Rotational Motion Lessons

Investigating the "Spinning Coin" (Euler Disk) Problem

Most everyone has spun a coin on its edge on a table top, and many find the result quite fascinating.  The coin gradually begins to fall on its side while spinning, makes a whirring sound with increasing frequency the longer it spins, and then abruptly stops.  The Swiss physicist, Leonhard Euler, studied this back in the 1700's.  An educational toy, referred to as Euler's disk can now be purchased on-line and in hobby shops specializing in science.  Such disks have been carefully engineered to spin for a much longer time than a coin.

While the physics of a spinning coin is similar to that of a gyroscope, the physics can get very complicated.  Complex mathematical theories have been deducted, under a variety of simplifying assumptions, to explain the fascinating behavior of a spinning coin.  Never-the-less, the spinning coin provides for interesting student discussion at all grade levels.

As the coin spins, its center-of-mass lowers.  This results in a loss of gravitational potential energy, which is converted into kinetic energy of rotation.  There are also dissipative forces including (1) rolling friction that affects the precession rate and (2) air resistance that becomes especially important just before the coin stops spinning.

OK, we can't very well attach PocketLab to a coin, but we can attach it to other larger disks that behave much like a coin.  The wood disk shown in the figure below was found to work very well with PocketLab.  It is about 5" in diameter and 3/4" thick, and was purchased at a local hobby store.  PocketLab was attached to the center of the disk using some Scotch Removable Poster Tape. The Z-axis is perpendicular to the face of the wood disk, while the X-axis and Y-axis lie on a plane parallel to the face of the wood disk.

PocketLab was set to provide angular velocity measurements at the highest rate permitted by the device running the PocketLab app.  The magnetic field was zeroed, and the wood disk was given a spin on a level surface.  The video below was obtained from the PocketLab app and shows the X, Y, and Z angular velocity vs. time graphs superimposed on a movie of the spinning disk.

Another movie, taken with an iPhone in SLO-MO mode, makes it much easier to study the motion of the spinning disk.  The movie appears below.

A graph of X, Y, and Z angular velocity from one run of the experiment is shown in the figure below.  The graph was created in Excel from data provided by the PocketLab app.
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The green curve shows how the rate of precession slows down as time progresses.  The red and blue curves show the speed-up of the wobbling that occurs as time progresses.  The horizontal portion of the red curve at the bottom left of the graph is created when the Y angular velocity exceeds the 2000 degrees/sec limit of the PocketLab.  The spinning stops abruptly at approximately 11.2 seconds into the movie.

What a great exercise for students to perform with PocketLab!  Set up, data collection, and entry-level discussion can easily be accomplished in one class period.

A Velocity Lab Experiment on Rolling Resistance

Rolling resistance is a force that opposes the motion when an object rolls along a surface.  In this experiment a coasting cylinder on a carpet gradually slows down and stops due to rolling resistance.  The primary factor affecting rolling resistance here is deformation of the carpet as the cylinder rolls.  Not all of the energy needed to deform the carpet is recovered when the pressure from the cylinder is removed.  In other words, the effect is non-elastic.  The purpose of this experiment is two-fold:  (1) to determine the force of rolling resistance and (2) to determine the coefficient of rolling resistance between the cylinder and the carpet.

The cylinder in this experiment is an unopened can of jellied cranberry sauce.  The author used this as it is reasonably heavy and there seemed to be little if any sloshing of the sauce in the can as it rolled.  The photo below shows the can resting on the carpet with PocketLab Velcro'd to the end of the can in such a way that the z-axis is the desired axis of rotation when used with the VelocityLab app.  The white arrow shows the direction of the initial push on the can.  After the initial push, the can eventually slows down and stops moving.  This is due primarily to rolling resistance, and using the can means that there is no friction in an axle that would also consume some energy.

The video below shows a typical run of the experiment.  Graphs of position, velocity, and acceleration versus time are synced with the video. As can be seen, the experiment is easy to set up, and the data is obtained quickly for analysis by student lab groups.

The figure below shows the free-body diagram of the cylinder as it slows down from rolling resistance.  The normal and gravitational force are equal in magnitude as the cylinder rolls on the level plane of the carpet.  From Newton's second law, the net force, ma, is equal to the force of rolling resistance.  The coefficient of rolling resistance is defined by the equation shown in the diagram as the ratio of the force of rolling resistance to the normal force.  The coefficient of rolling resistance is, therefore, a dimensionless quantity that can be thought of as the force per unit weight required to keep it moving at a constant speed on a level surface, assuming negligible air resistance.

The position, velocity and acceleration graphs in the figure shown below were obtained using Microsoft Excel from the VelocityLab.csv file produced by the VelocityLab app.  The acceleration of the cylinder can be obtained in two ways: (1) from the slope of the region of the velocity vs. time graph where the can is slowing down, and (2) by averaging the data points in the acceleration vs. time graph in the region where the can is slowing down.  Both methods give an acceleration of -0.363 m/s/s.  With the acceleration known, students can then address the two-fold purpose of the experiment.  (Note that the mass of the cylinder plus PocketLab, obtained from a balance, is 0.479 kg.)  First, the force of rolling resistance is ma = (0.479 kg)(-0.363 m/s/s) = 0.174 N.  Secondly, the coefficient of rolling friction = a/g = (-0.363 m/s/s)/(-9.81 m/s/s) = 0.0177.

An interesting optional exercise for the student lab groups would be to repeat the experiment with the can rolling on, say, a school hallway or on a gymnasium floor.  What are their predictions on the value of the coefficient of rolling friction in this case?  Does experiment verify their predictions?

Note that the VelocityLab.csv file used by the author is included as one of the attachments.  At the writing of this lesson, it should be noted that the acceleration in the VelocityLab.csv file is given in units of g's.  A column needs to be added giving acceleration in m/s/s by multiplying g's by 9.81.

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.

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.

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.

Rotational Dynamics of a Falling Meter Stick

There is a well-known problem in rotational dynamics that involves a meter stick.  The meter stick is held in a vertical position with one end on the floor.  It is then released so that it falls to the floor.  The end initially on the floor is not allowed to slip during the fall.  Students are asked to derive an equation that predicts the angular velocity of the meter stick just before it hits the floor.  The derivation involves many physics concepts including gravitational potential energy, rotational kinetic energy, conservation of energy, moment of inertia, and angular velocity, thus giving the student a good workout in the physics involved.

Now with the availability of The PocketLab, students can do a quick experiment to test the validity of their theoretical equation for angular velocity.  Finally, they can discuss some possible explanations for any differences between their theory and their experimental results.

The movie below shows a typical run of the experiment.  PocketLab, in its silicone protective case, has been taped to the center of the meter stick, though it could have been placed anywhere on the meter stick since the angular velocity is the same for the entire rigid meter stick.  It seems reasonable, however, to place PocketLab at the center-of-mass of the meter stick to keep the mass uniformly distributed about the center-of-mass.

The photo below shows a close-up of PocketLab taped on the meter stick after the meter stick has hit the floor. With the orientation of PocketLab shown, the meter stick has fallen in the XZ plane, making the Y angular velocity of interest in this experiment.

The graph below shows the Y angular velocity of the meter stick as it falls.  This graph was obtained with Excel from the gyroscope.csv file created by the PocketLab app.  It is seen that the angular velocity of the meter stick just before hitting the floor is about 271 º/s.  This agrees to within 13% of the theoretical angular velocity of 311 º/s.  The theory is found in a file attached to this lesson.

A PocketLab Experimental Analysis of a Yo-yo

The yo-yo, a toy with an axle connected to two disks and string wound on the axle, has been of fascination to many for centuries.  It also offers a perfect opportunity to study angular velocity when a PocketLab has been attached to it.  A graph of angular velocity vs. time of a yo-yo will require students to think carefully about the detailed behavior related to its motion.

The author worked with a purchased \$3 yo-yo, but found the results to be much clearer when attaching a PocketLab to a more substantial and heavier home-made yo-yo, as shown in the figure below.  The photo on the left shows PocketLab attached to one of the two wood disks, each disk about five inches in diameter and 3/4" thick.  The disks are connected with a short wood axle cut from a round dowel rod whose diameter is about 7 mm.  The disks and dowel rods were purchased at a local Michaels hobby shop.  The photo on the right shows the narrow gap between the disks with a string attached from the PocketLab Maker Kit.  The end of the string has been pressed into the axle hole on the disk and is held tight by the axle, unlike most yo-yo's that you buy in the store.  Store bought yo-yo's have the string tied to the axle loosely, so that all kinds of tricks can be performed by the practiced yo-yo-er.

The photo below shows the apparatus setup.  The free end of the string is looped over a rod attached to a ring stand.  The yo-yo is then dropped and allowed to "yo-yo" up and down on its own until it stops.  Based upon the orientation of the PocketLab app mounted on the yo-yo, the Z-angular velocity from the PocketLab app provides the data of interest in this study.  The photo shows the yo-yo when it has come to rest after completely unwinding.  The length of the string is about 1.1 meters.

The video below was taken with the PocketLab app superimposing angular velocity data and a graph.  The data rate was set to the maximum allowed for angular velocity and the angular velocity was zeroed with the yo-yo at rest before taking the video.

The graph below was created in Excel from Z-angular momentum data from the PocketLab app.  The gyroscope.csv file is attached for anyone interested in detailed data from the run.  Several points and lines on the graph below have been labeled for the purposes of discussion and analysis.  Students should be able to answer the questions after carefully viewing the video.  (Teachers--see the attached Teacher Document.)

Discussion Questions
1. What is the yo-yo doing at the points labeled with green dots?
2. What is the yo-yo doing at the points labeled with red dots?
3. In contrast to the green dots, what has caused the horizontal lines which have been labeled with the letter A?
4. What is the yo-yo doing on the lines labeled with the letter B?
5. What is the yo-yo doing on the lines labeled with the letter C?
6. Can you think of a way to determine the actual maximum angular velocities where you see the horizontal lines labeled A?

The graph below was obtained by copying/pasting the Z-angular velocity data into Logger Pro, an exceptional educational software program for analyzing data, and a product of Vernier Software & Technology (vernier.com).  It shows how the actual maximum angular velocities can be obtained where the horizontal lines labeled A are located.  Linear models are simply applied to the data points on either side of the A lines.  The intersection of these lines when extrapolated then tells us the actual maximum angular velocity.  This process seems justified because of the triangular shape of the green point peaks on the right of the Excel graph, where angular velocity has not exceeded the maximum allowable by PocketLab, i.e., 2000 degrees/second.

An Experiment in Rotational Dynamics that Emphasizes the NGSS Science and Engineering Practices

Here is a PocketLab based project that will get your physical science and physics students involved in many of the Next Generation Science Standards, particularly in the NGSS science and engineering practices.

Two wheels and a wood axle from the PocketLab Maker Kit are placed on a narrow inclined plane so that the red wheels overhang the sides of the inclined plane and the entire system rolls down on the wood axle without any slipping.  When the wheels and axle get near the bottom of the inclined plane, the wheels come in contact with the surface of the table top.  Challenge the students to hypothesize what will happen next.

The photo below shows a snapshot at the instant the red wheels contact a piece of cardboard on the table top.  Cardboard was used to provide more friction as the table top was quite slippery.  PocketLabs are mounted on both of the wheels to provide some symmetry, though only one of the PocketLabs is actually used.  A small piece of wood about the same mass as a PocketLab could be used as a replacement for the unused PocketLab.

The photo below is an enlargement that more clearly shows the two red wheels, axle, and mounted PocketLabs on the inclined plane and just reaching contact with the surface of the cardboard.

With hypotheses in hand, you can now either have the students design an experiment, without the use of PocketLab, to test their hypotheses, or show the video below.  This video shows what happens but does not provide any superimposed data from PocketLab.  If you use this video, then challenge the students to provide explanations for what happens when the wheels contact the surface.

Now it's time for the students to get quantitative by using PocketLab to collect some angular velocity data as the systems rolls down the incline and contacts the cardboard on the table.  Alternately, you can make use of the video below that contains combined video and data from the PocketLab app.  The orientation of PocketLab on the red wheel indicates that the Z angular velocity is of interest in the analysis.

The angular velocity vs. time graph below was made in Excel from Z angular velocity data in the csv file created by the PocketLab app.  The csv file used by the author is attached for your reference and for use by you and your students.

There are several discussion questions that relate to this graph:

1.  What do each of the points A, B, C, and D represent in the motion of the wheels and axle?
2.  Why is there a sort of sine wave feature in angular velocity from points A to B,  and from points C to D?
3.  What is the angular velocity of the wheels and axle system just before making contact with the table top?
4.  What is the speed of the center-of-mass of the system just before making contact with the table top?
5.  What is the angular velocity of the wheels and axle system at point C?
6.  What is the speed of the center-of-mass of the system at point C?
7.  Explain the physics of why the speed of the center-of-mass of the system increased upon contacting the table top.

Using VelocityLab in an AP/College Physics Experiment Involving Rotational Dynamics

This experiment is designed for AP Physics and college physics students.  It considers a solid cylinder of mass M and radius R that is rolling down an incline with a height h without slipping.  Using energy and dynamics concepts, students first derive equations for (1) the speed of the center of mass of the cylinder upon reaching the bottom of the incline, and (2) the acceleration of the center of mass of the cylinder as it rolls down the incline.  The free-body diagram at the center shows all forces acting on the cylinder as it rolls down the incline.

Then students use PocketLab and the VelocityLab app to perform an experiment that verifies their two equations.

The photo below shows the experiment setup as performed by the author.  An unopened jellied cranberry sauce can is used as the cylinder. A cranberry sauce can was selected as there is virtually no sloshing of the cranberry sauce as the can rolls down the incline, and the end of the can was a perfect size for mounting PocketLab using Velcro.  A plastic drafting triangle was used to hold the can still and then release it at the top of the ramp.  A pillow was used as a bumper at the bottom of the ramp to stop the can after reaching the bottom.

The movie below shows video combined with data for a typical run in which the radius of the cylinder was 7.4 cm and the height of the ramp was 10.2 cm.

The figure below shows graphs of position, speed, and acceleration obtained by using Excel to massage the data in the pos_vel_acc.csv file produced by VelocityLab.  The position graph shows the region in which the cylinder is rolling down the incline.  The position graph also indicates the length (1.164 m) of the inclined plane.  The speed graph shows that the speed at the bottom of the incline is 1.115 m/s.  The slope of the region where the cylinder is rolling down indicates the acceleration (0.580 m/s/s).  The acceleration can also be obtained directly from the  acceleration graph by averaging the acceleration points (0.589 m/s/s) during the time that the cylinder is rolling down the incline.  Both methods for determining the acceleration obtain close to the same value.

The theoretical equation for the speed at the bottom of the incline predicts that the speed should be 1.155 m/s.  Our experimental value of 1.115 m/s represents an error of only 3.6%.  The theoretical equation for the acceleration of the center of mass as the cylinder rolls down the incline predicts that it should be 0.572 m/s/s.  Our experimental value of 0.585 (averaging the value from slope of speed graph and mean from the acceleration graph) represents an error of only 2.3%.

The teacher is encouraged to view the attached pdf file that provides the theoretical equations and their derivations.

This lesson is a physics application of PocketLab that allows students to determine the radius of curvature of a gradual turn on a street.  A PocketLab mounted on the dashboard of a car records both the angular velocity and the centripetal acceleration of the car as it moves at a nearly constant speed around the curve.  All of the required data for an example problem are contained in files attached to this lesson.  Alternately, students can collect their own data.  If the latter approach is used, students should be cautioned to be safe: (1) follow all speed limits and traffic laws, and (2) have one person drive while another works with the PocketLab app.

The photos below give two views of PocketLab mounted on the dashboard of a car using double stick mounting tape.  The photo on the left shows that PocketLab has been mounted in such a way that it is level when the car is at rest on a level surface (in this case, the floor of a garage).  The photo on the right shows that PocketLab has been mounted so that the Z-axis provides the angular acceleration as the car moves in the XY plane.  Similarly, the Y-acceleration would then provide the magnitude of the centripetal acceleration as the car negotiates a turn on the street.

Below is a Bing map aerial view of the route (shown in red) driven by the car in the example problem of this lesson.  The start and end of the route are shown.  In addition the two major turns of interet are specified.  Turn #1 is a gradual right turn, and it is followed by a similar gradual left turn, turn #2.  These are the two turns whose radius we are interested in determining.

To get more of a feel for the actual ride, the video below was taken for this example.  It is instructive to follow the video while viewing the Bing map.  There are two initial sharp right turns at intersections, followed by a speed bump on the road, followed by turn #1, followed by another speed bump, followed by turn #2, followed by another speed bump, followed by a final right turn and stop onto a neighborhood street.  Again, turn #1 and turn #2 are the turns of interest, i.e., the turns whose radii are to be determined.

The graph below contains a detailed look at the data collected for the Z angular velocity of the car.  The three sharp right turns at intersections all have maximum angular velocity magnitudes of about 20 deg/s.  The data for turn #1 and turn #2 have been identified on the graph.  As there is some variation in the data, it is helpful to determine the mean angular velocity for each of the turns.  Vernier Software & Technology (vernier.com) has an excellent educational data analysis software called Logger Pro that was used to determine the means, shown on the graph.  Alternately, means could be calculated using a spreadsheet package such as Microsoft Excel.  The mean magnitude for the angular velocity is 5.432 deg/s for turn #1 and 5.146 deg/s for turn #2.

The graph below contains a detailed look at the data collected for the Y acceleration (centripetal acceleration) of the car.  There is significant variation of the data, so it is again necessary to compute means for turn #1 and turn #2.  These turns have been identified on the graph, and coincide in time (86 to 98 seconds for turn #1, and 116 to 127 seconds for turn #2) with the angular velocities from the previous graph.  The mean magnitude of centripetal acceleration for turn #1 is 1.264 m/s/s and for turn #2 is 1.093 m/s/s.

Attached is a pdf file for the teacher that shows the calculations for radii of turn #1 and turn #2.  The radii turn out to be 141 m and 135.5 m, respectivel, for the turns.  These are in good agreement with values that can be obtained from Bing or Google maps, when making use of the scale provided on these maps.

PocketLab Experiment on Centripetal Acceleration with a 3-speed Ceiling Fan

There are two approaches that the teacher can take to doing this experiment on centripetal acceleration with a three-speed ceiling fan and PocketLab.

The first choice is for those with an available three-speed ceiling fan.  In this case students can collect all data by actually performing the experiment themselves.  The PocketLab should be mounted to one of the ceiling fan blades with a very strong double stick mounting tape.  For safety, however, students should still wear goggles.  The author mounted PocketLab as close to the center pivot point of the blade as possible.  This was necessary to avoid exceeding the 8g limit for the PocketLab accelerometer at the highest fan speed.

The second choice is simply to use the data captured by the author from files attached to this project.  An empty data table on which students can use the data to make calculations related to the circular motion of the ceiling fan is found in the attached pdf file.  Accelerometer and gyroscope csv files are also attached, allowing students to obtain centripetal acceleration and angular velocity data, respectively.  Alternately, students can use the attached PocketLab movie to obtain this data.

The photo below shows the fan and mounted PocketLab.  An NSTA ruler has been added so that students can determine the radius of the circle in which PocketLab revolves.  From the photo, students should be able to (1) determine whether to use the x, y, or z component of acceleration to measure centripetal acceleration, and (2) determine whether to use the x, y, or z component of angular velocity to measure the angular velocity of the revolving PocketLab.
The figure below shows the base event timeline used when making the movie.  The fan is first turned on after which it reaches high speed after several seconds.  Then the fan is switched to medium speed, also reached after several seconds.  Then the fan is switched to low speed, reaching this speed after several seconds.  Finally, the fan is turned off, and begins to slow down.  The PocketLab accelerometer and gyroscope csv files will both show shapes similar to the figure below when selecting the appropriate x, y, and z components.
The movie used for all data collection can be seen below.

Using a 33-45-78 Turntable to Show that Centripetal Acceleration is Proportional to the Square of the Velocity and Inversely Proportional to Radius

PocketLab in conjunction with a 33-45-78 RPM turntable is an ideal setup for studying centripetal acceleration.  There are two videos that can be found in the Videos page of this web site.  They show that (1)  keeping radius constant implies that centripetal acceleration is proportional to the square of the velocity, (2) keeping velocity constant while varying the radius implies that centripetal acceleration is inversely proportional to the radius.

The PocketLab is placed in its silicone protective case to provide greater friction so that it doesn't slide off the turntable.  Y is toward the center of the turntable, and X is in the direction of rotational motion of the turntable. Single-graph mode is used with acceleration selected to be graphed.

From the accel X data in the videos students can then use Excel, Google Sheets, or any other analysis software to make graphs of (1) accel X vs. velocity-squared, and (2) accel X vs. the reciprocal of the radius.  Both of the graphs should be pretty close to straight lines, giving support for the desired outcomes.

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