4-May-2015: Angular acceleration

PURPOSE

The purpose of this experiment was to measure the angular acceleration of a rotating disk and derive its moment of inertia.

PROCEDURES

PART 1

Figure 1

We began by setting up the experiment as shown in Figure 1. The set-up consisted of a hanging mass connected to a set of disks with a string that was wrapped around a pulley. Compressed air was supplied to the system, which allowed the disk(s) to rotate on a virtually frictionless surface. Moreover, a rotational sensor kept track of how fast the disk(s) were rotating by counting the number of black and white stripes that passed by the sensor. There were total of 200 marks on each disk, so every time the sensor counted 200 marks, it "knew" that the disk(s) had completed a rotation.

Figure 2

Figure 3

Figure 4

For the first three experiments, we kept everything the same except the mass of the hanging object. We kept the bottom disk stationary and only allowed the top one to rotate. In addition, we used the smaller pulley (r = 1.301 cm) shown in Figure 1. The resulting graphs from using hanging masses of 25 g, 50 g, and 75 g, respectively, are shown in Figures 2 through 4. The effects of changing the hanging mass will discussed later.

Figure 5

For Experiment 4, we used a 25 g hanging mass and allowed only the top disk to rotate just as we did before. However, this time we increased the radius of the pulley from 1.301 cm to 2.500 cm. The motion of the disk is shown in Figure 5. Just as before, the part of the angular velocity versus time graph that slopes up represents when the hanging mass is accelerating downwards and the part that slopes down represents the hanging mass going upwards.

Figure 6

Figure 7

In Experiments 5 and 6, we used the larger pulley just as we did in Experiment 4 and kept the hanging mass at 25 g. The only variable we changed was the mass of the rotating disk(s). In both Experiments 4 and 5, we only had the top disk rotating. The only difference was that we used a steel disk in Experiment 4, while we used an aluminum disk in Experiment 5. In Experiment 6, we had both the top and bottom steel disks rotating. The resulting graphs from these experiments are shown in Figures 5, 6 and 7.

Figure 8

By analyzing the data table in Figure 8, it can be seen that changing different values has different effects on the angular acceleration of the disk(s). In Experiments 1 through 3, we kept all factors constant except the hanging mass. We increased the hanging mass 25 g at a time, which seemed to have increased the average angular acceleration by a proportional amount. For example, the average angular acceleration increased from 3.296 rad/s² to 6.588 rad/s², when we doubled the mass from 25 g to 50 g. The angular acceleration was multiplied by a factor of 1.999, which is almost exactly 2. The percent difference between the two values is only 0.0607 percent. This result is to be expected because the hanging mass is pretty much the only thing that is causing any torque on the disk except for friction, which seems to be negligible.

In Experiments 1 and 4, we observed how changing the radius of the pulley contributed to the angular acceleration of the disk with everything else constant. As we expected, we saw that the angular acceleration decreased as the pulley radius increased. When the radius increased from 1.301 cm to 2.500 cm, the acceleration decreased from 3.296 rad/s² to 2.228 rad/s². We expected this result because the increase in the pulley's mass and radius added to the moment of inertia of the rotating system since the moment of inertia of a disk is equal to ½MR².

On the other hand, in Experiments 4 through 6, we found the relationship between the mass of the disk and its angular acceleration. When we decreased the mass of the rotating disk from 1361 g to 466 g (from Experiment 4 to 5), the angular acceleration increased from 2.116 rad/s² to 5.900 rad/s². If we find the moment of inertia of each disk and multiply it by the angular acceleration, we end up with values of 0.006575 kg*m²/s² and 0.006277 kg*m²/s², respectively. The percent difference between these values is only 4.526 percent. Therefore, it can be seen that changing the mass of the rotating disk has a proportional effect on its angular acceleration.

Figure 9

In addition to the six experiments discussed above, we also performed another experiment in which we observed the relationship between the angular acceleration of the disk and the translational acceleration of the hanging mass. We accomplished this task by setting up a motion sensor directly below the hanging mass. We also taped an index card on the bottom of the hanging mass to enable the motion sensor to properly capture the mass' motion.

Figure 10

From the graphs in Figure 10, we found the two accelerations by looking at the slopes of their respective velocity versus time graphs. For example, the second graph in Figure 10 corresponds to the angular velocity of the disk plotted with respect to time. The average slope of this graph was 3.471 rad/s². The average slope of the mass' translational velocity versus time graph was 0.04374 m/s² (omitting the negative sign since it only dictates the direction of the acceleration). The angular acceleration of the disk (α) and the translational acceleration of the mass (a) are related by the equation a = rα, where r is the radius of the pulley. Using r = 1.301 cm = 0.01301 m, we solved for the right side of the equation, which was equal to 0.04516 m/s². The percent difference between the left side and the right side of the equation was only 3.241 percent. Therefore, we concluded that our results were valid.

PART 2

For the second part of the experiment, we derived an equation to find the moment of inertia of the disk(s) based on the angular accelerations that we found in the various experiments in Part 1. This equation is displayed in Figure 11. The r in this equation is the radius of the pulley and the α corresponds to the average angular acceleration of the disk(s), while the m refers to the hanging mass. We then found the moment of inertia's of the disk(s) by using the equation I = ½MR² and compared the values.

Figure 12

The data table in Figure 12 illustrates the resulting values from finding the moment of inertia's with the two different methods. As it can be seen from the image, the percent error in the first three experiments were extremely high. The percent error in the final three experiments were also higher than we would have liked, but were acceptable compared to the first three experiments. The reasons for these results will be discussed in the conclusion.

CONCLUSION

In Part 1 of the lab, we were having considerable success in getting the results that we wanted. For example, as mentioned before, when we doubled the hanging mass, the angular acceleration acceleration almost doubled as well. Another example is when we increased the mass of the rotating disk, its angular acceleration decreased by a proportional amount. However, when we got to Part 2 of the lab, we ran into some trouble. As it can be seen from Figure 12, the moment of inertia's found using the two different methods were very different from each other, especially in the first three experiments. Since the moment of inertia's found with the equation I = ½MR² did not have much room for error, we assumed that these were the more accurate values. It was difficult to pinpoint the cause for the moment of inertia's found using the equation in Figure 11 were so off from these values. The challenge was that these moment of inertia's were smaller than the expected values, which means the angular accelerations were bigger that they were supposed to be. It would have made more sense if the angular accelerations were lower than they were supposed to be because that could have been attributed to non-conservative forces such as friction. There is a possibility that we made an error in our calculations, but this seems unlikely because we carefully looked through our calculations for any errors. Therefore, it is difficult to say what was the cause for our values being so inaccurate.