The Physics of a Falling and Unrolling TP Roll

Yes, that's right--the physics of a falling and unrolling toilet paper roll.  This experiment will give students practice in rotational motion of an object and translational motion of its center-of-mass.  It will also involve both the kinematics and dynamics of the motion. While it can be done by use of the VelocityLab app, interpretation of the angular velocity data from the PocketLab app is much easier.
 
The figure below shows the apparatus setup for this lab experiment. A ring stand is on a table with a horizontal bar extended from the ring stand.  A PocketLab is attached with mounting tape to the side of the TP roll.  The PocketLab is oriented so that the roll is moving in the XY plane, thus making the Z angular momentum of interest in this experiment.  The first piece of the TP roll is taped to the horizontal bar and the roll is then allowed to drop to the table top, while the PocketLab app collects angular velocity data at the fastest rate possible.
 
Setup
 
The author found that the ideal distance for the fall is between 30 and 40 centimeters.  Distances larger than this resulted in exceeding the limit of 2000 º/s for PocketLab's gyroscope.  This is easily recognizable on the angular velocity graph, as the graph will show a horizontal plateau at 2000 º/s.
 
The angular velocity vs. time graph below was constructed in Excel from the Z-gyroscope data obtained from the PocketLab app.  The graph provides two important pieces of information for the student: (1) the angular acceleration while the TP roll is falling, which can be obtained from the slope of the graph region where it is falling, and (2) the angular velocity just before hitting the table top.
 
Angular Velocity Graph
 
From the two pieces of information in the above graph and measuring the vertical distance that the TP roll has fallen, students are then asked to compute:
 
(1) The final angular speed of the roll in radians/s.
(2) The angular acceleration of the roll while it is falling in radians/s/s.
(3) The translational acceleration of the center-of-mass of the roll in m/s/s.
(4) The final translation speed of the roll in m/s.
(5) Using a free-body diagram and Newton's Second Law of Motion as well as the mass of the combined TP roll and PocketLab, compute the tension in the sheets while the roll is falling.
 
As an optional exercise, students can be asked to derive an equation for the tension in the sheets as a function of the inner and outer diameters of the TP roll, the mass of the roll, and the acceleration of gravity.  This involves both the equation for net Force and the equation that relates the sum of torques to the moment-of-inertia of the TP roll and its angular acceleration.  The students can then compare the tension predicted by this equation to the tension they calculated in exercise #5.
 
A teacher document is attached with answers to the five questions and the optional exercise. 
 
  

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