Procedure / Set up:
The apparatus consists of a large disk on a central shaft and at the center of the disk are smaller cylindrical disks that serve as an axis of rotation.
As the cart is traveling down the incline, the disk exerts a frictional torque opposite the carts direction of motion. So before actually predicting the time it would take for the cart to travel 1m, we need to measure the frictional torque exerted by the apparatus. To accomplish this, we will need to determine the total moment of inertia and the angular acceleration of the apparatus. (T frictional = Iα)
Determining the total moment of inertia:
Measure and record the dimensions of the larger central disk and the two smaller side cylindrical disks. These values will be used to determine the volume of each piece. The smaller disks are labeled (1) and (3) and the larger disk is labeled (2).
The ratio of the volume for a piece divided by the total volume of the apparatus will give us the percentage of mass for that piece. For example if 86 % of the total mass of the apparatus belongs to the central disk, then we can take that value and multiply it by the total mass of the apparatus to find the mass of just that central disk. The total moment of inertia is the sum of the moment of inertia of the central disk plus the two moments of inertia for the smaller cylindrical disks.
Determining the angular acceleration:
We set up a camera so that its lens lines up directly with the apparatus' axis of rotation.
Our plan was to capture the disk as it rotates and use the Logger Pro video capture software to find the angular deceleration of the disk. To do this, we marked a point on the disk, recored the disk while it spun, and analyzed the video. During the analyzation process we plotted a point every time our marked spot on the disk went through one revolution or 2π radians. We did this for 15 revolutions or 30π radians. By plotting a point every revolution, Logger Pro would automatically create a column for the time is took to complete that revolution. We next created a column titled theta (ø). This column consisted of the number of radians traveled by the point every 2π revolutions/increments. We next plotted a graph of ø vs. t to find the angular acceleration of the disk.
The equation ø = At^2 + Bt + C is very similar ø = 1/2αt^2 + ωt, Therefore 2A is our actual value for angular acceleration, which equates to -1.075 rad/s.
Because T = Iα, we can now say our value for frictional torque is -0.02265 N*m.
Determining the time it takes the cart to travel down the incline:
The set up consists of the apparatus along with a track at the edge of the table, with the track inclined around 44º. A string is tied around the smaller cylindrical disk and then tied to a cart. The track is positioned so that it is parallel to the string.
Before running the experiment we are asked to predict the time it will take the cart to travel 1m. By drawing a free body diagram of the situation and using Newtons second law of rotation and along with kinematics, the time we determined it would take the cart to travel 1m was 9.42 seconds.
With our predicted value at hand we performed four trials and recored the times for each.
T1: 9.38 s
T2: 9.50 s
T3: 9.44 s
T4: 9.41 s
These values averaged to 9.432 seconds, which gave us a percent error of 0.127 %
Conclusion:
In this lab, our main goal was to determine the time it would take a cart to travel down an incline when attached to cylindrical rotating disk. We were able to find this value using concepts from Newtons second law of rotation, moment of inertia, kinematics, and rotational kinematics. The main steps were calculating the total moment of inertia of the apparatus, measuring the angular deceleration of the apparatus with Logger Pro video capture, and deriving an equation that would express the linear acceleration of the cart enabling us to find the time. The more difficult part of the lab was coming up with a method for finding the angular deceleration of the apparatus. Although our final error was relatively small, error could have been introduced from uncertainty in measurements or from unaccounted friction from the track.
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