May 4th, 2015: Lab 16: Angular Acceleration

Purpose:

Using known moment of inertia to compare it to the one obtained from using angular acceleration.

Experiment:
Part 1:
- Use the Pasco rotational sensor platform as shown below.

- Measure the diameter of bottom steel, top steel,  aluminum steel, small torque pulley, and large torque pulley. Use small calipers for the torque pulleys and the big one, as shown below, for the 3 disks.
 - When taking diameter measurements, make sure the two ends measure one farthest end to another.

- Open the hose clamp at the bottom of the sensor platform to enable one disk rotation operation. For experiment 1, place the bottom and top steels as labeled. On top of the 2 steels, place a small torque pulley attached to string. On top of the torque pulley, place a black pin,

- Connect the rotary sensor platform wire to LoggerPro. The top disk has 200 marks, therefore 200 counts/rotation. Go to setting and change the sensor's setting 200. The sensor can only read the top disk so make sure to plug in the right wire into LoggerPro. Also, set recording time to 20seconds, enough for a few times of hanging mass going up and down.
- Wrap the string around the torque pulley until the hanging mass is at its highest point.
- Turn on the compressed air at a sufficient amount for the the disks to rotates separately and smoothly without friction. Release the hanging mass then collect data on LoggerPro.
- Repeat these same steps for other 5 experiments using different rotating disks, torque pulley, or hanging mass (details in Data and Analysis).
- In addition, for one of the experiments, use the motion sensor, place it on the floor, and collect velocity, position, of the hanging mass as it goes up and down (place a card at the bottom of hanging mass so the motion sensor could see). This allows for the data collection of linear velocity and acceleration.

Data and Analysis:

Part 1:
Diameter (small torque pulley)= 0.025 +/- 0.001 m
Diameter (large torque pulley)= 0.005 +/- 0.001 m
Diameter (bottom steel)= 0.12651 +/- 10^-5 m
Diameter (Aluminum)= 0.12652 +/- 10^-5 m
Diameter (bottom steel)= 0.12649 +/- 10^-5 m

Mass measurements,

mass measurements

- As data from LoggerPro is concerned, we only need the angular velocity vs. time, since the slope of this will give out a much more accurate data than the actual angular acceleration vs. time graph.
- From the 6 experiments:

• experiment 1, 2, and 3: effect by changing hanging mass
• experiment 1 and 4: effect from changing size of torque pulley
• experiment 4, 5, and 6: effect from using different rotating disks

- The graphs shown below is from experiment 4. The other 5 experiments have graphs that look similar the this.

- The graph of Angle vs. time assures that the acceleration obtained from linear fit is constant. The displacement of angle for each cycle (going down+up) is approximately the same, given the same time interval. Therefore, we get a straight line graph for angular velocity vs. time.
- Our linear fit can be taken from the beginning point since the hanging mass was released before data collection takes place.
- Note that angular acceleration on its way up (indicated by negative slope) is larger than on its way down. Frictional torque (with direction opposing the downward fall direction) is the cause of this even though we assume there's no friction in the system.

- Average angular acceleration of α-up and α-down are taken so we could still assume,
 for this first part of lab, that there's no friction in the system.
- By comparing experiment 1, 2, and 3, we see that increasing the hanging mass is directly proportional to average 
angular acceleration of system. In experiment 1 and 4, we see that as we use larger torque pulley's radius, the 
acceleration increases. This statement agrees with τ-string= TR. As "R" is larger, there's less tension in the string, hence larger acceleration. Experiment 4, 5, and 6 show the comparison that as the mass of the rotational disk increases (larger moment of inertia), angular acceleration decreases. 
- To compare those experiments numerically,

• Ratio of expt. 2 to 1: m2/m1 = 2.03. Hence α is 2.03 times larger than α in experiment 1.
• Ratio of expt. 3 to expt. 1: m3/m1 = 3.02. Hence α is 3.02 times larger than α in experiment 1.
• Ratio of expt 4 to 1: R/r = 2. So α is 2 times larger with the bigger torque pulley
• Ratio of expt. 4 to 5: (M-top steel)/(M-Al)= 3.05. So α is 3.05 times larger than α using top steel
• Ratio of expt. 4 to 6: (M-top steel)/(M-top and bottom)= 0.503. So α using both steels is 0.503 less than α using top steel only.

- Picture below show calculations in detail, 

- To double check our data collection for α, we use motion sensor to measure velocity of
hanging mass, linear fit linear velocity vs. time, and get a-tangential to be 0.01324 m/sec^2 (descending).

- a-tangential (descending) = αr = (1.051)(0.025/2)= 0.01314 m/sec^2. Calculating error between the
two values give ~ 0.8%. So our α data collection agrees with the one we calculate from a-tangential.
Conclusion for part 1:
We see that hanging mass and radius of torque pulley is proportional to α, and mass of rotational disk
is inversely proportional to α.
Part 2:
- Assuming the system has no frictional torque, and α-up = α-down, we can use net torque and net force equation and derive for moment of inertia and α-disk as shown,

- From the same equation above, we get,

     where, 

• r = radius of torque pulley
• m = from hanging mass
• α = average from α-up and α-down

- If we were to count in frictional torque,

- From (7), we see that the derived equation for moment of inertia is very similar to "I" with no friction. Another point to consider is that when there's frictional torque, we have to have equation (1) and (2) because α-up and α-down are different.
- Using the measurement data the disks from part 1, we get,

expt 1-4: top steel disk; expt 5: Aluminum Disk; expt 6: top steel+bottom steel

-By theory, I of disk around its center of mass is (1/2)MR^2, 

- Comparing the actual to experimental. After doing uncertainty calculations of one experiment, we see that the actual moment of inertia is in the range of our calculations.

- Experiment 5's moment of inertia (Aluminum): 0.000949 kg.m^2 ; Our range of error ~ 0.5 to 4%.

Conclusion:

Within our range of measurement uncertainty, we could obtain a moment of inertia from angular 
acceleration (from Loggerpro) that's very close to the theoretical one calculated using (1/2)MR^2. Possible
errors include the fact that there are scratches on all the rotational disks,(providing friction), existing
moment of inertia in torque pulley, and pulley at the edge of the platform.