21-May-2017: Lab 16: Angular Acceleration

Lab 16: Angular Acceleration and Moment of Inertia

Joel Cook

Nina Song

Eric Chong

Lab performed May 8th and 15th, 2017

In this lab, we will be using an apparatus to observe the effects of changing the rotating mass, hanging mass, and pulley diameter on the angular acceleration of a system. The apparatus used allows us to change one variable for each trial to measure the effect on the system. For the second part of this lab, we used the data from part one to calculate the moment of inertia of each of the disks in the apparatus.

Part One:

Procedure:

The lab used the apparatus pictured below which consists of two disks and a pulley with a mass hanging over another pulley as pictured. The apparatus has an air system that allows for one disc to spinning with almost no friction on top of another disk or both disks to spin together as one mass. There are three discs supplied, one aluminum and two steel, to allow for different configurations. There are also two pulleys supplied to allow for the system to turn with the hanging mass at a different radius to the axis of rotation. A sensor attached to the apparatus is connected to a laptop and Logger Pro to allow for collecting angular speed and acceleration data.

We began the experiment by measuring and recording data for the diameter and masses of all of the discs and pulleys and the mass of the hanging mass. After reassembling the apparatus, we opened the air valve to start air flow and opened the hose clamp so that only the top disc of the system was rotating and the bottom disc was stationary. We wrapped the string of the hanging mass around the pulley and began data collection as we released the hanging mass. The hanging mass falls down slowly while unwinding the string on the pulley and rotating the disc. Once the mass unwinds fully, the disc continues to spin, winding the string around the pulley again and raising the mass. The mass continues to cycle up and down in this manner. We recorded angular acceleration for the portion of time when the mass is lowering and when the mass is rising.

Data and analysis:

The following table shows the angular acceleration for each trial in the up direction, down direction and average angular acceleration.

	Mass Hanging (g)	Torque Pulley Size	Disk Config.	Angular Acceleration Down (rad/s^2)	Angular Acceleration Up (rad/s^2)	Angular Acceleration Average (rad/s^2)
Hanging Mass Only	24.62	Small	Top Steel	0.5797	-0.6634	0.6216
2 x Hanging Mass	49.62	Small	Top Steel	1.177	-1.295	1.236
3 x Hanging Mass	74.62	Small	Top Steel	1.646	-1.953	1.800
Hanging Mass Only	74.62	Large	Top Steel	1.169	-1.284	1.227
Hanging Mass Only	74.62	Large	Top Aluminum	3.200	-3.631	3.416
Hanging Mass Only	74.62	Large	Top + Bottom Steel	0.5834	-0.6371	0.6103

When the hanging mass of the system is increased, the angular acceleration increases. As the table shows, when the mass of doubled and tripled, the angular acceleration roughly doubled and tripled.

When the size of the pulley increased, the angular acceleration decreased. As the table shows, when the diameter of the pulley was doubled, the angular acceleration decreased by approximately 1/4.

When the mass rotating decreased, the angular acceleration increased. The mass of the both steel disks is 2 times as large as one disk and the angular acceleration increased by two times. The mass of the aluminum disk is approximately one third of the steel disk and the angular acceleration increased proportionally to approximately 3 times the angular acceleration.

Part 2:

For the second portion of the lab, we used the data from part one to determine the moment of inertia (I) of each of the disks and disk combinations. It has been derived for us that the moment of inertia of a disk is the following:

I =        m*g*r         - m*r^2

      |α up| + |α down|

                  2

where m is the hanging mass, r is the radius of the torque pulley and α was calculated from the previous portion of the lab using Logger Pro.

Example:

Moment of inertia of one steel disk:

I (disk) =        m*g*r            - m*r^2

                |α up| + |α down|

                            2

I (disk) =        (0.07462)*(9.8)*(0.02485)           - (0.07462)*(0.02485)^2

                      |1.284| + |-1.169|

                                  2

I (steel disk) = 0.01477 kgm^2

I (two steel disks) = 0.02973 kgm^2

I (aluminum disk) = 0.00527 kgm^2

As expected, lighter the mass that's rotating is, the lower the moment of inertia will be. The moment of inertia of 2 steel disks was approximately twice as much. The aluminum disk had a mass of approximately 1/3 of the steel disk and had a moment of inertia of approximately 1/3 as much. 

Conclusions:

Both portions of the lab successfully demonstrated the relationships between mass, radius of rotation, and hanging mass on a rotating system. We found a directly proportional relationship between the mass of an object and the moment of inertia of the object. This makes sense because:

m (hanging mass)*g - Tension = m*α*r, so if the hanging mass increases, the angular acceleration will increase proportionally.

There was a directly proportional relationship between mass and angular acceleration as well. This also makes sense because the forces on the system will equal the product of mass of the rotating object, angular acceleration and the radius. With all other factors constant, if the mass increases or decreases, the angular acceleration of the mass must increase or decrease proportionally. For the radius of the pulley, there was an inverse relationship between radius and angular acceleration. When the radius doubled, the acceleration decreased by a fourth. This is expected because the radius term is squared. If the radius was increased by 1/4, the acceleration would decrease by approximately 1/16. For the mass of the rotating disk, the relationship between mass and angular acceleration was inversely proportional. When the mass was decreased to 1/3 of the original mass, the acceleration tripled. This makes sense because I = m*r^2 so if the mass doubles, the moment of inertia doubles. The lab was successful but there were some sources of error. There was error in the measurements taken which was mitigated by using the most accurate measuring tools available to us for the lab. The amount of air pressure on the apparatus was potentially inconsistent from one trial to the next due to the poor throttling characteristics of ball valves and not ensuring the air valve was opened to the same position each time. The affects of this were not tested to determine the impact on the experiment. This may have affected the amount of friction on the disks in each trial.