Click here to see a video of the experiment

RATIONALE:

Not all motion occurs in a straight line or even a curved path. Some very common motions are rotations, in which a rigid object turns about a fixed or moving axis. While it is conceptually possible to analyze the motion of such an object by applying the equations and methods of linear motion to each point on the object, the overwhelming complexity of such an approach renders it impractical for common use. Therefore, rotational motion is treated with the same principles of physics, but applied with a different coordinate system that simplifies the results considerably. It turns out that each principle or law of linear physics has an analogous equation in the realm of rotational physics. Thus, if one understands the principles of linear motion well, he should also understand the principles of rotational motion, because they are the same, and only the variable names change.

In this experiment, the student will investigate how the amount and placement of mass in a simple rotating system affects its rotational motion, and will be able to compare those effects with similar effects in linear motion systems.

When you have completed this experimental activity, you should be able to: (1) define the terms torque, angular acceleration, and moment of inertia; (2) write and apply Newton’s second law for rotational motion; (3) calculate the moment of inertia for a simple rotational system; (4) be able to experimentally determine the moment of inertia of a system; (5) understand the relationship between linear acceleration and angular acceleration; and (6) understand the differences and similarities of force and torque.

This experiment enables you to determine the moment of inertia of a variety of objects about axes of rotation that pass not only through the center of mass but through other points as well. The object is made to rotate about a vertical axis by applying a constant torque to it. This torque, G , gives rise to a constant angular acceleration whose value a is equal to the torque divided by the moment of inertia, I, as shown in equation (1):

A sketch of the apparatus is shown in the figure. The object whose moment of inertia is to be measured is a long thin rod, mounted on a spindle of radius r that rotates freely about a vertical axis. A light string that supports a weight passes over a pulley and is then wrapped around the spindle. The tension in the string provides the torque that produces the rotation of the system. The rod should be clamped so that it rotates about its center of mass. Weights can be clamped at various positions along the rod and the moment of inertia of the total system can then be determined.

The angular acceleration is measured indirectly. The weight falls with a linear acceleration a, and the value of a is determined by measuring the time required for the weight to fall a measurable distance s. For an object starting from rest, the distance it falls in time t is given by:

which allows the calculation of a as:

The angular acceleration can then be calculated since:

where r is the radius of the rotating spindle.

The first procedure is to determine the moment of inertia for the spindle and rod without any additional masses on the rod. Pick some kind of convenient reference point for the falling weight, so that you can start it from the same point each time, and measure the distance it will fall as accurately as possible with the meter stick. Use a falling mass of between 50 and 100 grams (including the weight hanger), and time its fall through the measured distance. Repeat three times, and use the average of the times in your calculations. Record all trials in your data table. Use the micrometer to measure the diameter of the spindle, and record both the diameter and radius of the spindle in your data table. In measuring the spindle, measure about 7 to 10 times, recording each measurement, and then use the average in your calculations. Since a very common error is to use the diameter rather than the radius in your calculations, be sure to compute and use the radius where it is required.

Now compute the linear acceleration of the falling mass using equation (3), and use that value in equation (4) to compute angular acceleration. The next step is to compute the torque, which can then be used in equation (1) to find moment of inertia.

Torque is a vector quantity whose magnitude is given by the product of the applied force and the perpendicular distance between the axis of rotation and the line of action of the force. In this experiment, the torque which causes the spindle to rotate is produced by the tension in the thread, which acts along a line tangent to the surface of the spindle. Since the perpendicular distance from the axis to the string is just the radius of the spindle, then the torque has a magnitude:

Now the tension in the string is given by:

since the falling mass is accelerated. Since you know both the mass and its acceleration a, the tension T can be easily computed, and then the torque can be found from equation (5). Find and record these values, then determine the moment of inertia, using equation (1).

Now add some mass to the rod by putting 100 gram masses near the two ends of the rod. Place them so that they are both the same distance from the axis of rotation, and measure that distance carefully. Repeat the observations and calculations of the previous part to determine the moment of inertia of the system in its new configuration. Follow the same systematic procedure as before. Record the value you get as an experimental value.

The value used for comparison is obtained by reviewing the theory of moment of inertia. It is shown in your textbook that the moment of inertia, I, of a system of discrete particles is given by:

where m is the mass of an object and x is the displacement of that object from the center of the spindle.

Thus the moment of inertia of a body depends not only on its total mass but also on how the mass is distributed in regard to the axis.

Choosing a different axis of rotation will result in a different moment of inertia, because the masses will not be the same distance from the new axis as from the old one.

The moment of inertia of your experimental system can be mathematically described as the sum of three terms, one for the rod and spindle, and one each for the two masses that you added. Thus we have:

where I_o is the moment of inertia of the rod and spindle, as determined previously. Calculate the moment of inertia of your system using equation (8), and compare it with the experimental value by using percent difference. What is the most likely source of error? In the final summary, report the average value.

Change the masses on the rod until the arrangement is non-symmetric. Don’t get too extreme in your arrangement, as this will cause a strong wobble in the apparatus which will affect the accuracy of your results. Repeat the observations and measurements in the same manner as for the symmetric arrangement, including calculations for both experimental and theoretical values of moment of inertia for the system. Again compare the corresponding values with percent difference. In the final summary, report the average value.

Report the values of inertia for each part. Compare these values. What conclusions can you draw from these comparisons? Comment on the additive nature of inertia.

1. How do the linear accelerations compare for the three parts?
2. How do the angular accelerations compare for the three parts?
3. How do the tensions compare for the three parts?
4. How do the torques compare for the three parts?
5. What conclusions can you draw from all of the comparisons?

Return to Table of Contents