MAY-04-2015 Angular Acceleration

Angular Acceleration

(Determining moment of inertia using angular acceleration)

Purpose:  The first half of this lab involves determining the angular acceleration (alpha) of various disks / disk combinations.  Having these values of angular acceleration, we can discuss numerically the differences in values when different parts of the system are altered (ie additional hanging mass, larger pulley radius, material of disk, etc..).  The second half of the lab is utilizing these angular acceleration values to determine the moment of inertia of the various disks / disk combinations and comparing them to the theoretical values using the known formula of the moment of inertia of a uniform disk (1/2MR^2).  

Part 1)

Procedure:  We first took relevant measurements of the various disks, pulleys, and the mass to be hanged via a string.  The apparatus is depicted below:

(Experimental Apparatus)

Listed below are the relevant measurements recorded:

(Measurements of experimental items)

In order to determine the angular acceleration of each setup, we used LoggerPro to record the angular velocity over time.  Performing a linear fit of this graph yields the angular acceleration as the hanging mass falls and rises.  The average of these two values is our target.  Below are two graphs of the first run of the experiment, with a setup of the original hanging mass (26 g), only the top steel disk spinning, and the hanging mass connected to the smaller torque pulley via a string.  The first graph gives us the angular acceleration as the mass falls, and the second as the mass rises:

(Angular Acceleration as mass falls)

(Angular Acceleration as mass rises)

Taking the average of these two values' absolute value, we have:

Average Acceleration = alpha = 1.1215 rad/s^2

Shown below are the varying values of alpha while running the experiment with various setups:

(Data from various Exp. setups)

Conclusions:  By analyzing the table provided above, we're able to draw some numerically specific conclusions:

 * By doubling the hanging mass, the angular acceleration is approximately doubled

 * By tripling the hanging mass, the angular acceleration is approximately tripled

 * By nearly doubling the torque radius, the angular acceleration is nearly doubled

 * By changing the rotating disk from steel to aluminum (nearly reducing the mass by a third), the angular acceleration is roughly tripled.

 * By roughly doubling the rotating mass (both steel disks), the angular acceleration is nearly halved.

Part 2)

Procedure:  In this portion of the lab, we'll utilize Newton's 2nd Law in order to determine the moment of inertia of each disk setup (ie top steel disk, top aluminum disk, top steel disk + bottom steel disk).

Depicted below is a free-body diagram of the first setup (Top Steel Disk), along with relevant 2nd Law equations required for determination of the moment of inertia using our recorded values of alpha:

(Moment of Inertia for Top Steel Disk)

The above picture displays that our experimental value for the moment of inertia of the top steel disk is acceptably near the theoretical value.  Performing similar calculations for the aluminum disk and top steel disk + bottom steel disk combination yield:

Aluminum Disk:

Experimental I = Ia = 1.10 X 10^-3 kg * m^2

Theoretical I = Ib = 0.92 X 10^-3 kg * m^2

Top Steel + Bottom Steel:

Experimental I = 6.20 X 10^-3 kg * m^2

Theoretical I = 5.38 X 10^-3 kg * m^2

Sources of Uncertainty:

Once source of uncertainty could be the way the angular velocity is recorded into LoggerPro, meaning the number of indentations counted by the system as one rotation could be offset.  Another is incorrectly recording the masses of the various pulleys or disks.  Yet another source is improperly measuring the diameter of the torque pulleys, as there are "lips" that surround the actual diameter of each one, this would in turn yield incorrect radius values used in deriving the moment of inertia values from our experimental data.