Begin Q&A Forum for 'ROTATIONAL INERTIA' Douglas, Number 6 of the post lab asks us to calculate the force of friction given the vertical intercept and a measurement for the bearings from the center of the wheel. I don't need linear inertia calculations, but instead to calculate rotational motion inertia for an object rotating on itself over an axis. Example: Have simple cylinder rotating on itself over vertical axis, and be able to Calculate or Plot out the rotational inertia for the Part. Rotary bearings were some of the earliest products we distributed. Proper bearing selection and implementation are among our core competencies. Our engineering team has extensive experience in designing systems and choosing bearings. Take advantage of their expertise to specify the best bearing for your application.

(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.

2. APPARATUS:

Welch rotation apparatus with ball-bearing shaft.
Moment of inertia rod attachment.
100 and 50 gram slotted weights.
1/10 second stop watch.
Weight hanger, string.

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 that 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 - τ_f = Iα .

τ_f 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 = α r .

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 - (τ_f r)/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, I_o.

I = I_o + 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 - τ_f r] - [I_o + mr²] .

A plot of MR² vs. (1/a) should be a straight line. The quantity [mr²g - τ_f r] is the slope of the line, from which the torque due to friction, τ_f, may be determined. The quantity [I_o + mr²] is the intercept on the MR² axis, from which I_o may be determined.

The quantity [mr²g - τ_f r] may be written [r(mgr - τ_f)], which more clearly shows that the frictional torque τ_f opposes the accelerating torque mgr.

This form of the equation suggests an appropriate
t, the time it takes to fall that distance
m, the total mass of the weights and hanger
the radius and length of the vertical steel shaft
the length and mass of the horizontal shaft
R, the distance of the weights on the horizontal arm from the rotation axis.
M, the total mass of the weights added on the horizontal arm.

The quantities to be determined are:

I_o, the moment of inertia of the rotating assembly
τ_f, the frictional torque in the bearing

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 that 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, I_o, when there's no added mass. The slope of the line allows a calculation of τ_f, 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) τ_f 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 too 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 τ_f 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?

7. QUESTIONS:

(1) Determine the fraction of mgh that 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).

Rotational Inertia Formula

(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 that 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.

Rotational Inertia Of Bearing Friction

Text and line drawing © 1998, 2004 by Donald E. Simanek.