04-May-2015: Angular Acceleration

Purpose: To apply a known torque to a disk (or disk combination) and measure its angular acceleration in order to calculate the object's moment of inertia. Then compare the results to the theoretical values for the objects' moment of inertia.

Procedure: The apparatus consists of an axis where 2 disks stack and rotate. Adjacent to it is a Pasco rotational sensor. On top of the disks, we attach a torque pulley with a string wrapped around it. The other end of the string goes over another pulley stationed at the edge and hangs a mass off of the table. Turning on the compressed air allows either just the top, or both disks to rotate, which causes the hanging mass to oscillate up and down. We had to clean the disks with alcohol so that we prevent friction from causing any erratic results.

Lab apparatus

Part 1: We wanted to perform 6 trials, using different sized pulleys and disks (or disk combinations), and record the angular acceleration produced by each set of circumstances. In addition we wanted to use one of our trials and look at v (hanging mass) vs. ω (disk) and a (hanging mass) vs. α (disk).

To begin the experiment, we had to first measure the various masses and diameters of the disks and pulleys, as well as the mass of the hanging weight. 

Measured values of objects using caliper and electronic scale

Next, we turned on the Pasco rotational sensor and connected it to our computer and opened up logger pro. Because there is no defined sensor for this rotational apparatus, we had to create something that can work with the equipment. We chose Rotary motion and set the equation in the sensor settings to 200 counts per rotation to match the 200 marks on the disks. Collecting data from a rotating disk gives us the graphs of angular position, angular velocity, and angular acceleration vs. time. The latter graph was useless due to the poor timing resolution of the sensors.

We checked the hose clamp on the bottom to make sure it was closed so that the bottom disk would rotate independently of the top disk when the drop pin was in place. Then we turned on the air, just enough to keep the disks smooth and spun the disks freely to test the equipment. Finally, with the string wrapped around the torque pulley and the hanging mass at its highest point, we began measurements and released the mass. Using the slopes on graphs of angular velocity gave we were able to measure the angular acceleration as the mass moved up and down.

It was important that we take the average angular acceleration of each trial because when the hanging mass went up, there is a frictional torque and torque from the string slowing down the the disk. And when the mass goes down, there is a frictional torque slowing down the disk and a torque from the string speeding up the disk. So we had to add up the angular acceleration going up and going down and divide it by 2.

The objective was to look at the effects of various changes in the experiment on the angular acceleration of the system.
     - Trial 1, 2, and 3: Effects of changing the hanging mass.
     - Trial 1 and 4: Effects of changing the radius at which the hanging mass exerts a torque.
     - Trial 4, 5, and 6: Effects of changing the rotating mass.

Below, we have the graphs for angular angular acceleration for trials 4 and 5.

graph of angular velocity vs. time with angular acceleration given as slope for trial 4

graph of angular velocity vs. time with angular acceleration given as slope for trial 5

Results: As you can see, the motion for both trials were very similar, except that the angular acceleration for trial 5 was about 3 times larger than that of trial 4. This was interesting because trial 5 had a disk with a mass roughly 3 times smaller than that of the disk in trial 4. 

Here is the data for all six trials we performed.

data set for 6 trials

We decided for the sixth trial we would look at v (hanging mass) vs. ω (disk) and a (hanging mass) vs. α (disk). In order to measure the hanging mass's linear velocity, we placed a motion sensor directly below the hanging mass.

lab apparatus + motion sensor

Below we have our graphs for angular velocity of the disk the and linear velocity of the hanging mass. Finding the slope of the graphs gave us the angular acceleration of the disk and the linear acceleration of the hanging mass. 

graphs of angular velocity vs. time and velocity vs. time

Conclusion part 1: After performing the 6 experiments and reviewing our results, we were able to see how a change in the mass of the disk, the size of the torque pulley, and the amount of mass hanging effect the angular acceleration of the disk(s). Changing the hanging mass from 25 grams to 50 grams caused the average angular acceleration to change from 0.454 rad/s^2 to 0.926 rad/s^2. Doubling the hanging mass doubled the average angular acceleration. And changing the mass from 25 to 75 grams effectively tripled the average angular acceleration. Changing only the size of the torque pulley, from 2.836 cm diameter to a 5.380 cm diameter pulley, resulted in a change in average angular acceleration from 0.454 rad/s^2 to 0.872 rad/s^2. And changing the mass of the disks, from a 1360 gram steel disk to a 466 gram aluminum disk, caused the average angular acceleration to increase from 0.872 rad/s^2 to 2.489 rad/s^2.

Part 2: Use our data from part 1 to calculate an experimental moment of inertia and compare it to a theoretical moment of inertia.

Because there is some frictional torque in the system, the angular acceleration isn't the same when the mass ascends and descends. Below is our derivation of how we will calculate an experimental moment of inertia. The theoretical moment of inertia is simply the moment of inertia of a disk. 

Deriving experimental moment of inertia 

Results: Using those two equations, we were able to input our measurements and compare the experimental and theoretical moments of inertia below. 

experimental vs theoretical moments of inertia

Conclusion part 2: Our results showed that our experimental moment of inertia for each trial was significantly different from the moment of inertia we should have gotten theoretically. 

Final Conclusion/Uncertainty: Part one of our experiment required that we graph the angular velocity of 6 different trials with various changes in either hanging mass, pulley size, and mass of disks to see how it effected the average angular acceleration of the disks. We also set up a motion sensor to compare the linear velocity of the hanging mass with the angular velocity of the disk and the linear acceleration of the hanging mass with the angular acceleration of the disk. The results showed that increasing the mass hanging and the size of the pulley increased the angular acceleration, while increasing the size of the disk decreased the angular velocity. Part two required that we use our results from part one to calculate our experimental moments of inertia using our derived expression and compare it to the results using the theoretical moment of inertia. These results however were not very accurate, which could be due to a few factors. 

One reason our results for our experimental moments of inertia may be so off can be because we may have not cleaned the disks well enough to produce a nearly frictionless experiment. Also knots that we could not get out of the string could have slowed down the angular acceleration of the disks as it spun. Finally, there are various factors including the age of the device we were using, the integrity of our measurements and measuring tools, and energy lost to friction from the pulley.