1-Jun-2015/3-Jun-2015: Simple harmonic motion

PURPOSE

The purpose of this experiment was to derive equations that modeled objects in simple harmonic motion. This allowed us to find the theoretical values for their periods, which we compared to the values we found experimentally.

PROCEDURES

Period of a Ring in Simple Harmonic Motion

Figure 1

Figure 2

For the first part of the experiment, we attempted to derive an equation for the period of a ring in simple harmonic motion. The first step we took in order to accomplish this task was set up the torque equation as shown in Figure 2 by using the diagram displayed in Figure 1. Using the moment of inertia of a ring rotating about its center of mass [I = 1/2 M(R² + r²), where R = 0.0696 m, r = 0.0579 m, and M does not matter because it ends up canceling out], we were able to find the moment of inertia of the ring when it was rotating about a pivot that was a distance Ravg = 0.06375 m (the average of the inner and outer radii of the ring) away from the ring's center of mass by using the parallel axis theorem (I = Icm + Md²). Next, we solved the equation for α and assumed that sinθ was approximately equal to θ for very small values of θ. We took these steps in order to put the equation in the general form of simple harmonic motion (α = -ω²θ). From the resulting equation, we were able to deduce that ω was equal to [(2gRavg)/(R² + r² + 2Ravg²)]^1/2 = 8.753 rad/s, which we plugged into T = 2π/ω to find the period. We found this value to be 0.7178 s,

Figure 3

Following the calculations, we set up the experiment as shown in Figure 3. We pulled the ring back at an angle and released it to let it oscillate. We captured the ring's period using a pre-programmed file in Logger Pro, which we found to be 0.72 s. This gave us a percent error of 0.306 percent, which is pretty much as accurate as an experiment can be. This was probably due to the fact that there wasn't much room for error since the only force affecting the ring's motion was gravity. The only factor that could have contributed to error in the results was the sway of the ring when we first released it from rest. However, this did not have a significant effect on the results because the ring stabilized itself after a few oscillations.

Period of a Triangle in Simple Harmonic Motion

Figure 4: Derivation of the moment of inertia's

We began the second part of the experiment by deriving the moment of inertia's of a triangle with the axis of rotation at its apex and a triangle with its axis of rotation at the center of its base. First, we found its moment of inertia with respect to its apex as shown in the top portion of Figure 4. We designated dM to be a thin bar with its center of mass a distance y away from the axis of rotation. Therefore, in order to find the moment of inertia of dM, we had to apply the parallel axis theorem (with d = y) on the moment of inertia equation of a thin bar rotating about its center of mass (I = 1/12 ML², where L = 2x in this case). We were able to find the moment of inertia of the entire triangle by integrating the resulting equation from 0 to H. We found this value to be I = 1/24 MB² + 1/2 MH². Then, we used the same dM to find the center of mass of the triangle. We did this by using the equation ycm = (∫dM*y)/(∫dM). We found the center of mass to be 2/3 H from the triangle's apex. From this value, we were able to apply the parallel axis theorem again (with d = 2/3 H) to find moment of inertia of the triangle with respect to its center of mass. We did this by turning the equation from I = Icm + md² into Icm = I - md². From this step, we were able to find the moment of inertia of a triangle rotating about its center of mass was I = 1/24 MB² + 1/18 MH². Then, using d = 1/3 H, we implemented the theorem a third time to find the moment of inertia of the triangle rotating about the center of its base. The resulting equation was I = 1/24 MB² + 1/6 MH².

Figure 5: Derivation of the period of the triangle rotating about its axis

Figure 6: Derivation of the period of the triangle rotating about the center of its base

After finding the moment of inertia of the triangle with respect to the two axes of rotation, we were able to derive equations for the period of the triangle in both orientations (Figures 5 and 6). Just as before, we first set up the torque equation and set everything equal to α. In addition, we assumed sinθ to be approximately equal to θ for very small values of θ. We did this to get the equation in the general form of systems in simple harmonic motion (α = -ω²θ). From the resulting equation, we concluded that ω was equal to the square root of (16gH)/(B² + 12H²) for the triangle rotating about its apex and the square root of (8gH)/(B² + 4H²) for the triangle rotating about the center of its base. Then, we found the period by dividing 2π by these values. The resulting equations were 2π[(B² + 12H²)/(16gH)]^1/2 and 2π[(B² + 4H²)/(8gH)]^1/2, respectively.

Figure 7: Theoretical values of the periods

After deriving the equation, we plugged in the appropriate values (B = 0.14 m, H = 0.147 m, and g = 9.81 m/s²) to find the theoretical periods. For the triangle rotating about its apex, we found the period to be 0.69081 s. On the other hand, for the triangle rotating about its base, we found the period to be 0.60238 s. The mathematical processes are displayed in Figure 7.

Figure 8: Rotating about its apex

Figure 9: Rotating about the center of its base

Following our calculations, we set up the triangle about its apex and released it from a slight angle to let it oscillate. Its period was measured using a pre-programmed set-up for a pendulum on Logger Pro. As it can be seen from Figure 8, the experimental value for the period was 0.695362 s, which was only 0.659 percent away from the expected value. Then, we set up the triangle about the center of its base and reran the experiment. The resulting data is shown in Figure 9. From Figure 9, it can be seen that the experimental period for the triangle rotating about its base was 0.607886 s. In this case, the percent error between the theoretical and experimental values was 0.551 percent. Once again, there was virtually no error in the results.

CONCLUSION

In this experiment, we found the moment of inertia of objects rotating about different points to set up torque equations. Then, we set everything equal to α to get the equations into the general form of simple harmonic motion (α = -ω²θ). This allowed us to figure out what ω was, which in turn allowed us to find the period of the motion. We then ran the actual experiment and compared the theoretical values to the experimental ones.

The theoretical and experimental values were almost identical. As mentioned before, the reason was because the only force affecting the ring's motion was gravity. Since the force of gravity is constant as long as we're close to the Earth's surface, there was not much room for error. One thing that was concerning at first was the sway of the objects when we first released them and let them oscillate. This ended up not having much of an affect on the results because the objects stabilized after a few oscillations. Another factor that I thought would affect the results was the friction in the pivot. This most likely did not alter the results significantly because the pivot had a smooth outer surface.

20-May-2015: Conservation of energy/conservation of angular momentum

PURPOSE

Figure 1

In this experiment, we attempted to predict a how high above the ground a system would reach after a meter stick rotating about a pivot collides with a piece of clay at rest (Figure 1). In order to do so, we employed a combination of conservation of energy and angular momentum equations.

PROCEDURES

Figure 2

Figure 3

For the first part of our calculations, we found the moment of inertia of the meter stick, which had a mass of 0.08 kg, rotating about an axis that was 0.475 m away from its center of mass (Figure 2). We used the parallel axis theorem (I = Icm + md²) and found this value to be 0.02472 kg*m². Next, we applied the conservation of energy equation to the meter stick rotating from the horizontal position to the vertical position as shown in Figure 3. We set the reference point as the meter stick's center of mass in its final position. Therefore, the meter stick had an initial gravitational potential energy, which eventually all turned into rotational kinetic energy. We set the initial gravitational potential energy (mgh) equal to the final kinetic energy (½Iω²) and solved for the final angular velocity (ω). We found this value to be 5.492 rad/s.

Figure 4

Afterwards, we applied the conservation of angular momentum equation to the meter stick-clay system before and after the collision. For the initial angular momentum, we found the product of the moment of inertia of the meter stick by itself and the angular velocity found earlier with the energy equation (ωo = 5.492 rad/s). On the other hand, the final angular momentum was equal to the product of the moment of inertia of both the meter stick and the clay, and the angular velocity right after the collision (ωf), which was unknown. We found the moment of inertia of the piece of clay by treating it as a particle and applying the equation Ic = mcd², as shown on the bottom of Figure 4. Then, we solved for ωf, which was 2.746 rad/s.

Figure 5

Figure 6

Then, for the final step of our calculations, we set up the conservation of energy equation by using the diagram shown in Figure 5. First, we established the pivot as the reference point and up as the positive y-direction. Then, we used trigonometry to figure out that the vertical distance between the pivot and the final positions of the meter stick and piece of clay were 0.475cosθ and 0.975cosθ, respectively. This resulted in the equation shown on the very top of Figure 6. Plugging in I = Isys = 0.02472 + 0.02472 = 0.04944 kg*m², ω = 2.746 rad/s, m1 = 0.08 kg (mass of meter stick), m2 = 0.026 kg (mass of clay), and g = 9.81 m/s², we were able to eventually solve for cosθ. We found this value to be 0.7001. We took the inverse cosine of this number to find the angle that the system made with the vertical, which we found to be 45.57°. We realized later that finding the angle was unnecessary because the value we really needed was cosθ.

Figure 7

From this angle, we used basic trigonometric relationships to find the vertical distance between the final position of the piece of clay and the ground, as shown in Figure 7. One thing to note is that we treated the piece of clay as a particle as we did throughout the rest of the calculations. Therefore, the dimensions of the clay piece were not considered in the derivations. From Figure 7, it can be seen that the theoretical maximum height of the stick-clay system was 0.2924 m.

Figure 8

Finally, we ran the actual experiment and found out how high above the ground the system actually reached after the collision. We did this by utilizing video analysis in Logger Pro. As it can be seen from Figure 8, We found the height to be 0.2904 m. This gave us a percent difference of 0.6840 percent. This accurate result was due to the fact that gravity was the only significant external force acting on the system throughout the entire process.

CONCLUSION

In this experiment, we broke down a collision into three different stages: motion before the collision, the collision itself, and the motion after the collision. We utilized conservation laws during these stages, which ultimately allowed us to estimate how high above the ground a meter stick and a piece of clay would rotate after a collision occurred between the two objects.

As mentioned earlier, the percent error was very low in this experiment because the only external force that had a considerable impact on the results was gravity. However, another reason the theoretical and experimental values were so close to each other was that we did not factor in the clay piece's dimensions into our calculations. If we had tracked the distance between the end of the meter stick and the ground (which is what our theoretical value actually represents) instead of the center of the clay piece, it would have been 1 to 2 cm smaller. Therefore, the percent error would have been approximately 3 to 7 percent higher. However, this is to be expected because there were other assumptions we made in our calculations, First, we did not account for the friction between the pivot and the meter stick as it was rotating. In addition, there was probably some angular momentum lost during the collision since it was not a perfectly head-on collision. This explains why the experimental value was smaller than the theoretical one.

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.