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:
F

F
1

F
2

F
3

...

0
net
(1)
Torques may be calculated about the axis of your choosing and because a meter stick is our
body, the length of the lever arm can be conveniently determined without resolving components.




...

0
net 1
2
3
(2)
Each torque is specified by the equation:
  Fd
(3)
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:
F

F


0



x
y
(4)
Figure 1: Experimental setup for Part II
The condition that both the net force along the x-axis (which is conveniently taken along the
incline) and the net force along the y-direction (perpendicular to the incline) must be zero yields
a relationship between the coefficient of static friction μ (where Ffr=μ FN) and the inclination
angle: μ=tan θ.
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:

 

0

mgr

F
R

0

mgr

(
M

m
)
gR
cos

0

fr
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 μ=tan θ will result in:
mgr

(
M

m
)
gR
sin

sin
 mr
(M

m
)R
(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.
2. 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 by string tied tightly around the meter stick.
3. Attach three masses to the meter stick using strings. Neglect the masses of the support strings.
4. Attach the force sensor cords to the Interface box as you have done in previous labs.
5. For today's lab you do NOT need to graph the force sensors over time, instead, drag the force
icons to the Digits icon in the "Displays" menu. 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.
6. Then hang a known mass on each force sensor to verify that it is reading correctly.
If you need to increase the precision of the display, select the downward-facing arrow on the
icon that says "3.14" and select Increase Precision. This will increase the number of significant
figures. Next, tare each force sensor without weight to establish zero.
7. Hang the masses (100, 200 and 500g according to the table in step 9) on the three remaining
strings and 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(g)
Force #
Force (N)
m1  100
m2  200
m3  500
F1
F2
F3
m1  g 
m2  g 
m3  g 
m4 
F4
(m
)g
4
meterstick
Position (m)
F5
F6
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)
Torque (N-m)
Axis at x=0m
Axis at x = 0m
F1
F2
F3
F4
F5
F6
Sum of Positive Torques
Sum of Negative Torques
%diff between positive and
negative torques
Sum of all Torques
Lever Arm (m) Torque (N-m)
Axis at x=0.5m Axis at =0.5m
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. 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.
2. 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
1
10
2
15
3
20
M
(kg)
mmin
mmax
maverage
σm
θºexperiment
σθ
For your Lab Report:
Include a sample calculation of the torques, and % differences in Part I for x=0 condition. Find
θexperimen±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?