0.4 kg
0.45 kg
0.425
0.025
0.044267
For your Lab Report:
Include a sample calculation of the torques, and % differences
in Part I for x=0 condition. Find θexperiment±σθ for all 3 cases
in Part II. To determine σθ use equation 6 and assume that the
uncertainty for the numerator (A=mr) is 7%, and for the
denominator (B=(M+m)R) is 4%. Compute θmax= sin-
1[(A+σA)/(B-σB)] , θmin= sin-1[(A-σA)/(B+σB)] and σθ=
(θmax - θmin )/2. Compare θexperiment to θtheory. Are they
consistent? Within how many sigmas?
0
=
=
=
å
å
å
t
y
x
F
F
q
sin
)
(
g
m
M
F
fr
+
=
R
F
mgr
R
F
mgr
fr
fr
=
Þ
=
-
Þ
=
å
0
0
t
q
sin
)
(
gR
m
M
mgr
+
=
R
m
M
mr
)
(
sin
+
=
q
=
1
m
=
´
g
m
1
=
2
m
=
´
g
m
2
=
3
m
=
´
g
m
3
=
4
m
=
´
g
m
meterstick
4
0
...
3
2
1
=
+
+
+
=
F
F
F
F
net
0
...
3
2
1
=
+
+
+
=
t
t
t
t
net
Fd
=
t
1. Abstract (1 points)
A brief statement summarizing what was done, why, and giving
the principal results. It should be complete enough so that one
need not read the paper to understand the abstract. Everything
in the abstract is repeated, but with more elaboration, in the
paper. The purpose of the abstract is to allow the reader to
determine whether or not it will be worth the while to read the
entire paper. All data result values may not be included in this
section but some statement as to the conclusiveness of the data
is required. It is best to write the abstract after the entire paper
has been written. It is impossible to write this section before all
analysis is completed.
2. Introduction (1 points)
The introduction provides the background and theory motivating
the experiment as well as your hypothesis which likely results
from such theory. Important physical principles that may be
used later in the paper should be explained in a general way.
Key derivations that lead to these results should be referenced
and included as appendices. Equations used in the body of the
report must be introduced in this section unless the equation is
not part of the theory at hand, for example a geometrical
equation used because of procedural steps. These types of
equations should be included in the Results portion of the lab
paper.
3. Description of the Experiment (2 points)
The experiment must be described thoroughly but concisely.
The description should cover all apparatus used (diagrams of
experimental arrangements, if helpful) and a short discussion of
techniques and procedures. This latter discussion only needs to
be sufficiently detailed to reveal both the strengths and
weaknesses of the work. Do not copy bullet lists from the lab
manual, this section must be your own work.
4. Results(These are results based on your experimental
measurements, NOT on theoretical predictions) (3 points)
Present your experimentally gathered data and observations of
that data in tabular and/or graphical form. Make sure to write
about each graph or table directly after it has been introduced.
Include a description of any mathematical manipulations of the
data such as how the associated uncertainty was calculated for
the measured values. Any sample calculations should be
included in the Appendices; the results of these calculations
should be included in the data table presentation beside the
relevant data. After each data presentation, i.e. data table or
graph, observations should be discussed. Data may not be
simply stapled to the back of the lab paper but included within
the body of the lab paper. If data is long, students may
abbreviate the data to relevant columns but entire data sets must
be referenced and included in an appendix unless otherwise
noted by the lab instructor. If multiple data runs were acquired
discussing of precision may appropriately be completed in this
section.
5. Analysis (3 points)
Analysis of the experimental and predicted results
should be presented here. Experimentally obtained graphs and
tabular data comparison to theoretical curves belong here.
Results stand or fall as supported by the data and analysis,
irrespective of your opinion. Accuracy analysis belongs in this
section because it compares experimental data to predicted
results.
6. Closing Remarks(1 point)
Draw conclusions about the results. While speculations are
sometimes appropriate in this section, opinion must be carefully
distinguished from conclusions that are supported completely by
evidence.
7. References(1 point)
List the sources (books, papers, URLs, etc…) you used when
writing your lab research paper. Please note that copying a
picture from a website or from the online manual requires a
citation. Please make sure to include all URL addresses used as
sources either of content or images during the paper writing
procedure. Make sure to avoid plagiarism when using any of the
sources.
1

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Physics 161Static Equilibrium and Rotational Balance Intro.docx

  • 1. Physics 161 Static Equilibrium and Rotational Balance Introduction In Part I of this lab, you will observe static equilibrium for a meter stick suspended horizontally. In Part II, you will observe the rotational balance of a cylinder on an incline. You will vary the mass hanging from the side of the cylinder for different angles. Reference Young and Freedman, University Physics, 12th Edition: Chapter 11, section 3 Theory Part I: When forces act on an extended body, rotations about axes on the body can result as well as translational motion from unbalanced forces. Static equilibrium occurs when the net force and the net torque are both equal to zero. We will examine a special case where forces are only acting in the vertical direction and can therefore be summed simply without breaking them into components: (1) Torques may be calculated about the axis of your choosing: (2)
  • 2. where torque is specified by the equation: (3) where d is the lever arm (or moment arm) for the force. The lever arm is the perpendicular distance from the line of force to the axis about which you are calculating the torque. Normally, up is "+" and down is "-" for forces. For torques, it is convenient to define clockwise as "-" and counterclockwise as "+". Whatever you decide to do, be consistent with your signs and make sure you understand what a "+" or "-" value for your force or torque means directionally. Part II: Any round object when placed on an incline has tendency of rotating towards the bottom of an incline. If the downward force that causes the object to accelerate down the slope is canceled by another force, the object will remain stationary on the incline. Figure 1 shows the configuration of the setup. In order to have the rubber cylinder in static equilibrium we should satisfy the following conditions: (4)
  • 3. Figure 1: Experimental setup for Part II The condition that the net force along the x-axis (which is conveniently taken along the incline) must be zero yields the relationship. (Prove this!) Without static friction the cylinder would slide down the incline; the presence of friction causes a torque in clockwise (negative) direction. In order to have static equilibrium we need to balance that torque with a torque in counterclockwise direction. This is achieved by hanging the appropriate mass m. Applying the last condition to the center of the cylinder will result in: where r, the radius of the small cylinder (PVC fitting), is the moment arm for the mass m and R, the radius of the rubber cylinder, is the moment arm for the frictional force which accounts for M and m. Combining this equation with the equation for Ffr from above will result in: (5) (6)
  • 4. By adjusting the mass m, we can observe how the equilibrium can be achieved. Procedure Part I: Static Equilibrium Figure 2: Diagram of Torque Experiment Setup 1. Weigh the meter stick you use, including the metal hangers. 2. Attach the force sensor cords to the Interface box as you have done in previous labs. Set up the hardware in Pasco by adding two force sensors for channels A and B. 3. For today's lab you do NOT need to graph the force sensors over time, instead, drag the Digits icon in the "Displays" area. This will give you a digital readout of the force sensor data at the given time. Create two Digit displays, one for each force sensor on the meter stick. Choose “Force Channel A (N)” and “Force Channel B (N)” in the two Digit displays. 4. Remove the meter stick from force sensor hooks, click on Record button, and tare each force sensor without weight to establish zero. Then hang a known mass on each force sensor to verify that it is reading correctly. Measure the mass on the scale first. If you need to increase the precision of the display, hover the mouse over the increase digits icon shown in the figure below. This will increase the number of significant figures.
  • 5. 5. Keep Record button on. Set up the meter stick and force sensors as shown in Figure 2. The meter stick will be suspended from a beam via the two force sensors. (These will also be used to determine the upward vertical forces at these positions.) The force sensors must be attached vertically anywhere that yields equilibrium using the metal hangers. 6. Attach three masses (100g, 200g and 500g according to the table in step 9) to the meter stick using metal hangers. Neglect the masses of the metal hangers in your torque calculations. 7. Balance your system by moving the three weights and watching the Digits displays. All forces must be vertical to avoid difficulties, so make sure the meter-stick is level and be sure the force sensors are pulling straight upward. Adjust the location of the masses so that the force meters read almost identical forces. 8. Record the position and mass on the meter-stick for each mass. 9. Using the following data sheet to record the results, calculate the sum of the masses responsible for the positive forces and the sum of those responsible for the negative forces. (Forces 5 and 6 are the force meters.) Check to see if the sums are equal. Mass(kg) Force # Force (N) Position (m)
  • 6. 100 = 0.1 F1 980 = .98 0.149 200 = 0.2 F2 1960 = 1.96 0.211 500 = 0.5 F3 4900 = 4.9 0.685 143.04 = .14304 F4 1401.792 = 1.401
  • 7. 0.5 F5 -4.60 0.091 F6 -4.50 0.944 10. Using the zero position (x = 0 m) of the meter stick as the axis of rotation and counterclockwise torques as positive, determine the sum of the torques acting in both directions and record them on the data sheet. Check for equality between positive and negative sums within the calculated uncertainties. Now imagine the lever arm is located at the axis point in the middle of the meter stick (x = 0.50 m) and recalculate torques. Check for equality between positive and negative results, within calculated uncertainties. Torque Calculation Table Lever Arm (m) Axis at x=0m Torque (N-m) Axis at x = 0m Lever Arm (m) Axis at x=0.5m Torque (N-m) Axis at =0.5m F1 0.149 0.14602 0.351 0.34398
  • 8. F2 0.211 0.41356 0.289 0.56644 F3 0.685 3.3565 0.185 -0.9065 F4 0.5 0.7005 0.5 0.7005 F5 0.091 -0.4186 0.409 -1.8814 F6 0.944 -4.248 0.444 1.998 Sum of Positive Torques 4.61658 3.60892 Sum of Negative Torques -4.6666 -2.7879 %diff between positive and negative torques
  • 9. 5.002% 82.102% Sum of all Torques -0.05002 0.82102 Part II: Rotational Balance The experiment uses a mass hanging on a string over a cylinder to exert a constant torque on the system resulting in a rotational balance of the system. 1. Measure the outer radius of the cylinder and the radius of the PVC fitting around which the string is wrapped. 2. Set up the apparatus as shown in Figure 3 and make sure the mass attached to the string is not touching either the board or the side of the cylinder. Set the angle of incline to 10o. 3. Place the string around the PVC fitting and make sure that it goes around the fitting at least for two turns. 4. Place enough mass at the end of the string so the cylinder does not roll down the incline. Record this mass as mmin Figure 3 5. Start adding more mass to the mass found in the previous step until the cylinder starts rolling up the incline. Record this mass as mmax. 6. Find the average of mmin and mmax. Let σm= ( mmax -
  • 10. mmin )/2. Place all these values in the following table. Repeat the experiment for θ= 15 and 20 degrees. Run θºtheory M (kg) mmin mmax maverage σm θºexperiment σθ 1 10 0.25697 0.1 kg 0.25 kg 0.175 0.075 0.02878 2 15 0.25697 0.15 kg 0.25 kg 0.2 0.05 0.031088 3 20 0.25697 0.4 kg 0.45 kg 0.425
  • 11. 0.025 0.044267 For your Lab Report: Include a sample calculation of the torques, and % differences in Part I for x=0 condition. Find θexperiment±σθ for all 3 cases in Part II. To determine σθ use equation 6 and assume that the uncertainty for the numerator (A=mr) is 7%, and for the denominator (B=(M+m)R) is 4%. Compute θmax= sin- 1[(A+σA)/(B-σB)] , θmin= sin-1[(A-σA)/(B+σB)] and σθ= (θmax - θmin )/2. Compare θexperiment to θtheory. Are they consistent? Within how many sigmas? 0 = = = å å å t y x F F q sin ) ( g m M
  • 15. 1. Abstract (1 points) A brief statement summarizing what was done, why, and giving the principal results. It should be complete enough so that one need not read the paper to understand the abstract. Everything in the abstract is repeated, but with more elaboration, in the paper. The purpose of the abstract is to allow the reader to determine whether or not it will be worth the while to read the entire paper. All data result values may not be included in this section but some statement as to the conclusiveness of the data is required. It is best to write the abstract after the entire paper has been written. It is impossible to write this section before all analysis is completed. 2. Introduction (1 points) The introduction provides the background and theory motivating the experiment as well as your hypothesis which likely results from such theory. Important physical principles that may be used later in the paper should be explained in a general way. Key derivations that lead to these results should be referenced and included as appendices. Equations used in the body of the report must be introduced in this section unless the equation is not part of the theory at hand, for example a geometrical equation used because of procedural steps. These types of equations should be included in the Results portion of the lab paper. 3. Description of the Experiment (2 points) The experiment must be described thoroughly but concisely. The description should cover all apparatus used (diagrams of experimental arrangements, if helpful) and a short discussion of techniques and procedures. This latter discussion only needs to be sufficiently detailed to reveal both the strengths and weaknesses of the work. Do not copy bullet lists from the lab manual, this section must be your own work.
  • 16. 4. Results(These are results based on your experimental measurements, NOT on theoretical predictions) (3 points) Present your experimentally gathered data and observations of that data in tabular and/or graphical form. Make sure to write about each graph or table directly after it has been introduced. Include a description of any mathematical manipulations of the data such as how the associated uncertainty was calculated for the measured values. Any sample calculations should be included in the Appendices; the results of these calculations should be included in the data table presentation beside the relevant data. After each data presentation, i.e. data table or graph, observations should be discussed. Data may not be simply stapled to the back of the lab paper but included within the body of the lab paper. If data is long, students may abbreviate the data to relevant columns but entire data sets must be referenced and included in an appendix unless otherwise noted by the lab instructor. If multiple data runs were acquired discussing of precision may appropriately be completed in this section. 5. Analysis (3 points) Analysis of the experimental and predicted results should be presented here. Experimentally obtained graphs and tabular data comparison to theoretical curves belong here. Results stand or fall as supported by the data and analysis, irrespective of your opinion. Accuracy analysis belongs in this section because it compares experimental data to predicted results. 6. Closing Remarks(1 point) Draw conclusions about the results. While speculations are sometimes appropriate in this section, opinion must be carefully distinguished from conclusions that are supported completely by evidence. 7. References(1 point) List the sources (books, papers, URLs, etc…) you used when writing your lab research paper. Please note that copying a picture from a website or from the online manual requires a
  • 17. citation. Please make sure to include all URL addresses used as sources either of content or images during the paper writing procedure. Make sure to avoid plagiarism when using any of the sources. 1