Lab#16: Angular Acceleration

Angular Acceleration

Samuel
Ellis
Mia
11-2-2016

Purpose:

To use our knowledge about torque and apply it on the rotating object to measure the angular acceleration and its inertia and how the radius affect torque.

Theory:

There is a direct relationship between radius and apply force. We substitute different radius and forces to find their downward and upward speed, and compare it with radius, mass and acceleration. According to our formula, the final answer should come out exactly or somewhere similar.  

Procedure:

1. Measure each of the following to at least three significant figure

    - diameter and mass of top steel disk

    - diameter and mass of bottom steel disk

    - diameter and mass of top aluminum disk

    - diameter and mass smaller torque pulley

    - diameter and mass of larger torque pulley 

    - mass of hanging mass supplied with apparatus

2. Plug the power supply into the Pasco rotational sensor. If there is a cable with the yellow paint or tape, connect only that cable to the Lab Pro at Dig/Sonic 1, so the computer is reading the top disk. If there cables are the same, connect them both. You will need to discern which is measuring the top disk and ignore the other sensor. 

3. Set up the computer. Open LoggerPro. There is no defined sensor for this rotational apparatus so we will need to create something that works with this equipment. Choose Rotary Motion. There are 200 marks on your top disk, so you need to set up equation in the Sensor setting 200 counts per rotation. When you collect data, you can see graphs of angular position, angular velocity and angular acceleration vs. time. The graph of angular acceleration vs. time is useless due to poor timing resolution of the sensors. 

4. Make sure the hose clamp on the bottom is open so that the bottom disk will rotate independently of the top disk when the drop pin is in place. 

5. Turn on the compressed air so that the disks can rotate separately. You will not so much air that you pop the hose form the air source, but enough to keep things smooth. Set the disks spinning freely to test the equipment. 

6. With the string wrapped around the torque pulley and the hanging mass its highest point, start the measurements and release the mass. Use the graphs of angular velocity to measure the angular acceleration as the mass moves down and up. 

(Measure hanging string does not touch the edge of table and make sure clean disk with alcohol before started experiment) 

Data:

hanging mass- measure by electronic scale

α(down, up) - measured by LoggerPro 

α_average - measured by me

EXPT 1 -- hanging mass only - 24.9 grams, small torque pulley, Top steel, α(down, up)= 1.098,1.205(rad/sec^2), α_average= 1.151

EXPT 2-- 2 x hanging mass - 49.9 grans, small torque pulley, Top steel, α(down, up) = 2.206, 2.388(rad/sec^2), α_average = 2.297

EXPT 3 -- 3 x hanging mass - 74.9 grams, small torque pulley, Top steel, α(down, up) = 3.315,3.557(rad/sec^2), α_average = 6.872

EXPT 4 -- hanging mass only - 24.9 grams, large torque pulley, Top steel, α(down, up) = 2.125,2.343(rad/sec^2), α_average = 2.233

EXPT 5 -- hanging mass only - 24.9 grams, large torque pulley, Top aluminum, α(down,up) = 5.927,6.638(rad/sec^2), α_average = 6.2825

EXPT 6 -- hanging mass only - 24.9 grams, large torque pulley, Top steel + bottom steel, α(down, up) = 1.069,1.777(rad/sec^2), α_average = 2.246

Measured Data:

Top steel disk: 126.3 cm, 1353.3 grams

Bottom Steel disk: 126.3 cm, 1345.8 grams

Top aluminum disk : 126.3 cm, 465.8 grams

Small torque pulley: 27.8 cm, 10.0 grams

Large torque pulley: 53.2 cm, 36.3 grams

Graph and Calculated Data:

(Above, is the Angle vs Time(top graph) and Velocity vs Time(bottom graph). For Angle vs Time, you can see that as time increase, the angle of the disk is also increase. In addition, instead of increasing linearly, the line is more a curve like line. The reason behind that is because the velocity is changing constantly, since the disk is constantly changing its rotation and hanging mass gives acceleration to the disk. Therefore, it create a curve like graph. For Velocity vs Time graph, the disk is changing its direction, that's how it get those up and down line.)

Analysis:

According to the result from our data, there should have some kind of relationship from different hanging mass, rotating mass and radius. 

EXPTS. 1, 2, and 3: Effect of changing the hanging mass

The difference between 1, 2, and 3 is that the hanging mass is increase by 20 grams. The resultant angular acceleration is quite different. 

EXPT 1 = 1.152 rad/s^2, mass = 24.9 grams

EXPT 2 = 2.297 rad/s^2, mass = 49.9 grams

EXPT 3 = 3.436 rad/s^2, mass = 74.9 grams

Result, angular acceleration increase as hanging mass increase.

EXPT 1 and 4: Effect of changing the radius and which the hanging mass exerts a torque

With same amount of hanging mass and disk, we used different torque pulley. 

EXPT 1 (small torque pulley)= 1.152 rad/s^2, mass = 24.9 grams

EXPT 4 (large torque pulley) = 2.233 rad/s^2, mass = 24.9 grams

EXPT 4, 5, 6: Effect of changing the rotating mass

With same amount of hanging mass and same length of torque pulley, we changed the rotating mass(same radius but different mass)

Steel disk: 1353.3 grams

Aluminum disk: 465.8 grams

EXPT 4 = Top steel, 2.233 rad/s^2

EXPT 5 = Top aluminum, 6.2825 rad/s^2

EXPT 6 = Top steel + bottom steel, 1.123 rad/s^2

Part 2 

We compare theoretical value to experimental value ( it should come out close or similar value) by using I_disk = (mgr)/alpha - mr^2. When using this equation, we assumed there's no friction between two disk and string on pulley. 

Experimental value = (mgr)/alpha - mr^2

Theoretical value = 1/2* mr^2

EXPT 1

Experimental value = 0.00294 --> ((0.0139*9.8*0.0249)/1.1515) - 0.0139^2 * 0.0249

Theoretical value = 0.00269 -->  1/2 * 1.353*(0.1263/2)^2

EXPT 2

Experimental value = 0.00295 --> (0.0139*9.8*0.0499)/2.297 - 00139^2 * 0.0499

Theoretical value = 0.00269 -->  1/2 * 1.353*(0.1263/2)^2

EXPT 3
Experimental value = 0.00295 --> (0.0139*9.8*0.0749)/3.436 - 0.0139^2 *0.0749
Theoretical value = 0.00269 -->  1/2 * 1.353*(0.1263/2)^2

EXPT 4
Experimental value = 0.00288 --> (0.0266*9.8*0.0249)/2.233 - 0.0266^2 *0.0249
Theoretical value = 0.00269 --> 1/2 * 1.353*(0.1263/2)^2

EXPT 5
Experimental value = 0.0010 --> (0.0266*9.8*0.0249)/6.2825 - 0.0266^2 *0.0249
Theoretical value = 0.0011-->1/2* 0.4658* (0.1363/2)^2

Conclusion:

As we observed and analyzed the relationship between pulley radius, hanging mass and disk to angular acceleration, we found out that they all related to each other. When we increase hanging mass but pulley and disk remain the same, our angular acceleration increase. In addition, when we change small pulley to bigger pulley, the angular acceleration is also increase. But when we combined to steel disk with steel disk, the resultant acceleration become really. The reason behind that is because steel disk is heavier than regular disk that we performed previous experiment. When heavier object spin with each other, they will have more frictional force acting between them. When frictional force increase, the angular acceleration decrease, that's why our resultant alpha is the smallest out of 6 experiment.