Tuesday, May 26, 2015

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.

Posted by at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)