Abstract

In this experiment, three objects of differing shapes were tested to determine their moments of inertia. The moment of inertia of the first object, a rectangular bar, was calculated to be 0.0127 kg·m², a 0.765% difference from the theoretical value of 0.0126 kg·m². The second object, a solid disk, had an experimental value of 0.0084 kg·m², differing from the theoretical value of 0.0091 kg·m² by 7.85%. The final object, a hollow cylinder, had an experimental moment of inertia of .0042 kg·m², a 14.6% difference from the theoretical value of 0.0049 kg·m².

Exercise 1: Rectangular Bar

Exercise 2: Solid Disk

Exercise 3: Hollow Cylinder

Conclusions

In each experiment, variable masses were hung from a string passing through a pulley and wrapped around the axis of rotation of the objects being tested. The torque supplied by the hanging mass allowed the determinations of the moments of inertia of a rotating bar, disk, and hollow cylinder. The moment of inertia of a rectangular bar was calculated to be 0.0127 kg·m², differing by 0.765% from the theoretical value, 0.0126 kg·m². The solid disk had an experimental moment of inertia of 0.0084 kg·m², differing from the theoretical 0.0091 kg·m² by 7.85%. The hollow cylinder had an experimental moment of inertia of .0042 kg·m², a 14.6% difference from the theoretical 0.0049 kg·m².

The measuring devices used in the experiment added some degree of uncertainty to measurements. The triple beam balance had an uncertainly of 0.00002 kg. Distances measured on the meter stick are accurate to ±0.02 cm. Distances from the calipers are accurate to ±0.005 cm. Computer measurements of angular acceleration are accurate to ±0.001 rad/s².

In performing this experiment, there was a slight problem in carrying out the procedure which may have contributed to the errors in the results. The directions asked that the string should be wrapped around the middle spool. As an oversight, the string was instead wrapped around the top spool of half the radius. Since the error of the measurements seems to increase with increasing mass, it is possible that because of the significantly lower torque applied to the system the frictional forces and the ignored moment of inertia of the pulley combined to create a rather large error.

In addition, the final experiment with the hollow cylinder additional error may have been incurred through wobble in the object. The groove in which the cylinder was to be placed was too large for the object to be placed securely. Therefore, the estimation of its location had to be approximated, resulting in a rotational axis which was offset from the object’s center of mass, if only by a little. This would have increased the moment of the object, further increasing error.

Questions

Q 1 Given that the rotating shaft is made of stainless steel (density=7.6 g/cm³), justify why the moment of inertia of the spool was ignored during the theoretical treatment of the rotating system.

Stainless steel has a relatively low density, meaning that the rotating shaft had very little mass. A small mass combined with a very small radius leads to a negligible moment of inertia which can be ignored in the theoretical calculation.

Q 2 Derive the equation for the tension of the string.



1. F_T - m_hg = - m_ha
2. F_T = m_hg - m_ha
3. F_T = m_h (g - a)

Link to textual explanation of this derivation

Q 3 Torque on the on rotating bodies in this experiment is given as τ = r_s × F_T. What is the direction of the torque?

The torque, as a cross product, is perpendicular to the force and radius vectors, passing through the point of rotation.

Q 4 If the mass of the rectangular bar were to increase by a factor of 10, how would the bar's moment of inertia change?

The bar’s moment of inertia would also increase by a factor of 10.

Q 5 Say this experiment was repeated exactly as before, but instead of wrapping the string around the middle spool, it was wrapped around the large spool.

1. The force on the spool would remain the same.
2. The torque of the spool would double with the doubling radius.
3. The moment of inertia of the solid disk would remain constant.
4. The system’s angular acceleration would increase because of the greater torque applied.

Q 6 If the solid disk was twice as thick but its mass was unchanged, how would its moment of inertia be changed?

The moment of inertia does not depend on the thickness, so it would remain the same.

Q 7 In this experiment, we ignored the moment of inertia of the pulley. Is this a good assumption? Why or why not?

The mass of the pulley is insignificant compared to the hanging mass or the rotating masses. Therefore, its moment of inertia would not have greatly affected the acceleration of the system, and could be ignored.

Q 8 When comparing the theoretical and experimental values of the moments of inertia, did you determine the percent difference or the percent error? Why?

The percent difference was compared because even the theoretical value was based on empirically obtained numbers, not a fundamental accepted value.

Q 9 Should we calculate the moment of inertia of the rectangular bar by assuming it is a long thin rod rather than a bar with a discernible width? In other words, which geometry yields a lower error value?

The rod approximation assumes that the mass of the object is centered on a single axis. The beam which was tested had mass spread over a larger volume, so the formula for a bar is more accurate.