13-May-2015: Finding the moment of inertia of a uniform triangle about its center of mass

Lab Partner: Jenna Tanimoto

PURPOSE

The purpose of this experiment was to find the moment of inertia of an object through experimental methods.

PROCEDURES

Figure 1

We began this experiment by constructing the set-up as shown in Figure 1. The set-up was very similar to the set-up used in Lab 16: Angular acceleration, except for the pulley connected to the disks. This pulley had an attachment that allowed us to place an object that would rotate with the disks. We first ran the experiment without anything added to find the moment of inertia of the system without the triangle. We would later subtract this from the moment of inertia found in the next two parts to find the moment of inertia of the triangle by itself.

Figure 2

Figure 2 shows the resulting graphs. From the second graph in Figure 2, we were able to find the angular acceleration of the system by looking at the slopes of the graph. We assumed that the parts of the graph that were sloping down represented the system when the hanging mass was going up and vice versa for the parts of the graph that were sloping up.

Figure 3

Next, we added a right triangular plate to the system by attaching it to the pulley at its center of mass, as shown in Figure 3. We first had the longer side as the height and the shorter side as the base. Then in the second part of the third part of the experiment, we changed the orientation of the triangle so its base was bigger than its height.

Figure 4: Triangle with h > b

Figure 5: Triangle with h < b

Figures 4 and 5 represent the disks' motion with the triangle in the two different orientations. We found the angular acceleration of the disk for each trial using the same process as the first part of the experiment (from the slopes of the graphs). All the values that we found are summed up in Figure 6 below. 

Figure 6

From the values on the left side of Figure 6, we found the average angular acceleration for the three cases (no triangle, triangle h>b, triangle h<b), which were 6.640 rad/s², 5.266 rad/s², and 4.192 rad/s², respectively. The dimensions of the triangle and the distance between the y-axis and the center of mass of each triangle are also shown in Figure 6.

Figure 7

Figure 7

After finding the average angular acceleration of each case, we were able to derive the moment of inertia using the same equation that we used in Lab 16 (Figure 7). The r in the equation refers to the radius of the pulley, while m refers to the hanging mass. The α represents the average angular acceleration of the disk. We found the moment of inertia of the system without the triangle (I = 0.0009078 kg*m²) to subtract it from the moment of inertia with the triangle (I = 0.001149 kg*m² for h>b and I = 0.001447 kg*m² for h<b) to find the moment of inertia of the triangle by itself. This process is shown in Figure 8 with the triangle in the different orientations. The Isys represents the moment of inertia of the entire system including the triangle and Io represents the moment of inertia of the rotating system without the triangle. For the triangle with its height bigger than its base, the moment of inertia was 0.00024 kg*m². On the other hand, the triangle with its base greater than its height had a moment of inertia of 0.000539 kg*m². This makes sense that the triangle with its base greater than its height had a bigger moment of inertia because more of its mass is concentrated farther away from the axis of rotation.

Figure 9

Figure 10

After finding our experimental values, we used calculus to derive a formula to find the moment of inertia of a triangle rotating about the y-axis as shown in Figure 9. We ended up with the formula I = 1/6MB². Then, we found the moment of inertia about the triangle's center of mass (for both cases) by using the parallel-axis theorem as demonstrated in Figure 10. The d is the horizontal distance between the y-axis and the triangle's center of mass.. These distances are illustrated on the bottom right corner of Figure 6.

Finally, we compared these values to the experimental values displayed in Figure 8. For the triangle with h>b, we got an theoretical value of 0.0002439 kg*m² and an experimental value of 0.00024 kg*m². The percent error between these values is 4.167 percent. For the triangle with h<b, we got a theoretical value of 0.0006047 kg*m² and an experimental value of 0.000539 kg*m². The percent difference between the expected and experimental values was 10.86 percent. The percent error for the second case was slightly higher that we would have liked.

CONCLUSION

In this experiment, we were able to get some reliable results. However, values found for the triangle with h<b was not as close as to the theoretical values as we would have liked. With a percent error of 10.86 percent, the error was not alarmingly high, but it was too high to just overlook. We believe that there could have been a few factors that contributed to this slight error. First, the hanging mass could have weighed more than we had assumed that it weighed based on the value marked on the mass itself. This would explain why the experimental moment of inertia was smaller than the theoretical one because the angular acceleration of the system was higher than it was supposed to be. One factor that we would have expected to contribute to the error in the experiment was the negative torque caused on the system by friction. However, this seems not to be our case because this would have made the experimental value being higher than the theoretical value because the angular acceleration would have been slower than we expected.

11-May-2015: Moment of inertia and frictional torque

PURPOSE

The purpose of this experiment was to use our torque equation to ultimately derive an acceleration for an object and compare it to its experimental value.

PROCEDURES

Figure 1

Before setting up our experiment, we made some calculations to find the moment of inertia of an inertia wheel. We did this by imagining that the inertia wheel was composed of two cylinders and a disk. We found the moment of inertia of each component and added them up to find the total moment of inertia of the entire inertia wheel. However, in order to do, we had to first find the mass of each component. We did this by measuring their diameters and thicknesses and calculating their volumes. We then used the ratios of their volumes with respect to the total volume of the inertia wheel and set them equal to the ratio of their masses over the inertia wheel's total mass. With these masses, we were able to find the moment of inertia of each component by assuming that they were all disks and using the equation I = ½MR². When we added these individual moments of inertia, we got a value of 0.020439 kg*m². This whole process is shown in Figure 1.

Figure 2

We began the procedures of this experiment by giving the inertia wheel a spin and allowing it to come to a stop by itself. We captured the motion of the wheel with a camera (Figure 2) and used video analysis to measure the negative angular acceleration caused by the axle.

Figure 3

When we analyzed the video, we used the piece of tape on the wheel as a reference point and placed a data point on the top right corner of it every time the wheel completed a rotation (Figure 3). We set the origin at the axle and rotated the xy-plane until the y-axis was somewhat lined up with the data points, as shown in Figure 3.

Figure 4: Angular displacement vs time

Next, we created a "Manual Column" in Logger Pro called "theta" and plugged in 0, 2π, 4π, and 6π, which corresponded to each time the inertia wheel made a rotation. Then, we plotted "theta" versus time as shown in Figure 4. We applied a "Quadratic Fit" to the graph. We noticed the resulting equation was one of the constant angular acceleration equations that we learned in class i.e. Δθ = ωot + ½αt². From this relationship, we were able to deduce that A was equal to ½α. Therefore, we found α by multiplying A by 2, which was -1.0322 rad/s². From this value, we were able to find the negative torque caused by the wheel's axle by multiplying it by the wheel's moment of inertia (I = 0.020439 kg*m²). We found this value to be -0.02114 N*m.

Figure 5

In order to predict the time it would take the cart to travel 1 m down the track, we applied Newton's second law of motion on the cart in the x-direction and the torque equation on the wheel as shown in Figure 5. We assumed that the only two forces acting on the cart in the x-direction were the tension (T) in the string and the horizontal component of the cart's weight. We set up the force equation according to this assumption and set everything equal to T. Then, we set up the torque equation on the wheel. Besides the negative torque caused by the axle, the only other force contributing to the torque on the wheel was T. In addition, we related the angular acceleration (α) to the translational acceleration of the cart (a) with the equation a = rα. Then, we set the resulting torque equation equal to T as well and set the force and torque equations equal to each other. This got rid of T and we were left with only a as an unknown. This allowed us to solve for a, which we found to be 0.02101 m/s². We used this value in the kinematic equation to solve for t (the time taken for the cart to travel 1 m down the track), which was 9.756 s.

Figure 6

After completing our calculations, we set up the track as shown in Figure 6. We wound up the string on one of the smaller "cylinders" and released the cart to allow it to accelerate. We used our phones to measure the time it took for the cart to travel down 1 m. We measured this time to be 10.58 s. Compared to the theoretical value found earlier, the percent error was 8.445 percent. This was a bit higher than what we would have liked (~5 percent). However, we concluded that the results were acceptable because there were several factors that probably slowed down the cart that we did not consider in our calculations.

CONCLUSION

As mentioned before, the percent error was slightly higher than we would have liked. This was a result of several factors. One of the biggest sources of error probably came from the way we measured the time it took for the cart to move down 1 m. First of all, the response time of starting and stopping the "stopwatch" added to the overall time. In addition, it was difficult to determine exactly when the cart passed the 1 m mark since the cart was moving at a relatively fast speed by the time it reached that point. Another factor that contributed to the error came was the friction between the string and the cylinder as the cylinder was assumed to be a frictionless "pulley" in our calculations.

Despite these sources of error, our experiment was a moderate success as our percent error was not extremely high. Furthermore, we gained valuable experience in implementing what we learned about moment of inertia's and torque in a real-life experiment. It was interesting to see that our calculated values were close to the experimental values.

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.

27-Apr-2015: Conservation of momentum and energy

PURPOSE

The purpose of this experiment was to combine the conservation laws of momentum and energy to find an unknown velocity of a system.

PROCEDURES

Figure 1

In this experiment, we used the set-up shown in Figure 1. We pulled on the trigger (circled in green) to shoot a ball into a concave in a block (circled in red) to give the block a velocity and raise it above its initial position. The block swung on a set of strings, which made an angle with the vertical as the block was raised. This angle was measured on the part of the apparatus circled in blue. In addition, we measured the mass of the ball and the block, and the length of the strings. All the measurements are illustrated below in Figure 2. M was the mass of the block, while m was the mass of the ball. L was the length of the strings and θ was the angle that the strings made with the vertical.

Figure 2: Measurements with uncertainties

Figure 3: Derivation of initial velocity of ball

The uncertainty of each measurement was also included in Figure 2. These uncertainties were later used to find the propagated uncertainty of the ball's initial velocity. In order to find the propagated uncertainty, we had to first find an equation for the velocity. We applied the conservation of momentum equation to the system before and after the ball was embedded in the block. Initially, only the ball had momentum because the block was at rest. After the collision, the ball and the block moved as a single unit and had the same final velocity. This was supposed to mimic an inelastic or a plastic collision. Then, we applied the conservation of energy equation to the system. We took the system's initial position right after the collision as the reference point at which point it did not have any gravitational potential energy. We took the maximum height that the system reached as its final position. At this point, the system was momentarily at rest before swinging back down. Therefore, it only had a final gravitational potential energy and no kinetic energy. We then plugged in the final velocity that we found in the momentum equation as the initial velocity of the energy equation to find the resulting equation shown in Figure 3. This was the ball's initial velocity after being shot out of the spring gun, which we found to be 329.88 cm/s.

Figure 4: Partial derivatives with respect to each of the four variables

After finding an equation for the ball's initial velocity, we proceeded to take the partial derivative of the equation with respect to each of the four variables: M, m, L and θ. This process is shown in Figure 4.

Figure 5: Partial derivative with respect to M and m after plugging in values

Figure 6: Partial derivatives with respect to L and θ after plugging in values

Then, we plugged in the measured values to each of the four resulting equations (Figures 5 and 6). One thing to note is that we converted the mass measurements into kilograms and the length measurements into meters to be consistent in our calculations. Alternatively, we could have just kept the original units and converted g (the acceleration due to gravity) into cm/s² (we could have kept the masses in grams either way because they ended up canceling out). 

Figure 7: Propagated uncertainty

Finally, we multiplied the absolute values of the resulting values to the uncertainties of the corresponding measurements, as shown in Figure 7, to find the propagated uncertainty of the ball's velocity. The resulting value was in meters, which we converted back to centimeters since the ball's velocity was in centimeters as well. As it can be observed from the image, we found the propagated uncertainty to be 5.07 cm/s. We can conclude from this result that the error was not too large since the uncertainty is much smaller than the calculated velocity, which was 329.88 cm/s. 

CONCLUSION

In this experiment, we implemented our knowledge on the conservation of momentum and energy in order to find the initial velocity of an object. It was good practice in applying theoretical knowledge in a real-life situation. We exercised our ability to find the propagated uncertainty of a derived value. Practicing the steps in order to accomplish this allows us to fully incorporate the process in our minds.

A source of error could have come from the fact that we ignored the tension in the strings when we applied the conservation of energy equation. This most likely contributed to the ball's calculated initial velocity being smaller than it actually was. Another source of error could have came from the apparatus used the measure the angle. There was probably a little bit of friction at the pivot of the marker. This could have also led to a smaller calculated value versus the actual value.

22-Apr-2015: Collisions in two dimensions

PURPOSE

The purpose of this experiment was to analyze two dimensional collisions to see if momentum and kinetic energy are conserved.

PROCEDURES

Figure 1: Set-up of experiment

Figure 2: Balls used to simulate collisions

This experiment was conducted with the set-up displayed in Figure 1. The camera (circled in red) was used to capture the motion of two metal ball before and after a collision on a glass surface (circled in blue). As shown in Figure 2, two of the balls that we used in this experiment were the same size and mass, while the third was smaller and lighter. In the first part of the experiment, we used the two balls that were equal in mass. We placed one of them near the center of the surface and rolled the other towards it to cause a collision.

Figure 3: Graphs of position vs time

After capturing the motion of the balls with the camera, we used video analysis on Logger Pro to
track their positions in the x- and y-directions throughout the entire process. Then, we plotted their positions with respect to time. The location marked by the black circle indicates when the collision occurred. Therefore, any graphs before the black circle refer to the initial velocities, while the the graphs after the black circle correspond to the final velocities. The red graph in Figure 3 correspond to the velocity of the rolled ball in the x-direction, while the blue graph correspond to its velocity in the y-direction. The green and brown graphs correspond to the stationary ball's velocity in the x- and y-directions, respectively (notice that the green and brown graphs begin after the collision since the ball was at rest until that event).

Figure 4: Calculations of initial and final momentum

Figure 5: Calculations of initial and final kinetic energy

After finding the velocities, we calculated the momentum of the system before and after the collision in both the x- and y-directions (Figure 4). As it can be seen from the figure, the initial momentum of the stationary ball was zero in both cases since it did not have any velocity. When we completed the calculations, we compared the values on both sides to see if they were equal. The percent error in the x-direction was only 5.97 percent, while the percent error in the y-direction was 35.97 percent. Although the percent error was technically extremely high in the y-direction, we can assume that momentum was mostly conserved since the momentum values in the y-direction were much smaller than the values in the x-direction. The reasons why these values were not a hundred percent accurate will be discussed later in the conclusion.

In addition to seeing if the momentum was conserved, we also saw if the kinetic energy was conserved (Figure 5). In this case, we did not have to break the kinetic energy values into components because energies are scalars. This time the percent error was much higher at 25.25 percent. However, this is to be expected because some energy was most likely lost in the form of heat and sound when the balls collided.

Figure 6

For the second part of the lab, we rolled the less massive ball at a larger, stationary ball. We repeated the process from the first part of the lab and used video analysis to track the motion of the balls. We once again graphed their positions in the xy-plane with respect to time and applied "Linear Fit" on each plot of points to find the velocities. This graph is shown above in Figure 6. The moment at which the balls collided is marked with the black circle.

Figure 7

Figure 8

Using the initial and final velocities we found by analyzing the graph in Figure 6, we were able to find the initial and final momentum of the system in the x- and y-directions. We were also able to find the initial and final kinetic energy. The mathematical process is shown above in Figures 7 and 8. This time the percent error in the conservation of momentum equation in the x-direction was a bit higher at 12.28 percent. This may be due to the friction between the stationary ball and the surface, which was probably more significant than Part 1 of the experiment because the smaller ball most likely did not exert as much force on the stationary ball as the bigger ball of Part 1 did. In addition, the percent error in the y-direction was 43.72 percent. As mentioned before, this error can be ignored because the momentum in the y-direction was very small compared to the momentum in the x-direction. Moreover, the percent error for the kinetic energy values was 38.49 percent. Again, this is to be expected because some of the initial kinetic energy probably dissipated during the collision in the form of heat and other forms of non-conservative work.

CONCLUSION

From this experiment, we learned firsthand that momentum is conserved in two dimensional collisions, while kinetic energy is not. This is because some of the energy is lost in the form of heat and sound. In addition, we gained valuable experience in using video analysis to track the motion of objects in two dimensions. This will be useful skill that we can employ in the future.

Some error could have come from the video analysis since it was difficult to accurately track the motion of the balls. Therefore, the velocity values were most likely a bit off. Moreover, we did account for the fact that there were some friction between the balls and the surface that they were rolling on. This could have also contributed to the error.