22-May-2017: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque

22-May-2017
Purpose: Demonstrate and calculate the moment of inertia, and frictional torque, of a large metal disk on a central shaft.

Summary:

1) We first took measurements of the various parts of the apparatus above. We took the radius of the metal disk and the metal shaft, and recorded the mass printed on the disk, and from this we calculated the mass of the individual parts of the apparatus, which allows us to calculate the moment of inertia of the disk.

2) Using a stopwatch (when the video capture failed), we timed how long the blue tape on our disk would take to come to rest once I gave it an initial spin, and then recorded where it stopped. This would allow us to find the angular velocity, and subsequently the angular acceleration, which would let us solve for the frictional torque exerted by the shaft. We did this multiple times until we got values for angular velocity that was reasonable.

Down the slope

3) Then, we propped up a wooden board in front of the disk at an angle, and connected a dynamics cart to the shaft of our apparatus with a string.

4) Using our calculated value of frictional torque in step 2, we set up equations for the cart and the disks' torque, which lets us solve for the acceleration of the cart, and subsequently for the time it might take for the cart to travel down the ramp 1 meter. This calculation of time would be our theoretical value.

5) To get the experimental value of time for comparison, we simply let the cart roll down the ramp, and our group recorded the time it took for it to travel 1 meter down the ramp. Our experimental value of time was the average of all our measurements.

Analysis:

In order to calculate the moment of inertia for the disk, and subsequently the frictional torque, following my measurements of the disks' radius and mass, solving for its inertia was a matter of using the form of inertia for a circular disk of mass M and radius R. We took the radius of the metal disk and the metal shaft, and recorded the mass printed on the disk, and from this we calculated the mass of the individual parts of the apparatus - the large disk, and the shaft to calculate the moment of inertia of the disk.

Calculating m_1/m_3, mass of the left//right hand shaft, and m_2, mass of the disk using the total mass M printed on the disk.

Calculating the inertia of the disk using m_2.

Then, I had to somehow find the frictional torque of the disk and shaft. To do this, I needed the moment of inertia of the disk, and its angular acceleration, which would be equal to whatever torque the friction was causing. Because I just calculated inertia, I had to find an angular acceleration. To do this, I first thought about using video capture to plot the position of the blue tape over time, which in theory should give me an angular acceleration, presumably less than 1 rad/s^2, because I was not spinning the disk nowhere near that fast. However, the video capture gave me values for alpha that were ridiculously wrong (about 5.1 rad/s^2) - so I turned to using a stopwatch and calculating omega and alpha by hand.

To calculate this, I first found how long it took the disk to stop after its initial spin, how many times the blue tap crossed the top of the disk, and lastly, where the blue tape was stopped relative to the vertical. Determining how many radians the disk spun divided by our average time^2 would give us the angular acceleration, which came out to a much more reasonable 0.5874 rad/s^2.

Calculating for the frictional torque, I stated that the product of the disks' inertia and its angular acceleration is the frictional torque, because once I gave the disk a constant spin, no other force but friction would have been acting perpendicular to the disk. Using my calculation for Omega, solving for the frictional torque gave me a value of .012278 kg*m^2/s^2.

frictional torque

In the next part of the lab, in order to verify just how accurate our calculations for inertia and frictional torque have been so far, we needed to set up an equation that would allow us to predict how long it would take for the dynamic cart to roll 1 meter down the ramp.

It is important to solve for the cart's linear acceleration (which is also equal to the disks'), because we can then use kinematics to solve for time t:

I started by writing my axis in the x and y directions, and using newton's second law I came up with an expression for the dynamic cart and the tension in the string as they both moved down the ramp:

mgsin(0) - T = ma

Then writing out an expression for I*Alpha using torque:

T*r - τ = Iα

Because I want to solve for linear acceleration, 'a,' I convert Alpha to 'a' in my torque expression, and solve for 'a' by first solving for T in my second law expression, and substituting this into my torque expression gives me a value of the cart's linear acceleration to be 0.03233 m/s^2, and a short kinematics calculation gives me an expected time of 7.865 seconds on the ramp as our theoretical value.

Calculating the theoretical time

Lastly, our experimental value for time was determined with three of us timing how long it takes for the dynamics cart to roll down the ramp inclined at 47°, and finding the average of our three values (provided they weren't too different from each other, which they weren't). Our average recorded time came out to be a remarkably close 7.68 seconds, with a percent error of only 2% compared to our  theoretical value. Clearly, our calculations had been very accurate, and demonstrated that the calculations of inertia and torque simply worked.

Experimental results for time

Measured Data

Time measurements

Calculated Data:

Conclusions:

Based on the accuracy of my calculated result of time compared with the experimental result of the dynamics cart rolling down the slope, I conclude that the moment of inertia, and frictional torque, of a large metal disk on a central shaft can be calculated with a high degree of accuracy, and this does indeed work.
Regardless, there were a lot of uncertainties in this lab.
Apart from the usual, trivial uncertainties (like the scales having an uncertainty of a tenth of a gram, or the calipers a tenth of a millimeter) that would hardly affect all of our calculations, the first one to note was the mass printed on the metal disk. While we assumed and trusted what it said, the fact that we were not able to verify its accuracy leads me to assume that our subsequent calculations for each of the individual components of the apparatus and our frictional torque - also have an uncertainty as a consequence.
Next, when calculating omega, because we did not use logger pro's video capture -- we timed it instead -- our measurement of the period and the radians it travels in that time have significant uncertainties of their own. Obviously, this means that my results for the angular acceleration, and subsequently the inertia of my disk, also have an uncertainty.
Lastly, when rolling the dynamics cart down the ramp, we recognize that some energy is lost whilst the wheels make contact with the ramp, and that this would mean our recorded values of time would be inaccurate as well, because we treated the cart as if it were just a sliding mass at some angle in our calculations.
Ultimately, however, our percent error demonstrates that these were not significant enough to detract from our conclusion.