22-May-2017:Lab 17: Moment of Inertia of Triangle

Lab 17: Moment of Inertia of a Right Triangle

Joel Cook

Nina Song

Eric Chong

Lab finished May 22, 2017

In this lab, we were tasked with finding the moment of inertia of a triangular metal plate using a rotation apparatus. In theory, by using the moment of inertia for a triangle rotating about one edge, and the parallel axis theorem, we can calculate the moment of inertia when the axis of rotation is about the center of mass. We will be verifying this theory by measuring the angular acceleration of the system with and without a triangular plate installed to calculate the moment of inertia of the plate.

Theory: Since there is going to be friction in the system, the magnitude of the angular acceleration as the mass ascends will not be equal to the magnitude as the mass descends. To account for this, we will use the equation for moment of inertia using average angular acceleration that was derived in Lab 16: Angular Acceleration. 

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

               ( |α(up)|+|α(down)|)

                              2

As previously derived, the moment of inertia for a triangle rotating about the center of mass, 

I = 1/18 * mass * base^2

Procedure:

For this experiment, we used the apparatus shown below:

Air flows between the rotating discs to reduce the friction and allow the system to rotate freely about the axis. The hanging mass supplies the tension force on the disc through the string which causes the disc to rotate. We connected the apparatus to Logger Pro and a laptop to take readings for the angular acceleration of the system as the discs rotate clockwise and counterclockwise. The string was wrapped around the pulley and the mass was released to begin rotating the system. Measurements were taken while the system was rotating. 

After we measured the angular acceleration for the system without the triangular plate in place, we mounted the plate as shown below:

The triangular plate has a mounting hole located 1/3 of the width and length of the plate. For a right triangle of uniform density, the center of mass is 1/3 from the perpendicular edges of the triangle, for both the height and width. Marks on the triangle allow for measuring the angle at which the plate is mounted. The plate was mounted exactly perpendicular for both cases.

The last portion involved taking measurements again with the plate mounted as shown below:

Data/Analysis:

Mass hanging: 25g

Mass of plate: 456g

Vertical base length: 98.2mm

Horizontal base length: 149.1mm

Radius of disc: 0.02485m

The following data was graphed during the experiment. For our purposes, we analyzed the graph of velocity vs time for each trial. By taking a linear fit of the angular speed vs time graph, we were able to calculate the angular acceleration of the system as it rotates. The graphs show the acceleration both in clockwise and counterclockwise direction. 

The first graph pictured shows the angular acceleration for the apparatus with no plate attached. The acceleration recorded was -6.548 rad/s^2 and 5.855 rad/s^2, respectively.

The second graph pictured shows the angular acceleration for the apparatus with the triangular plate mounted vertically. The acceleration recorded was -5.179 rad/s^2 and 4.516 rad/s^2, respectively.

The last graph pictured shows the angular acceleration for the apparatus with the triangular plate mounted horizontally.  The acceleration recorded was -4.102 rad/s^2 and 3.648 rad/s^2, respectively.

By calculating the moment of inertia of the system without the plate and with the plate and subtracting the results, we can find the moment of inertia for the triangular plate in each mounting configuration.

The moment of inertia for the apparatus without the plate mounted:

I = (0.025 kg)*(9.8 m/s^2) (0.02485 m)    -  (0.025 g)*(0.02485 m)^2        = 0.00097 kgm^2

                 (5.855 + 6.548) rad/s^2

                                     2

Moment of inertia for system with vertically mounted triangle = 0.000124 kgm^2

Moment of inertia for system with horizontally mounted triangle = 0.000156 kgm^2

To find the moment of inertia for just the triangular plate we take the difference as described previously:

Vertical mounting:

0.000124 - 0.00097 = 0.00027 kgm^2

Horizontal mounting:

0.000156 - 0.00097 = 0.00059 kgm^2

These values can be compared against the moment of inertia of triangle with axis through the center of mass calculated from the equation:

I(cm) =  1 *Mass*Base^2

              18

Vertical: I =  1  * (0.456)*(0.0982)^2   =  0.000244 kgm^2 

                     18

1 - 0.000244/0.00027 *100 = 9.63% difference

Horizontal:  I =  1  * (0.456)*(0.1491)^2   =  0.00056 kgm^2 

                     18

1 - 0.00056/0.00059 *100 = 5.08% difference

Conclusion:

Overall, the lab was successful. We were able to show that calculating the moment of inertia of the apparatus with and without the triangular plate and subtracting the two gave us the moment of inertia of the triangular plate. We confirmed that the value obtained experimentally is within 10% of the value obtained theoretically. As is true in any experiment, there are errors and uncertainties in the lab. The tools used to measure lengths and masses could be more precise, there is going to be friction between the rotating discs, and the pulley is not massless or frictionless. We attempted to mitigate these factors as much as possible by using the most precise equipment available to us and taking friction into account. Our results suggest that our efforts were successful. 

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.