Paul J. Dolan, Jr.* and David Sturm, 5709 Bennett Hall, Dept. of Physics and Astronomy, University of Maine, Orono, ME 04469-5709, liquidhelium@hotmail.com.
Abstract
We describe an experiment that is performed in the first
semester of the introductory course at the University of Maine in which
students calculate, and then experimentally investigate, the rotational
inertia of an irregular (and changeable) object. The experiment utilizes
a Force Table as a base and the Pasco Smart Pulley© photogate as a
sensing device.
Introduction
Rotational Inertia is one of the more interesting, and sometimes more difficult, topics that introductory Physics students encounter. Experiments in which students can quantitatively measure I, and also carefully measure how the value of I changes as the positions of the masses change, can greatly aid their understanding of the topic, and their appreciation for the fact that ‘physics works’. The experiment that is done here allows students to both calculate the rotational inertia of a somewhat irregular object, as the positions of the masses are changed, and to verify their calculations experimentally. Careful students can obtain a discrepancy of 5% or less in their final result.
The apparatus was first proposed by Professors Charles
Smith and Gerald Harmon several years ago, prior to the introduction of
computer-assisted data collection, and has undergone several iterations
to reach its current form. The apparatus is shown in figure 1. It consists
of a bearing inside a spool; the shaft around which the spool rotates fits
into the center hole of the standard Force Table. So long as the friction
in the bearing is reasonably constant, its actual value does not matter,
as this will be directly measured in the experiment. An aluminum bar, with
pairs of holes, spaced at 2 cm intervals out from the center, is attached
to the top of the spool; the spacing of the holes has been chosen so that
the actual rotational inertia of the bar is 95 % of the value of a solid
bar of the same size and masses. Pairs of masses (about 400 g) fit into
these holes. Moving the positions of these masses allows one to change
I, without changing the total mass. Torque may be applied via string is
wrapped around the spool (radius 3 cm), which is attached to a standard
50 g weight hangar. The string passes over the force table pulley; a small
riser block is used to insure that the force is applied horizontally and
tangent to the spool. Each setup has been made to be identical, so that
a direct comparison of all students’ data is possible.
Theory
First, students are asked to calculate the total rotational inertia of the system: spool plus bar plus masses, for five possible positions of the masses (R). Critical dimensions are measured with a vernier caliper with a dial gauge, to an accuracy of .001 cm.
The rotational inertia of the bar is given by:
where m_{o} is the mass of the bar, l its
length, and w its width. The rotational inertia of the cylindrical masses
is given by
where M is the mass of
one cylinder, and ? is its diameter. The second term is the contribution
from the parallel axis theorem, and is the one portion of I that will be
varied experimentally; in fact, this comprises the major contribution
to the total rotational inertia. The rotational inertia of the spool is
not simple to calculate, by the students at this level, as the spool is
non-uniform. This has been measured to have a value of
Experiment
The angular velocity of the bar (w ) is measured with the photogate for a Pasco Smart Pulley, and is plotted as a function of time. The quantity in which we are interested is the angular acceleration, i.e., the slope of w vs. t. One might be concerned that friction could play a major role in the experiment. However, this contribution is explicitly measured. The students first measure w vs. t for a ‘freely spinning’ apparatus. As observed over several rotations, the apparatus will slow down, and thus they can find the angular acceleration due to friction (a_{f}) (which is, of course, negative). Having done this for each position of the cylindrical masses, the students proceed to apply a mass to the string (a 50 g weight hangar is sufficient), and to find the angular acceleration (a ) of the apparatus ‘under load’.
As with all good experiments, the students do each measurements
several times, and use the average. Use of the Smart Pulley facilitates
this.
Data & Analysis
When one constructs the free body diagrams for the forces and torques on the spool and on the weight hanger, one finds that the total moment of inertia is related to the mass of the hangar (m) as
(4) I = m(g - ra )r/(a - a_{f})
where m is the applied ‘load’ (the weight hangar), and g is the acceleration due to gravity. We have found it convenient in this experiment to use CGS units.
Measurements of the physical parameters of the problem
are given in Table I. Typical data for the average acceleration of the
rotating object, both freely (a_{f})
and ‘under load’, are shown in Table II. Typical data, both theoretical
and experimental, for the rotational inertia is also shown in Table II.
One can see that there is remarkable close agreement between the values.
Discussion
Except for the smallest spacing of the masses, the discrepancy
between the experimental and theoretical values is well under 5 %. There
are two major contributions to the uncertainty in the experiment, beyond
the usual measurement uncertainty, which is quite small. The first is the
value of I_{s}. This value is an average experimental value taken
from all the apparati in use. As the moveable masses dominate I as R increases,
one would expect any discrepancy in I_{s} to be most noticeable
at the lowest values of R. The other uncertainty enters in the measurement
of a_{f}, which is assumed to be independent
of w . In fact, this is not the case, and it
is seen that a_{f} increases as w
increases, which is to be expected. The variation in a_{f}
between ‘slow’ and ‘fast’ rotation rates is also largest at the lowest
value of R, so that once again this effect should cause a greater discrepancy
at the lower R values. In contrast, the variation in the measured value
of a ‘under load’ is very small, typically 0.01
rad/s^{2}.
Conclusion
This experiment has proven to be very successful in allowing
students to calculate and directly measure the rotational inertia of a
(somewhat) irregular object, as the positions of the masses is varied.
Students come away with a greater appreciation of this often difficult
topic. The apparatus used is not especially difficult to construct, and
has the added advantage that it makes good use of an underutilized piece
of equipment that most physics departments own, the force table.
Bar: length(l) = 30.00 cm, width (w) = 2.551 cm, height = 0.965 cm, mass (m_{o}) = 185.6 g
I_{b} = 13320 g-cm^{2}
Spool: height = 5.30 cm, diameter = 2.999 cm, mass = 183.2 g
I_{s} = 2500 g-cm^{2}
Cylindrical masses: (average) mass (M) = 399.8 g, (average) radius (r ) = 2.5605 cm, height = 10.05 cm
‘Load’ (weight hangar), mass (m) = 50.3 g
Table II: Theoretical values of I, Experimental values
of I and a .
Distance from center (R) (cm) | (Average) Acceleration due to friction (a_{f}) (rad/s^{2}) | (Average) Acceleration ‘under load’ (a ) (rad/s^{2}) | Theoretical Rotational Inertia (g-cm^{2}) | Experimental Rotational Inertia (g-cm^{2}) | Percent Discrepancy |
3.0 | -0.1994 | 5.858 | 25637 | 23968 | 6.5 % |
5.0 | -0.1414 | 3.783 | 38431 | 37234 | 3.1 % |
7.0 | -0.0939 | 2.500 | 57621 | 56556 | 1.8 % |
9.0 | -0.06166 | 1.726 | 83208 | 82259 | 1.1 % |
11.0 | -0.04579 | 1.231 | 115192 | 115348 | -0.1 % |