Lab 18 (17th): Moment of Inertia and Frictional Torque

Title: Moment of Inertia and Frictional Torque

Purpose: Determine the frictional torque acting on a large metal disk rotating about a cylindrical shaft. Verify by observing the motion of a cart accelerating down a ramp.

Theory: The moment of inertia of an irregular large object composed of a number of smaller shapes with known inertia can be found by summing the inertia of its components about the axis of rotation. As the system being observed is not ideal and does experience friction, the magnitude of frictional torque may be found by finding the object's angular acceleration and using the equation t = I * a. To confirm this value of frictional torque, a theoretical linear acceleration of a cart down a ramp may be calculated using kinematics equations. The theoretical time the cart would take to travel a distance of 1 meter may then be compared to actual measured times.

Apparatus:

• Rotating disc setup
• Slow motion video capture
• Vernier caliper
• Logger Pro
• Low friction cart
• Cart track
• String

Procedure:

Using the calipers, find the diameter and thickness of the 3 cylindrical sections of the rotating setup. Find the total mass of the disc which is stamped along one face of the large central cylinder.  Find the inertia of the three components using volume to find its mass and the equation I = 1/2MR^2. Use slow motion video capture to record the motion of the disc, following one point through approximately 4-7 slow rotations. In Logger Pro video analysis, record the time for the completion of each revolution by following the path of a single marked point. Set one rotation equal to 2 pi radians. Perform a second degree polynomial fit on a graph of angle rotated in radians vs time. The constant value of the x^2 term is 1/2 the angular acceleration. Using this value of angular acceleration, calculate a predicted value for the time it takes for a cart to travel 1 meter with a string attached to the smaller cylinder of the disc. 

Data and Graphs:

Photo of experimental disc apparatus:

Photo of video analysis:

Photo of graph (Note: this is the incorrect photo, a photo of the actual graph used was not taken):

Photo of cart and track setup

Table of collected data and calculated values:

Mcart	0,526	kg
Incline	45,6	°
Mdisc	4,808	kg
Diameter (m)	Depth (m)	V (m^3)	M (kg)	I (kgm^2)	Total I	α (rad/s^s)	Frictional τ (Nm)
0,2006	0,0156	0,0004930	4,135	0,02080	0,02088	0,7676	0,01603
0,0314	0,0513	0,00003973	0,3332	0,00004106
0,0314	0,0523	0,00004050	0,3397	0,00004186

Predicted T 8,01 s Actual T (s) Average T %Deviation %Error 7,83 8,018 -2,349 0,104 8,07 0,644 7,92 -1,226 8,16 1,767 8,13 1,393 8,00 -0,229

Data Analysis:

The derivation for the formula used to calculate linear acceleration is shown below:

Linear acceleration was then used in kinematics equations to produce a theoretical time.

Conclusion:

Our value of frictional torque of 0.01603Nm is very near the true value of frictional torque as the actual travel time of the cart differed from the expected travel time by only 0.104%. However, it was observed that the lubrication of the disc was a significant factor in determining the frictional torque as the disc, prior to additional lubrication, produced noticeable sound when rotating, signifying a greater loss of energy to friction. Even so, our data is valid because the two parts of the experiment, the determination of frictional torque and the comparison using a cart, were conducted in relatively quick succession and without additional use of the disc between experiments. Other factors that introduce error include the non-ideal nature of the low friction cart as it is not entirely frictionless. The string connecting the cart to the disc was unlikely to be perfectly parallel to the track and thus the actual torque applied by the cart on the disc may be lower than predicted, increasing the time taken to travel 1 meter down the ramp. This experiment had very few major potential sources of error and as such should produce reliable data. However, it was noted that some time measurements deviated rather largely from the average given the nature of this experiment, with one data point having 2.35% deviation. It would be possible to improve on the experimental procedure by hanging a mass vertically instead of sliding a cart to eliminate the presence of external friction forces and ensure that motion of the hanging mass is always perpendicular to the moment arm of the disc.