Lab #8 Discussion
In this lab we used torque to experimentally discover the mass of a single paper clip. Torque is calculated as
the product of the magnitude of an applied forces times the perpendicular distance from its line of action to the
designated pivot point.
τ = FA⊥
The equation shown above is read as “torque equals the product of force times its moment arm.”
In the experiment you used a meter stick as your rigid beam. The initial step in Part I of the lab was to place
the knife-edge at the stick’s center of mass and observe that the meter stick remained level, or balanced. Many
of you became acutely aware of the sensitivity of the apparatus when trying to locate this precise balance
position.
image courtesy of miniscience.com
Once that was accomplished you then hung 5 washers exactly 10 centimeters from the knife-edge. This
immediately destroyed the stick’s equilibrium. The torque provided by these 5 washers would provide a
constant, consistent counterclockwise torque while you collected the next 10 data points.
Your data consisted of 10 different combinations of number of clips (N) and their positions. To determine the
moment of arm of each combination your subtracted the position of the knife-edge (center of mass) from the
clips’ position.
Hopefully you quickly discovered that as the number of paper clips increased, their moment arm decreased.
That is, the weight of the clips was inversely proportional to their distance from the knife-edge, or moment arm.
To determine your value for the mass of a single paper clip in Part I of the experiment you then plotted a line of
1
. You were asked to plot a line using all of your data in order to obtain the best value from the
A⊥
behavior of all of the data; not just from one point. Moreover, if you had plotted N vs A ⊥ your graph would have
N vs
been an hyperbola since the weight of the clips is inversely proportional to their distance from the knife-edge.
To rectify your data to be linear, we plotted N vs
1
.
A⊥
Now that you have a graph of your data, how do we experimentally determine the mass of a single clip?
Remember that the weight of the washers provided a ccw torque while the weight of each combination of clips
provided a cw torque.
τ cw = τ ccw
N ( mclip g ) L = mwashers g (0.10)
N ( mclip ) L = mwashers (0.10)
N=
0.10 mwashers
mclip L
⎛ 0.10 mwashers
N =⎜
⎜
mclip
⎝
⎞1
⎟⎟
⎠L
This equation represents the same information as the equation of the trend line which can be rewritten as
⎛1⎞
N = 3.9368 ⎜ ⎟ + 0.0447
⎝L⎠
By examining the parallel structure of these two equations we can see that the
numerical slope of our trend line represents the expression – that is, the
coefficients of 1/L are equal.
3.9368 =
0.10 mwashers
mclip
Since you measured the mass of your washers, the only unknown in this
equation is the mass of a single paper clip.
In Part II of your experiment, you offset the knife-edge by 6 centimeters so that the weight of the meter stick
(focused at the stick’s center of mass) provided the clockwise torque and the weight of paper clips provided the
counter-clockwise torque to bring the stick back into equilibrium.
τ ccw = τ cw
N ( mclip g ) L = mstick g (0.06)
N ( mclip ) L = mstick (0.06)
In this equation, you recorded the number of clips, N, the positions of the clips and the knife-edge, and the
mass of the meter stick. Once you calculated the moment arm for the clips, the only unknown in the equation
was the mass of a single paper clip; giving you a second experimental value for a single paper clip’s mass.
Since we do not know the actual mass of a single clip, you were asked to compare your two experimental
values by calculating the percent difference.
Exercise #1: After completing this lab, it should
be easy for you to now recognize that a triplebeam balance acts on the principle of torque and
rotational equilibrium. When an unknown mass
(#1) is placed on the pan its center of mass
should be located in the center of the pan. This
will insure that its moment arm agrees with the
manufacturer’s specifications of the balance.
Each of the counter-masses (#2) should always be settled into a
slotted position on their respective beams, once again to insure
that their moment arms are in agreement with the manufacturer’s
specifications.
When balanced, the counter-clockwise torque created by the
unknown’s weight will equal the combined clockwise torques
provided by the weights of each of the counter-masses. When the
torques match, the pointer on the beam will be “zeroed” and the
mass of the unknown can be found by adding together the “mass
locations” of the counter-masses. Remember that this device
measures gravitational mass since it can only operate correctly in
the presence of gravity since it is actually matching the torques of
the respective weights (forces).
Image of the balanced Diet Mountain Dew courtesy of Rutgers
(http://genchem.rutgers.edu/balance3b.html)
When the torque produced by the weight of the unknown mass exceeds the combined torques created by the
weights of the slotted counter-masses, the manufacturer provides an additional “500-gram” and “1000-gram”
cylinders which can be placed on the pegs on the end of the balancing beam to increase the amount of “mass”
the balance is able to measure.
Suppose the moment arm for the “peg” at the end of the balancing beam is 40 cm and the moment arm for the
center of the pan is 15 cm. How much is the physical mass of a “1000-gram” cylinder?
Why are you told that should NEVER carry a triple beam balance by its beam?
Exercise #2 The only triple beam balance in the room is broken and a team of students is assigned the task of
determining the mass of a meter stick. To assist them, they have a set of hooked masses (ranging in value
from 10 grams to 90 grams), a lever stand support, and a knife-edge clamp. Patiently, they determine that the
center of mass of the stick to be located at 50.3 cm. Mimicking the lab they just completed in class, they offset
the stick’s center of mass by 10 cm and try to rebalance the beam with slotted masses.
Finally they obtain equilibrium with the following data from left-to-right. 20 grams at 10 cm, the stick’s center of
mass at 50 cm, the knife-edge at 60 cm, 70 grams at 75 cm, and 30 grams at 95 cm.
Show the equations and necessary calculations to determine the meter stick’s unknown mass.