D-8 ROTATIONAL INERTIA
(1) To demonstrate the application of energy and momentum principles to a rotating system.
(2) To measure the moment of inertia of various mass distributions by a dynamical method and to verify the equation I = mr2 for moment of inertia.
Welch rotation apparatus with ball-bearing shaft.
3. BRIEF SUMMARY OF THEORY:
The moment of inertia of a body is the measure of its rotational inertia about an axis. It is determined by the body's mass distribution about that axis. The importance of the moment of inertia lies in its effect on the body's dynamical behavior. In particular, a torque applied to the body will cause an angular acceleration of the body given by
τ = Iα
where τ is the applied torque, I is the body's moment of inertia, and α is its angular acceleration. [Both τ and I must be expressed about the same axis.]
The moment of inertia of a localized or "point" mass about an axis is
I = mr2
where m is the mass and r is the perpendicular distance measured from the axis to the mass. The moment of inertia of a distributed mass is found by subdividing the body into infinitesimal point masses and integrating over the entire volume of the mass. Most textbooks have tables showing moments of inertia of bodies of simple geometric shape. You will need to consult this table, and you will also need to be able to apply the parallel axis theorem.
4. EXPERIMENTAL STRATEGY:
The apparatus has a vertical shaft (D) rotating on high quality ball bearings (B). A horizontal arm attaches to the top of the shaft. Slotted weights (C1 and C2) may be placed on this arm at various radii, and secured with wing nuts. You will dynamically measure the moment of inertia of weight distributions placed on this arm, with respect to the vertical rotation axis. You will also check these results against calculations made by use of the textbook formulae for moments of inertia of simple shapes.
The rotation is produced in a manner which allows measurement of the acceleration of the system. A string is wrapped around the vertical shaft, run over the pulley (H) to a weight hanger. Weights (W) are placed on the hanger and allowed to fall a measured distance. The time of fall is measured.
The falling weight covers a distance y in t seconds, so
y = (1/2)at2
Use Eq. 3 to calculate the linear acceleration of the falling weight. The net force on the falling body is
mg - T = ma .
The net torque on the rotating shaft is therefore
τ = Tr - τf = Iα .
τf is the retarding torque due to all frictional forces, primarily in the pulley and the main bearing.
The angular and linear accelerations are related by
a = αr .
From Eq. 4 through 6 the tension T and the angular acceleration α may be eliminated. The result may be solved for the moment of inertia, I:
I = [(g/a) - 1] m r2 - (τfr)/a
Eq. 3 may be used to find a, experimentally. Eq. 7 then gives the moment of inertia in terms of measured quantities.
The total moment of inertia, I can be written as the sum of the inertia due to the added weights, MR2, and the inherent inertia of the vertical shaft and cross-arm, Io.
I = Io + MR2
In order to investigate Eq. 8 systematically, rearrange it algebraically into a form suitable for graphical analysis. Leaving the details as an algebra exercise, the result is:
MR2 = (1/a)[mr2g - τf r] - [Io + mr2] .
A plot of MR2 vs. (1/a) should be a straight line. The quantity [mr2g - τf r] is the slope of the line, from which the torque due to friction, τf, may be determined. The quantity [Io + mr2] is the intercept on the MR2 axis, from which Io may be determined.
The quantity [mr2g - τfr] may be written [r(mgr - τf)], which more clearly shows that the frictional torque τf opposes the accelerating torque mgr.
This form of the equation suggests an appropriate data-taking strategy. To obtain a straight line graph, the quantities in square brackets must be kept constant. All of the quantities in square brackets are constants except m, but m may be easily kept constant while varying M and R.
5. EXPERIMENTAL PROCEDURE:
The moment of inertia of the vertical shaft and the horizontal arm can be measured, but cannot easily be directly calculated from geometry. One might be tempted to directly measure their moment of inertia by accelerating the cross arm with no added weights on it. This would require a very small m on the hanger to achieve a time of fall long enough to measure precisely. It would also make the applied force on the shaft so small that the torque of friction might cause systematic error.
Another serious concern is whether the torque due to friction is constant for all cases. Friction, as you know, usually depends on the force pressing surfaces together (the loading force.) Here the loading force is the total force on the bearing, which is the total weight of the shaft, cross-arm and any masses on the arm. The only way to hold the friction constant is to keep M and m both constant, allowing R to vary.
So you will hold m constant in all cases, for the reasons given above. Then systematically study cases with different values of M and R.
It is always a good idea to have a clear strategy in mind for analyzing the data before you begin to collect data. Then you will be sure to have sufficient data in the correct form to analyze.
Your objective is to experimentally verify that the moment of inertia of a mass moving in a circle of radius R is proportional to the mass and proportional to the square of the radius of the circle. Therefore it is good strategy to study a sufficient number of cases where M is held constant and R varies, and still another set of cases where R is held constant and M varies. You should study the next section, DATA ANALYSIS, before actually taking any data!
In general, the procedure is this: Place an appropriate number of slotted weights in the weight hanger so the acceleration is slow enough to time precisely. Record:
h, the distance the hanger "falls"
The quantities to be determined are:
Io, the moment of inertia of the rotating assembly
Place equal masses at equal radii on either side of the horizontal arm, for good static and dynamic balance. Tighten the wing nuts securely.
Consider the meaning of R. It must be measured from the rotation axis to the effective center of the mass. Where is the effective center? It is not the center of mass. It is the location of a point mass which would have the same moment of inertia as the distributed mass. The "effective" radius is called the "radius of gyration". it is somewhat larger than the distance to the center of mass.) If you have had calculus you should derive an equation for the radius of gyration for a cylindrical mass that is oriented in the same way that your cylindrical weights are. If you have not had calculus, just measure from the geometric center, and realize that you are introducing a small systematic error by doing so.
(1) Take data for various values of the mass M, holding R constant at a value of about 12 cm. Use this data to make a plot of M vs. (1/a). Include the mass of the wing nuts in M.
(2) Take data for various values of R with constant M. Make a plot of R2 vs. (1/a).
6. DATA ANALYSIS
(1) Look at the MR2 vs. (1/a) graph from procedure step (1), where R was held constant. From equations (8) and (9) you expect that the graph should be a straight line (with some deviations due to indeterminate errors). If the deviations are consistent with your error estimates, draw the best straight line through the data. From the intercept of this line on the MR2 axis, calculate the moment of inertia the shaft and rod, Io, when there's no added mass. The slope of the line allows a calculation of τf, the torque due to friction. Be especially careful to determine whether the values of torque of friction are meaningful; that is, calculate the limits of error on the value.
(2) Look at the R2 vs. (1/a) plot of data from procedure (2), where R was held constant. It should also be a straight line. Does it have the same slope and intercept as the other graph? Should it? If it doesn't, what does this tell you?
(3) You may also plot all of the data on a single graph. Suggestion: Plot R2 vs. (1/a) with M as a parameter. You would expect that for each value of M you'd get a different straight line plot, but all the straight lines would be parallel. Does this happen?
(4) You may also plot all of the data, plotting M vs. (1/a) with R as a parameter. Discusss the significance (if any) of the slopes and intercepts of this graph.
(5) τf is an interesting quantity in this experiment. It will, of course depend on your particular apparatus and the quality of the main bearing. It may be too small to affect your measurements, and therefore to small to measure, but don't assume that. Let your data analysis reveal whether it is significant. If it is affecting your data in a measurable way, you may discover that the friction is not a constant, but is dependent on the total mass loading of the bearing (2M). To show this dependence may be a real challenge!
(6) The equations used in this experiment do not separate the various frictional effects. The quantity τf actually includes the friction in the pulley, which probably is larger than that of the main bearing. You would expect this pulley friction to depend on the mass loading of the pulley, which is proportional to the falling weight, mg. Does your data support this?
(1) Determine the fraction of mgh which is dissipated by frictional effects.
(2) A student assumes that the tension in the string is just equal to the weight hanging from it. (This is, of course, not so.) How much error will this mistake cause (in %) in this student's experimental determination of the moment of inertia?
(3) Show the full derivation of equation (7).
(4) How large would the systematic error be in the moment of inertia if the mass of the wing nuts had not been included in the calculation?
(5*) Derive an equation for the moment of inertia, expressed as a function of the experimentally measured quantities listed in procedure (1).
(6*) Derive an equation for the total energy which was lost to frictional heating, in terms of the experimentally measured quantities listed in procedure (1).
(7*) Equation (8) was used to calculate moments of inertia. That equation did not explicitly include the amount of frictional energy loss. Yet friction certainly does affect the data—the falling body would accelerate a bit faster if friction were removed. Then why can friction be "ignored" when using equation (8)?
(8*) Treating the rotating weights as cylinders, develop an equation to experimentally determine their radius of gyration.
Text and line drawing © 1998, 2004 by Donald E. Simanek.