May 13th, 2015: Lab 17: Moment of Inertia (at center of mass) of a Uniform Triangle

Purpose:

Determine I-cm of a triangle by using Parallel Axis Theorem and compare this to the experimental value obtained from the measured angular acceleration.

Experiment:

- Use the same tools from angular acceleration lab. Ideally, use the same kit from last lab so the measurements of rotational disks, hanging mass, and torque pulleys stay true for this lab as well.
- Measure the mass and dimensions of the triangle.

- Make sure the platform is leveled. Wrap the string-hanging mass around large torque pulley till the mass is at maximum height. On top of the pulley, screw in the metal rod, as shown below.

- Set up Loggerpro and other needed equipment just like angular acceleration lab. Make sure the sensor setting is 200 counts/revolution.
- Turn on the compressed air, so the the two disks turn independently. Release the disks and let them rotate. Collect data on Logger pro. Let the hanging mass move up and down three times so there's 3 α-up and 3 α-down. We're going to take average of these to omit any friction in the system.
- Repeat the same steps to taking data for system with the triangle in two different orientations as shown below,

               

            

Data and Analysis:

Theoretical inertia of the triangle:

- Coming up with an expression for an inertia of a triangle spinning around its edge is easier because we can take "dm" of the triangle and claim that each of that strip has dI = (1/3)*dm* x^2. Therefore we get the inertia to be (1/6)MB^2. The Parallel Axis Theorem allows us to move axis of rotation to the triangle's center of mass. Using equation for center of mass, we get the distance from center of mass of x-axis to be 1/3 of the base.

Calculation for center of mass of a uniform triangle

- Since it is a uniform triangle, we can use area density to express "dm".

- We can repeat the same process for another orientation although we have to keep the variables unchanged for each dimension. Using the derived equations, we could plug in the mass, base and height that we measure to get "I-cm" for both cases.

Experimental moment of inertia for the triangle:
Data:

Triangle's mass= 0.454 +/- 0.001  kg

base= 0.0987 +/- 0.00001 m

height= 0.14756 +/- 0.00001 m

hanging mass = 0.0245 +/- 0.001 kg

r-large torque pulley= 0.025 +/- 0.001 m

- Since the descending and ascending angular acceleration are not the same, we have to take their average so that we don't have to deal with frictional torque in the system.
- The way we obtain α is the same; take linear fit of omega vs. t graph.
- Due to the existing inertia in the rod used to support the triangle, we have to calculate it separately then subtract from the inertia of triangle+rod. α is also partially determined by the inertia of the spinning disk, so we don't have to calculate that inertia separately. The only inertia we're adding to the system is the triangle.
- Shown below, in order, α of system with rod only, horizontal, and vertical orientation:

- Take the average of these 3 cases give us,

Ascending α has (-) sign so we have to take absolute value

- We then use this obtained value for the equation derived below. 

       

- Moment of inertia is inversely proportional to α.

Conclusion:

Our error and uncertainty is rather large. Mass measurement would be one of the biggest contribution here since our equipment's uncertainty is 1 gram. As far as collecting α is concerned, friction due to the rotating disks is our most reasonable explanation for error because a slight change in α, even to 2 decimal places, change our error calculation by significant amount. Another source might also be from the amount of compressed air that was in the system. Too little air, indirectly, can cause α to be smaller. Nonetheless, within our range of error and uncertainty, our experimental calculations are close to calculations using Parallel Axis Theorem.

May 11th, 2015: Lab 18: Moment of Inertia and Frictional Torque

Purpose:

Measure the system's angular acceleration, moment of inertia, and derive an equation to find frictional torque in the metal disk system using these measurements.

Experiment:

- The apparatus is composed of 3 cylinders. On it, there's a stapled mark that indicates the total of the 3 cylinders combined (in grams). Record this number.

- Use small and large calipers to measure diameter, and height (or thickness) of the 3 cylinders.
-Use these measurements to calculate moment of inertia of each cylinder, then the total inertia.
- To determine the angular acceleration, set up video capture to record to the disk after it's given a spin and slows down on its own due to friction.
- Place a tape on tip of disk so we could add new point series later during video analysis. In Logger pro open video capture and set it to record to 15-20 seconds. Go to camera settings, "Adjustments", "Image", and adjust contrast and lighting accordingly.  Adjust the camera visibility to have the radius of the large disk in the middle. Test out by start recording video. Make sure that the tape is seen all the time.
- For the experiment, give the disk a spin, then record video. The video should still record until the disk stop on its own.

Data and Analysis:

Calculating Moment of Inertia
- Since only the total mass is given and we can assume that the cylinders have uniform volume density, we can find each of their individual mass.
- We use volume of cylinder = πR²H, and the relationship of ρ=m/v . Note that we need m1+m3 together and m2 alone. 

Calculating frictional torque (τ_f )

- After given a spin, the only torque acting on the system is frictional torque. So for the net torque equation, we can write: _Στ: τ_{f = (I-total) α}

_- For video analysis, set origin for x and y-axis (ideally at the center of the disk), set scale of the radius, add new point series and start adding points taking the tape as the point of interest. Continue adding points until the disk stops.

_{- The point series will give x and y- displacement. From this we could take the derivatives to get x and y-velocity, by going to new calculated column.}

- Since we want to relate our data to rotational motion, we could create another new calculated column for angle. From our x and y-position, use arctan("y" / "x") to get the angle.

- Up to this point, we could check to see if our data is reasonable by graphing angle vs. time and take linear fit,

- Although there are plenty of discontinuity in our graphs, we see that angular velocity decreases with time, which is explained by the fact that the disk slows down.

- Next up, create another new calculated column for tangential velocity, that is the linear velocity at the edge of disk. Using Pythagorean Theorem gets us V= sqrt[(V-x)^2 + (V-y)^2]; and we just found V-x and V-y.

- Up to this point, what we're trying to do is to get Logger pro to give us angular acceleration (alpha). And we know that ω vs. t gives us the slope= alpha. To find ω, use the relationship between rotational and translational motion, one of which is ω= V/R. where R= radius of the disk and V is the one obtained from Pythagorean Theorem.

- Our final data table should look like,

- We could also take derivatives 2 times to get alpha from our angle. But seeing that angular acceleration varies so much although it would've stayed constant, we stay with the idea of using slope of ω vs. t. 

- Graphing  ω vs. t and taking linear fit gives, α = -0. 4246 rad/sec^2,

- So _Στ: τ_{f = (I-total) α = (0.02146)(-0.4246) = -0.009118 N.m;} where negative sign indicates opposite direction of motion. We take I-total because the whole system is bound to frictional torque.

Disk-Cart system

- After finding the frictional torque, use this to derive an equation for linear acceleration for metal disks-cart system, where cart is attached by a string and wrapped around one of the smaller radius of the 3 cylinders.

- During this set up, make sure the string is parallel to the track and does not slip as it unwinds from the disk. Keep the angle(track with horizontal) unchanged throughout the experiment.

- As usual, we can obtain two equations one from rotational dynamics and another from Newton's Second Law, sum of forces. Since α= (a-center of mass )/radius, replace this and solve for "a-center of mass".

 

-From this linear acceleration, use kinematics equation to find the time it takes for the cart to travel 1 meter down the track, released from rest. Since we know distance and acceleration the cart travels, we can find Δt.

- The calculation below show how we get the equation of linear acceleration for the cart. Following the lab handout, we set our initial conditions to be when angle (of the track with the floor)= 40°, m-cart = 0.500kg, and r = 0.0158m and find that it takes 8.20 seconds for the cart to travel 1 meter down the track.    

- τ-string = TR; where R= radius of the disk the string is wrapped around. Since the whole rotating apparatus is acceleration, we take I-total to be on the other side of our net torque equation.

- Using our real data, m-cart = 0.504kg; angle = 47.8°, same radius, Δx = 0.98m, we get t =0.72s

- Experimental data, where Δx = 0.98m (the cart doesn't quite begin at the top nor get to the very bottom of the track)

• trial 1: t = 7.42 s
• trial 2: t = 7.44 s
• trial 3: t = 7.47s

- Hence, error of the last two trials range between 0.27% and 0.67%. The experimental trials could only have larger values than the calculated one, because there's always neglected friction on the track that could slow the cart down, therefore longer time it takes to get to the bottom.
- If we wrap the string around the larger radius and let go of the cart, it will take longer for the cart to get to the bottom of the incline. This observation is made clear in the derived equation. When you increase the radius, the acceleration gets smaller.

Conclusion

In this uniform rotating apparatus, we could find frictional torque using angular acceleration and calculated moment of inertia. This result could be further used to determine the linear acceleration. Assuming constant linear acceleration, we could use kinematics to solve other possible problems given. For part 1, where frictional torque has to be determined, errors include the fact there's drop frames and the metal supporter gets in the way when points series are taken. Besides the poor video quality, there's also usual measurement uncertainty in the diameters. For part 2, we also have measurement uncertainty in angle, cart's mass, and Δx. Within our range of error and uncertainty, we can conclude practicality in our data analysis.

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.