Finding the moment of inertia of a uniform triangle

Objective:

• Finding the moment of inertia of a uniform right triangle thin plate around its center of mass, for two perpendicular orientation of the triangle.

Physical thinking:

• Approach:
• We need to find the moment of inertia of a right triangle rotates around the center of mass, but we don't approach to solve the problem from the center of mass. Here is why:
• If we rotate around the center of mass, we will miss some piece of the whole right triangle while integrating.
• The limit for integration is not simple. So, we need another approach to solve this problem
• Now, we can approach the problem from the theorem we learn in class which is called parallel axis theorem. Basically, the theorem states that

• Since finding the moment of inertia when rotating around center of mass takes more work and may not work out, we can shift the pivot point to the edge of the triangle. Based on the parallel axis theorem, if we can find the moment of inertia when the pivot point is at the edge and know the distance from the center of mass to the edge, we can figure out the moment of inertia when the pivot point at center of mass.

First, finding the moment of inertia when the triangle rotates around its height (longer leg).

• Choose a small representative disk dm inside the triangle. 
• Write an expression for dm
• Sum all of small disks of the triangle by integrating from 0 to H.
• When we're all done finding the moment of inertia when the pivot point is at the edge, we can apply the parallel axis theorem above to find the moment of inertia when the pivot point is at the center of mass.
• Here is the work.

Figure 1: Deriving an equation for the moment of inertia of the triangle rotating around its height.

Second, finding the moment of inertia when the triangle rotates around its base (shorter leg).

• Doing the same procedure like above, and below is the work.

Figure 2: Deriving an equation for moment of inertia of the triangle rotating around its base.

Now we start an actual experiment to find out the experimental value of moment of inertia. The goal is to compare the value we got from our calculation and from the experiment to see how well we run the experiment.

Set-up:

• The apparatus of the experiment includes the rotation stand, steel and aluminum disk, hanging mass, torque pulley, triangle holder and triangle, and lastly Logger Pro.
• Putting the aluminum disk on top of the steel disk into the rotation stand, make sure that only the top disk rotate this time by letting the air run through the end. 
• The large torque volley is put on top of aluminum disk and connects to the hanging mass. 
• The triangle holder is used instead of a drop pin to hold the triangle.
• Connect the apparatus to the Logger Pro and set up the rotation sensor to read 200 counts per revolution.

Below is the apparatus

Figure 3: Apparatus of the experiment. 

The left one is when the triangle rotates around its long leg (or height)

The right one is when the triangle rotates around its short leg (or base)

• Also using caliper and balance to measure the mass and dimension of triangle, hanging mass, torque pulley and steel and aluminum disk.

Figure 4: Measuring dimensions of components of apparatus.

Procedure:

• First of all, the mass and dimension of components of apparatus are measured. The mass of the triangle is 456g, its height(longer leg) and base(shorter leg) are in turn 14.936cm and 9.844cm. The hanging mass is 25.0g. The diameter of the large pulley is 5.02cm. The mass of aluminum disk is 466g and its diameter is 12.630cm. The mass of the steel disk is 1348g and its diameter is 12.628cm. Lastly, the mass of the triangle holder is 26g.
• We actually do three trials: the first one is without the triangle, the second one is when the triangle rotates around its shorter legs, and the last one is when the triangle rotates around its longer legs. 
• Turn on the air supplier to let the disk rotate, make sure the hose clamp opens so that the bottom and top disk rotate independently. As the disk rotates, the hanging mass starts going down and goes back up again. Logger Pro starts collecting the data since the hanging mass is released. The graph obtained in Logger Pro will be the graph of angular velocity vs. time. By taking the value of the slope, we can find the angular acceleration. However, to minimize the effect of friction torque on the system, we take the average angular acceleration as the mass goes down and back up. Likewise, we in turn find the the average angular acceleration when the triangle is attached. Below is the graph of angular velocity vs. time of three trials.

Figure 5: Graph of angular velocity vs. time when rotating without triangle. 

Figure 6: Graph of angular velocity vs. time when rotating around the shorter leg (base).

Figure 7: Graph of angular velocity vs. time when rotating around the longer leg (height).

Discussion:

• The equation to calculate the experimental moment of inertia is similar to the one we used in angular acceleration lab, so we don't have to derive one more time. Equation (1) is shown below
  

• The average angular acceleration is calculated by summing the angular acceleration when the hanging mass goes down and goes back up, then divide by two. The values are recorded in the table below

Figure 8: Result table of average angular acceleration.

• Using equation 1, we can calculate the experimental moment of inertia of each trial. Below is an example how to calculate it.

• Result table for experimental moment of inertia.

• Then, by subtracting the moment of inertia of the system with the triangle by the system without the triangle, we can find moment of inertia of the triangle for each orientation. Below is an example how to calculate and result table after we're done.

Figure 9: Result table of theoretical moment of inertia of the triangle.

• The experimental value of moment of inertia was found, so we need to calculate the theoretical moment of inertia to compare the result and check how accurate the experiment went. 
• Using the dimensions of the triangle and two equations derived when applying parallel axis theorem above, plugging in the number and find the experimental moment of inertia. Here is the calculation.

Figure 10: Theoretical moment of inertia when triangle rotating around its height.

Figure 11: Theoretical moment of inertia when triangle rotating around its base.

• Last but not least, we calculate the percent error to check whether the result comes out like what we expected (within 10%). If so, we performed our experiment well and two equation we derived are correct. Here is the formula we will use to calculate the percent error (formula (*))

• Apply the formula (*) to calculate the percent error when triangle rotating around its long leg.

• Apply the formula (*) to calculate the percent error when triangle rotating around its short leg.

• As you can see, our percent errors for both orientations are fairly small and within 10%. This means that we run the experimental very well and two equations we derived are correct.

Some source of errors may happen:

1. Uncertainty in measurement of dimensions of components of the apparatus.
2. Uncertainty in measurement of angular displacement due to the rotational sensor.
3. Friction may affect the angular acceleration even though it is already minimized by taking the average angular acceleration.
4. We may let too much air run through the system than we actually need.

Conclusion:

• In this lab, we perform an experiment to find out the moment of inertia of the triangle about its center of mass rotating both orientation: one is around its height, and the other one is around its base. 
• We first successful come up with two equations to calculate the theoretical values of moment of inertia. 
• When the triangle rotates around its base, we find: 

• When the triangle rotates around its height, we find:

• The percent error of this experiment comes out very well. The percent error of the triangle when rotating around its base is 2.32% and that of the triangle when rotating around its height is 0.76%. Both are small and within 10% regardless of errors may happen during the experiment. 

Moment of Inertia and Frictional Torque

Objective:

• Determining the moment of inertia of a rotating disk and angular deceleration under the effect of friction torque.
• Calculating the time the cart takes to go down an inclined plane.

Physical thinking and derivation:

• We are asked to find the time the cart needs to go down 1m on an inclined plane theoretically, the most reasonable approach for this problem is to use torque clockwise is equal torque counterclockwise, force applied vertically up equals force applied vertically down, and force applied horizontally left equals force applied horizontally right. 
• In order to use those three equations above, we need to determine the moment of inertia of the rotating disk. 
• The rotating disk includes three parts with a large metal disk in the center, and two small shafts on each side. The large metal disk is cylindrical shaper, so do these two shafts on the side, but they are not symmetrical. An equation to calculate moment of inertia of the cylinder is half of the product of mass and radius squared. (Equation 1)

• Therefore, in order to calculate moment of inertia, we need to measure the radius of each disk and figure out their masses. We use a big caliper to measure the diameter of the large disk, and use small caliper to measure the diameters of two shafts on the side.
• Since three disks are connected to each other as a system, we can't actually measure the mass of each one respectively. The approach to solve the mass of each disk is to find the percent volume of each one, and multiply it with the total mass of the system. The volume of each one can be calculated using an equation below.

• Since we find the mass of a large disk and two small shafts on each side, we can use equation (1) to find the moment of inertia of each one, then add them up to get the moment of inertia of the whole system.
• Below is our calculation

Figure 1: Recording dimensions and calculating the percent volume of the large disk and two shafts.

Figure 2: Calculating the mass, moment of inertia of each disk, then find the moment of inertia of the whole system.

Calculating the angular deceleration due to the friction torque:

• Before we actually do the experiment with the cart, we run a first trial without the cart to figure out how we will calculate the angular acceleration. The reasons we need angular acceleration of rotating disk itself because we take the friction torque into account. Friction makes rotating disk slow down until it stops. Also, the friction torque is product of moment of inertia and angular acceleration. Therefore, we need to figure out angular acceleration in order to find friction torque (moment of inertia is already calculated before).
• From the 2-D collision lab, we know that capturing and analyzing video of two objects collide helps us obtain the value of linear position and velocity. We will use the same idea for this experiment. We will capture and analyze the video of the rotating disk rotates and stops by itself. The only force applied to rotating disk to make it stop is friction. Thus, we can figure out the angular acceleration of friction torque.
• Since analyzing the video, we will get the linear position and velocity. From graph of linear velocity vs. time, we can obtain the value of linear acceleration. Linear acceleration is related to angular acceleration through the following equation.

• To get the value of angular acceleration, we divide linear acceleration by the radius of the rotating disk.

Setting up the apparatus to obtain the angular acceleration.

• Let the rotating disk sit on the table, make sure that the table is leveled.
• Setting up the camera such that the video capturer parallels to the center of the rotating disk. Adjust the camera setting until we are satisfied with the quality. Connect the camera to the Logger Pro.
• We also mark a tape on the rotating disk to make it easy for analyzing it later.
• Give the rotating disk a gentle spin and let it go until it stops. Start capturing the video and analyze it later.

Below is the apparatus:

Figure 3: The apparatus of the experiment.

Data collection and analysis.

• Since we have the video, we analyze it by dotting the tape until it stops, setting scale and origin. The scale is the radius of the large disk, which is 0.09818m.

• Every dot on the disk gives us a set of values in Logger Pro, and here is the data table of linear position and velocity.

Figure 4: Data table of linear position and velocity

• Since the linear velocity is breaking down into two components x and y, we need to find the resulting velocity by taking square root of x-componet velocity squared and y-component velocity squared.

• Then graphing the resultant velocity vs. time, and linearizing the graph to obtain the linear acceleration.

Figure 5: Linearizing the graph of linear velocity vs. time.

• Finally, divide the linear acceleration by the radius of rotating disk to get the value of angular acceleration. Here is the work.

• Wow! Now we can use that value to calculate the friction torque which is our main goal from the beginning.

Actually lab:

• The main objective of this lab is to compare the theoretical and experiment value of time needed for a cart to roll down an inclined track for a distance of 1 meter. 
• We can obtain the experimental value of time by using the stopwatch on our phone. For theoretical value, we will obtain through our calculation.

Calculating the experimental value of time.

• To find the experimental value of time, we apply kinematic equation since the acceleration is constant.

• As the cart starts at rest, we don't worry about initial velocity. Therefore, all we need is to find the linear acceleration. The most reasonable approach is to acceleration is through the second Newton Law, which is F = m*a
• We know the mass of the cart, thus we just need to analyze those forces acting on the cart. There are three major forces: the gravity, the normal force, and tension. Since the cart is on the inclined plane, the gravity is broken down into two component x and y. Adding those forces in the direction of the cart (moving down the inclined plane), we have:

• We also notice that the tension is still unknown. Therefore, we will apply the definition of torque to figure out the tension, then use that value along with other known values to calculate the acceleration. 
• Substituting the value of acceleration back into equation (*) to find the theoretical value of time.
• Calculation shown in the figure below.

Figure 6: Finding the experimental value of time.

t=7.91s

Setting up an actual apparatus of the lab.

• Let an inclined track rest at the edge of the table. The inclined track makes with the floor a certain angle needed to measure.
• Connect a 500-gram dynamics cart to the rotating disk through a string. Make sure that the string is parallel to the ramp. 
• When everything is set up, let the cart roll down a distance of 1 meter and uses a stopwatch to determine the actual time. 

Figure 7: The apparatus of the experiment.

Discussion:

• From calculation, the time the cart will take to roll down a distance of 1 meter is 7.91m/s. For the actual time we obtain from experiment is 8.09 m/s. The percent error is calculated below. 

• We can calculate the uncertainty of this experiment, then compare it to the percent error to see how well we run the experiment. For the angle, we can check the uncertainty if we are off 1 degree. We have sin(49.6) = 0.762 and sin(50.6) = 0.773. The uncertainty of the angle is equal ((0.773 - 0.762) / 0.773 * 100%) = 1.4%. While the radius of the small disk number 1 give us the uncertainty (0.01 / 3.14 * 100%) = 0.3%. Additional, the uncertainty in moment of inertia is equal the sum of uncertainties from radii (0.3% + 0.3% + 0.05% = 0.65%. Thus, the total uncertainty is 1.4% + 0.3% + 0.65% = 2.35%. Moreover, we can add some more uncertainty from dotting the points when analyzing the video or timing the actual time when the cart rolls down a distance of 1 meter. This refers that the actual uncertainty of this experiment is greater than 2.35%. However, our percent error already falls within this region, which means we runt the experiment pretty well. 

Conclusion:

• The main goal of this lab is to compare the theoretical and experimental value of time when the cart rolls down a distance of 1 meter. However, we break down the lab into three small sessions:
• First session: we calculate the mass of each disk in order to calculate the total inertia of rotating disk.
• Second session: we figure out the angular acceleration of rotating disk by analyzing the video when the rotating disk rotates and stops by itself (of course under the effect of friction)
• Third session: we calculate the theoretical value of time, obtain the experimental value from an actual experiment. Compare two values to see how well we do the experiment.
• For the first session, the inertia of rotating disk is equal 0.019986 kg*m^2
• For the second session, the angular acceleration is equal -0.9048 rad/s^2
• For the last session, the percent error is 2.2%.
• There are some sort of uncertainties in this experiment:
• Uncertainty from the angle measured
• Uncertainty from the radii measured.
• Uncertainty from timing the time the cart takes to roll down a distance of 1 meter.
• Uncertainty in dotting points when analyzing the video to figure out angular acceleration.

Angular acceleration: Part 1 and Part 2

Objective:
Part 1:

• Examining the effects of various changes in the experiment on the angular acceleration of the system
• The change in hanging mass
• The change in radius 
• The change in rotating mass

Set-up:

• What we need in this experiment is a device that help us figure out the relationship between torque and angular acceleration. 
• This device includes two steel disks, one aluminum disk, small and big torque pulley, hanging mass, drop pin, string, hose clamp, and of course, Logger Pro

The apparatus of experiment is shown below

Figure 1: The apparatus of the experiment.

Procedure:

• First, we measure the diameter and mass of those following things:
• Top steel disk
• Bottom steel disk
• Top aluminum disk
• Smaller torque pulley
• Larger torque pulley
• Hanging mass
• Using a small caliper to measure the diameter of small and large pulley, using a big caliper to measure the diameter of steel and aluminum disk, using a small balance to measure the hanging mass and big balance to measure the mass of the steel and aluminum disk (shown below)

 Figure 2: From left to right: device (large balance) measuring the mass of steel and aluminum disk, device (small balance) measuring the mass of pulley, and device (caliper) measuring the diameter of disk and pulley.

• Plug the power supply into the Pasco rotational sensor. Connect the cable with yellow paint to Logger Pro at Dig/ Sonic 1. Run a trial to make sure that the Logger Pro will read the top disk as it rotates.
• Since there is no defined sensor for this rotational apparatus, we will create something that works for this experiment. We will choose Rotary Motion, and in Sensor Setting, we set 200 counts per rotation. As a result, we will be able to get the graph of angular position, velocity, and acceleration vs. time as these disks rotate. 
• Also, making sure that the hose clamp at the bottom open so that the bottom disk will rotate independently of the top disk when we put the drop pin in.
• Turn on the air supplier to pull the air into the system, however, make sure that the air isn't so much. Otherwise, the experiment will be erratic. 
• Wrap a string around the torque pulley and let the hanging mass be held at the highest point, then start to release it. By linearizing the graph of angular velocity vs. time, we can find the value of angular acceleration.
• We will test the effects on the angular acceleration of the system of increasing the hanging mass, the radius of the torque pulley, and the rotating mass.

Physical thinking:

• In this experiment, there is a small part of friction that we may take into account because it may affect our result. When we release our hanging mass, it starts to go down and go back up until it reaches to the highest point as before. Therefore, we will break it down into two cases and examine the effect of friction
• Case 1: When the hanging mass going down
• The disk speeds up, thus, the net torque is calculated like below

• The net torque thus is less than the ideal one, then it in turn leads to the smaller value of angular acceleration
• Case 2: When the hanging mass going up
• The disk slows down, thus, the net torque is calculated like below

• The net torque is greater than the ideal one, then it in turn leads to the bigger value of angular acceleration
• To avoid more or less than the actual value of angular acceleration, we will get the sum of absolute value of both cases, then divide the sum by two, which is the average of both cases. 

• Therefore, we will linearize graph of angular velocity vs. time when the hanging mass goes down and goes back up again to find angular acceleration in order to find the average.

Data Collection and Analysis:

• Data table after measurement of dimensions of components of apparatus.

Figure 3: Recording dimensions of components of apparatus.

Experiment 1, 2, and 3: Effects of changing the hanging mass

• The first three experiments will dedicate to figuring out the effect of changing the hanging mass on the angular acceleration. In other words, everything except hanging mass will keep the same, and we keep adding around 25g to the hanging mass throughout these three experiments. 
• As we release the hanging mass and collect the graph of angular velocity vs. time, below is what we got.

Figure 4: Graph of angular velocity vs. time for the first three experiments

• We then linearize the graph and obtain the value of angular acceleration when the hanging mass goes down for three trials.

Figure 5:  Linearizing graph of angular velocity vs. time when the hanging mass goes down.

Angular acceleration (experiment 1) = 0.5892 rad/s²

Angular acceleration (experiment 2) = 1.081 rad/s²

Angular acceleration (experiment 3) = 1.826 rad/s²

• We do the same thing when the hanging mass goes back up for three trials. 

Figure 6: Linearizing the graph of angular velocity vs. time when the hanging mass goes up.

Angular acceleration (experiment 1) = 0.6581 rad/s²

Angular acceleration (experiment 2) = 1.171 rad/s²

Angular acceleration (experiment 3) = 1.941 rad/s²

• Calculating the average angular acceleration.
• Experiment 1: Average = (0.5892 + 0.6581) / 2 = 0.6234 rad/s²
• Experiment 2: Average = (1.081 + 1.171) / 2 = 1.126 rad/s²
• Experiment 3: Average = (1.826 + 1.941) / 2 = 1.884 rad/s² 

Experiment 1 and 4: Effects of changing the radius and which the hanging mass exerts a torque

• In this experiment, we will keep the same apparatus except changing the torque pulley. One we will do with small torque pulley, and the other we will do with the large pulley.
• Releasing the original hanging mass and obtain graphs of angular velocity vs. time for both experiments. Then, linearizing them to find the angular acceleration.

Figure 7:  Linearizing graph of angular velocity vs. time when hanging mass goes down

Angular acceleration (experiment 1) = 0.5892 rad/s²

Angular acceleration (experiment 4) = 1.176 rad/s²

Figure 8: Linearizing the angular velocity vs. time when the hanging mass goes up.

Angular acceleration (experiment 1) = 0.6581 rad/s²

Angular acceleration (experiment 4) = 1.265 rad/s²

• Finding the average angular acceleration 
• Experiment 1: Average = (0.5892 + 0.6581) / 2 = 0.6234 rad/s²
• Experiment 4: Average = (1.176 + 1.265) / 2 = 1.220 rad/s²

Experiment 4, 5, and 6: Effects of changing the rotating mass

• We keep the same original apparatus, but we will change the disk for each experiment. The first one we rotate with only the top steel disk, the second one we rotate with the top aluminum, and the last one is the top and bottom steel disks.
• Releasing the hanging mass and obtain graphs of angular velocity vs. time as before.
• Analyzing the relationship between angular acceleration and the rotating mass.

Figure 9: Linearizing the graph of angular velocity vs. time when hanging mass goes down.

Angular acceleration (experiment 4) = 1.177 rad/s²

Angular acceleration (experiment 5) = 3.283 rad/s²

Angular acceleration (experiment 6) = 0.5871 rad/s²

Figure 10: Linearizing the graph of angular velocity vs. time when hanging mass goes up.

Angular acceleration (experiment 4) = 1.265 rad/s²

Angular acceleration (experiment 5) = 3.581 rad/s²

Angular acceleration (experiment 6) = 0.6376 rad/s²

• Calculating the average angular acceleration
• Experiment 4: Average = (1.176 + 1.265) / 2 = 1.220 rad/s²
• Experiment 5: Average = (3.283 + 3.581) / 2 = 3.426 rad/s²
• Experiment 6: Average = (0.5871 + 0.6376) / 2 = 0.6134 rad/s²

Discussion:

• Looking at the first three experiments 1, 2, and 3, when the hanging mass is almost doubled from experiment 1 (0.0247 kg) to experiment 2 (0.0447 kg), the average angular acceleration is almost doubled (increasing from 0.6234 rad/s² to 1.126 rad/s² . When the hanging mass is about tripled from experiment 1 (0.0247 kg) to experiment 3 (0.0747 kg), the average angular acceleration is approximately tripled (increasing from 0.6234 rad/s² to 1.884 rad/s² . Therefore, we can conclude that the angular acceleration is directly proportional to the change in hanging mass.
• Looking at experiment 1 and 4, when the radius of torque pulley is approximately doubled (increasing from 0.0126m to 0.0251m), the average angular acceleration is doubled as well (increasing from 0.6234 rad/s² to 1.220 rad/s²). Therefore, we can conclude that the angular acceleration is directly proportional to the change in radius of torque pulley.
• Looking at experiment 4, 5, and 6, when the rotating mass is nearly doubled from experiment 4 (increasing from 1.348kg to 2.704kg) to experiment 6, the average angular acceleration decreases by half (decreasing from 1.220 rad/s² to 0.6134 rad/s² ). When the rotating mass is decreasing by about three times from experiment 4 (1.348kg) to experiment 5 (0.466kg), the average angular acceleration increases by almost three times. Therefore, we can conclude that the rotating mass has an inverse relationship with the angular acceleration.

Extra: 

Objective:

• Verify the relationship between linear velocity and angular velocity (omega).

Set-up:

• We use the same apparatus from previous experiment; however, in order to record the linear velocity, we will set up a motion sensor on the ground such that as the hanging mass goes up and down, the motion sensor can help us record the linear velocity.
• We will use the large pulley and original hanging mass for this experiment. 
• diameter of large pulley = 5.02cm
• Hanging mass = 24.7g

Physical thinking:

• From lecture in class and notes from physics books, we know that linear velocity is the product of the radius of the object rotating and the angular velocity. 

• The approach is to find the ratio between linear velocity and angular velocity, if the ratio is approximately equal the radius of the object rotating, which means that the relationship between both of them is verified.

Data Collection and Analysis:

• Following the same procedure explained above, we obtain the graph of angular position, velocity, and acceleration vs. time. Additionally, since we set up a motion sensor to measure linear velocity, we also obtain graph of linear velocity vs. time. However, we can set up Logger Pro to show us the graph of angular and linear velocity in order to find out their relationship.

Figure 11: Graph of linear and angular acceleration.

• Setting up a New Calculated Column to evaluate the ratio of linear velocity and angular velocity.

Figure 12: Setting up an equation in Logger Pro to evaluate ratio of angular and linear velocity.

• We then graph ratio vs. time, and examine it in Logger Pro. Here is our result.

• As you can see, the first mean value of ratio is 0.02543m (the negative sign just means that the hanging mass goes down). The mean value of ratio when the hanging mass goes up is 0.02529m. Both values are very close to the actual radius of the large pulley which is 0.0251m. Thus, we can conclude that linear velocity is the product of the radius of the object rotating and the angular velocity. Wow! Our main goal is proved.

Part 2: 

Objective:

• Comparing the experimental and theoretical value of moment of inertia of sis experiments to see how well the experiment goes, and also determine how far our experimental value is off from theoretical value.

Physical principles:

• Since, the apparatus of all six experiments are disks, we can find the theoretical moment of inertia by taking half of product of the mass and the radius of object rotating squared.

• For experimental value of moment of inertia, we consider the friction torque applied to the system, which makes the angular acceleration of the system when the hanging mass is descending is not the same as when it is ascending. Therefore, to minimize the effect of friction torque, we take the average value of angular acceleration. From that, we can calculate the moment of inertia using this following equation:

Calculations:

• Calculating the theoretical and experimental value of moment of inertia and percent error.

Figure 13: Calculation shown for experiment 1

Figure 14: Calculation shown for experiment 2

Figure 15: Calculation shown for experiment 3.

Figure 16: Calculation shown for experiment 4.

Figure 17: Calculation shown for experiment 5.

Figure 18: Calculation shown for experiment 6

Discussion:

• Result table for percent errors

Figure 19:  Percent error for all six experiments

• All percent errors from six experiments are within 5%, which is in the reasonable range for percent error. The percent error changes through six experiments because the uncertainty isn't consistent. Some experiments involve more uncertainty than the others. For example, the experiment 6, we need to measure the dimension of hanging mass, large torque pulley, top steel and bottom steel, while in experiment 4, we only need to measure the dimension of hanging mass, large torque pulley, and top steel. Therefore, there are more uncertainties in experiment 6 than these of experiment 4. 
• Friction also affects the result. The inconsistency of friction leads to the varying percent errors of six experiments. 

Some more sources of uncertainty/error in this experiment:

• Uncertainty in the measurement of dimension of disk, torque pulley, and hanging mass.
• Friction may cause our result to become erratic.
• Poor timing resolution of the sensors may affect the graph of angular position, angular velocity, and angular acceleration.
• Uncertainty in the measurement of the angular velocity of the rotating mass by the rotational sensor
• Uncertainty in the measurement of the linear velocity of the hanging mass by the motion censor.

Conclusion:

• In this experiment, we divided the lab into two parts:
• Part 1: Examining what factors affect angular acceleration and also verifying the relationship between linear and angular velocity.
• Part 2: Comparing the theoretical and experimental value of moment of inertia
• In part 1: We have found that the changing in the hanging mass, in radius of pulley, and in rotating mass have an impact on the angular acceleration.
• The change in hanging mass is directly proportional to the angular acceleration
• The change in radius of torque pulley is directly proportional to the angular acceleration.
• The change in rotating mass is inversely proportional to the angular acceleration
• The relationship between linear velocity and angular velocity is verified, which is linear velocity is the product of the radius of object rotating and angular velocity.
• In part 2: We compare the value of theoretical and experimental value of moment of inertia. The percent error of all six experiments are within 5%, which indicates that our experimental values are not far off from theoretical values. The errors involved are explained by uncertainties in measurements and also due to friction. To avoid that kind of error, we need better equipments.