Download Torque: Equilibrium of a Rigid Object Experiment 11

Name__________________________________Lab Partners____________________________
Instructor Name__________________________
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Torque: Equilibrium of a Rigid Object
Experiment 11
INTRODUCTION
When several forces act on an object, there are generally two effects: the object’s center of
mass translates and the object rotates about some axis. Under certain conditions, the object can be
in translational and rotational equilibrium at the same time. The purpose of this experiment is to
study the conditions for equilibrium of a rigid object when several forces act on it. You will learn
to calculate torques about any axis.
THEORY
The measure of a force's tendency to turn or rotate an object is known as the torque or moment
of the force. The torque  about an axis caused by a force F is defined as the product of the
magnitude of the force and the perpendicular distance from the axis of rotation to the line of action
of the force. Figure 11.1 illustrates some of the terminology that applies to calculating a torque.
The defining equation for torque is

 = F r sin  = F r┴,
(1)
where r is the distance from the point of application of the force to
the axis of rotation, r┴ is the perpendicular distance from the axis of
rotation O to the point of application of the force (also called the
lever arm or moment arm), and  is the angle between the line of
action of the force and the line drawn from the axis of rotation to the
point of application of the force. The standard convention is that
positive torques produce (or tend to produce) counterclockwise
rotation, and negative torques produce (or tend to produce)
clockwise rotation. Note that this convention is consistent with the
trigonometric convention for measuring angles.
Figure 11.1 Variables
for calculating torque.
An object is in equilibrium if it has neither translational nor angular acceleration. Two
conditions must be met for an object to be in equilibrium: (1) the vector sum of all forces acting
on the object must be zero, and (2) the algebraic sum of all torques about any axis must be zero.
Any axis may be chosen for computing the torques; it is not necessary to choose the axis about
which the rotation occurs or tends to occur.
The preceding brief discussion contains the basic physics for analyzing the results of this
experiment. There is, however, one more concept of practical importance for understanding the
experiment. The weight of an object will cause a torque about any axis except an axis through the
object's center of gravity (or center of mass). The center of gravity and the center of mass coincide
when an object experiences a uniform gravitational force, so that near the surface of the earth, the
Physics 1121 Experiment 11 Page 1
two are the same. The center of mass is defined as the point at which the object's entire mass may
be considered to be concentrated. For many objects that are uniform and symmetric, the center of
mass is located at the center of the object.
EXPERIMENT NO. 11
The photograph below shows a torque bar with the left and right ends labeled L and R,
respectively. The positions of each of the four bolts that hold the protractors are labeled A, B, C,
and D. Please note that the bar is positioned so that point A is closest to the left end.
Figure 11.2. Torque bar.
1. Write the equations associated with the two conditions for static equilibrium.
Condition 1: Equations from PreLab.
Condition 2: Equation from PreLab.
You will use these conditions to calculate the net force and net torque on a rigid object in static
equilibrium.
2. Measure the total length of the torque bar and the distances of points A, B, C, and D from the
left end.
Total length of torque bar = _______________
Distance to A (xA) = _______________
Distance to C (xC) = _______________
Distance to B (xB) = _____________
Distance to D (xD) = _____________
3. Set up the torque bar as shown in Figure 11.3 on the following page using the two 5.0-N
calibrated springs connected at points A and D. Suspend the torque bar from the support rod. Slide
the springs attached to the support bar apart so that the angles between the springs and the torque
bar are unequal and larger than 50 degrees.
Physics 1121 Experiment 11 Page 2
Figure 11.3. Experimental setup of the torque bar.
4. The angles indicated on the protractor A and D represent the angles the springs makes with the
bar. Record A and D.
A = __________
D = __________
5. What are the forces acting on the torque bar? Label these forces on the diagram.
6. Is the bar in static equilibrium? Explain.
7. The bar may not be horizontal. Record the angle of the torque bar relative to the horizontal, B
or C as measured on protractor B or C relative to the 90o reading. Use the protractor that rotates
most freely to measure the angle. You may need to hang one or two large paper clips at point B or
C to stabilize the protractor. Circle B or C below (which ever you use) and record its value.
B or C = _______________
You will need to use B or C to correct to A and B to take into account the fact that the torque
bar is not horizontal.
Now you will set up equations for the two conditions for static equilibrium.
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8. Write the equations for Fnet x and Fnet y for the bar in static equilibrium using the coordinate
system shown in Part 5. Note: Exercise some caution since the protractors measure the angles
between the torque bar and the objects attached to them, not angles measured with respect to
horizontal and vertical. (Correct for B or C.)
Fnet x = 0 =___________________________________________
(2)
Fnet y = 0 =___________________________________________
(3)
9. You will need to determine the magnitude of the weight of the torque bar, Fg. Record the force
of each spring on the torque bar as it is seen on the spring scale. These values represent the
measured values for FA and FD.
FA measured = _____________________
FD measured = ____________________
10. Use the Eqns. (2) and (3) to determine Fg and the mass of the bar m. Show your calculations
below.
Fg = _________________________________________
m = _________________________________________
Now determine the net torque about the pivot point O for the bar. Recall, the equation for the
torque about O is given by
o = Fr┴ = Fr sin 
where r┴ is the perpendicular distance from the axis of rotation (pivot) to the line of action of the
force F, r is the vector from the axis of rotation to the point of application of the force F, and  is
the angle between r and F. (See Figure 11.1 for a review of the variables.)
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11. Complete the table below for the torque produced by each force.
Indicate if the torque is positive or negative for rotation about point O.
Determine the angle  between the line of action of the force and the line drawn from the axis of
rotation to the point of application of the force.
In symbolic form, write the equation for r in terms of xA, xB, xCM, and xD.
In symbolic form, write the equation for the torque associated with that force.
Force
Sign of
the torque
 (degrees)
r (m)
Equation for the torque
FA
FB
Fg
12. Write the equation for the net torque about point A on the bar when the bar is in static
equilibrium and calculate the distance of the center of mass (xCM) from point A. (Note that point
O on the illustration is an arbitrary point for discussion.)
net = ∑ = 0 = _________________________________________________________
Distance of the center of mass (xCM) from point A = ________________________
Compare this value to the value (xC – xA) and comment about the position of the CM.
Physics 1121 Experiment 11 Page 5
(4)
13. Using equations (2) and (3), write a symbolic expression for FA and FD.
FA = ________________________________________
FD = ________________________________________
14. Substitute all known quantities into these equations and obtain a value for FA and FD. These
values represent the calculated values for FA and FD. Show your work below.
FA calculated = ________________________
FD calculated = ________________________
15. Determine the percent error in the calculated values, using the measured values as the standard
values. Please show your work below.
% error in FA = ___________________
% error in FB = ___________________
Physics 1121 Experiment 11 Page 6
16. Remove the calibrated spring from position D and attach a knife-edge clamp as close to point
D as possible. Rest the end of the torque bar on the knife-edge support. (You may need to hold the
knife edge support down because of the horizontal force now exerted on it.)
17. Sketch a diagram of your experimental setup and identify the forces on the torque bar on the
diagram.
18. Record the force measured by the calibrated spring attached at point A, the angle A on the
protractor, and the position of the knife edge from the right end of the torque bar.
FA = _______________
A = _______________
Distance from right end of torque bar to knife edge = _______________
19. Record the angle of the torque bar relative to the horizontal, B or C, measured on protractor
B or C relative to the 90o reading. Again, use the protractor which rotates most freely to measure
the angle. You may need to hang one or two large paper clips at point B or C to stabilize the
protractor. Circle B or C below (which ever you use) and record its value.
B or C = _______________
Use the static equilibrium conditions to determine the horizontal and vertical forces exerted on the
torque bar by the knife edge.
20. Write the equations for translational equilibrium.
21. Write the equation for rotational equilibrium about the knife edge.
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22. Use the equations from Parts 20 and 21 to write a symbolic expression for FV and FH.
FV = __________________________________________________________________
FH = ___________________________________________________________________
23. Determine the value for FV and FH.
FV = ____________________________
FH = ___________________________
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Questions
1. In the drawing to the right, a 90-cm long metal
bar having a mass of 0.5 kg is supported by two
strings as shown.
a) Determine the value for the tension in each
string.
b) Calculate the position of the center of mass from the left end of the torque bar.
2. Athletes sometimes do an exercise called "curling weights". This exercise consists of holding
the arm steady from the shoulder to the elbow while pulling a weight to the chest. Calculate the
torque about the elbow caused by a 40-pound weight that makes an angle of 55 degrees with
respect to the body.
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