11/7/16 - Angular Acceleration (Lab 16)

PURPOSE:

To determine and compare the moments of inertia for a set of rotating disks.

THEORY:
Rotational inertia (aka, moment of inertia) refers to an object's resistance to undergo an angular acceleration. The harder it is for an object to spin and keep spinning, the more rotational inertia it has. In this experiment, we applied torque to an object that can rotate, measured the angular acceleration of the object, and used this information to find its moment of inertia using both Newton's laws and a calculus derived equation. 

APPARATUS:
Note: The apparatus was assembled beforehand for this lab.

Pasco rotational sensor attached to Lab Pro/Logger Pro setup

EXPERIMENTAL PROCEDURE:
Part 1:

1. Measure diameter and mass of relevant materials. (Listed under Data/Graphs)
2. Plug the power supply into the rotational sensor.
3. Connect the sensor to Lab Pro/Logger Pro.
4. Set up the computer and calibrate the sensor on Logger Pro to read 200 ticks per rotation.
5. 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.
6. Turn on the compressed air such that the disks rotate separately.
7. With the string wrapped around the torque pulley and the hanging mass at its highest point, start the measurements and release the mass. 
8. Use a linear fit of the angular velocity vs. time graph to measure the angular acceleration as the mass moves up and down.
9. Draw some conclusions about your data and how the values are related.

Part 2:

1. Derive an expression for the moment of inertia of the disks using Newton's laws.
2. Calculate the experimental values for moment of inertia of the disks for each experiment using the equation: Idisk = (mgr/α) - mr^2.
3. Calculate the theoretical values for the moment of inertia of the disks using (1/2)MR².

DATA/GRAPHS:
Part 1:
***Diameter and mass of:

• Top steel disk: 126.3 mm, 1358.4 g
• Bottom steel disk: 126.4 mm, 1345.7 g
• Top aluminum disk: 126.3 mm, 465.1 g
• Smaller torque pulley: 49.7 mm, 18.1 g
• larger torque pulley:24.8 mm, 65.1 g

mhanging = 44.1 g

***Uncertainty in mass: +/- 0.1 g
    Uncertainty in diameter: +/- 0.1 mm

Recorded Data

Trial 1 - Angular speed vs time

ANALYSIS:
Part 1:

Data Analysis & Conclusions - Part 1

For part 1 of the lab, we conducted six experiments that demonstrated how changing the hanging mass, radius of the torque pulley, and the mass of the rotating disks affected the angular acceleration of the system. For example, in experiment 1,2, and 3, we found that when we doubled the hanging mass, the angular acceleration doubled as well. When we tripled the hanging mass, the alpha of the system also tripled. In experiments 1 and 4, we discovered that doubling the radius of the torque pulley doubled the alpha of the system. In experiments 4,5, and 6, we concluded that increasing the mass of the rotating disk decreased the alpha of the system. More specifically, when we tripled the mass of the rotating disk--with the alpha of the system with aluminum disk as our baseline--it reduced the alpha of the system to one third of its initial value. Moreover, when we increased the mass of the rotating disk sixfold, the alpha of the system receded to one sixth of its initial value.

Part 2:

Data Analysis & Conclusions - Part 2

Moment of Inertia Results - Part 2

After gathering data, our group calculated the theoretical and experimental values for the moment of inertia of the disks. We used Newton's laws to derive an equation for the experiment values and we used calculus to derive a simpler equation for the theoretical values. While the expressions we used to determine the moments of inertia look radically different from one another, their values were nearly identical. In fact, according to our data, the largest margin of error we encountered was about a three percent difference.

CONCLUSION:
Our experimental results for this lab were satisfactory because they lied within a reasonable margin of error to our theoretical values. As previously stated, the highest margin of error we encountered in our calculations was a three percent difference. While the small level of error in our moments of inertia indicate that our group's uncertainty in our calculations were fairly minimal, it is still in our group's best interest to pursue the most precise measurements and accurate calculations whenever possible. The first step in doing so involves recognizing the sources of uncertainty within our methodology. For example, one of the largest sources of uncertainty in our lab stemmed from human error, such as imprecisely measuring the radii of the disks. Another source of error in our lab came from assumptions we made about our procedure. For instance, we assumed that there is no frictional torque in the system. This assumption implies that the angular acceleration of the hanging mass is the same regardless whether it is ascending or descending. In reality, however, the system does indeed have some frictional torque because there is some mass in the frictionless pulley. Therefore, the angular acceleration of the system is not exactly the same when the mass is descending as when it is ascending.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.,
with special guests: Christian R. & Haokun Z.