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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.
 
Setup
 
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.
 
CloseUp
 
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.
 
angvel
 
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.
   

VelocityLab Investigation of Damped Harmonic Motion

This investigation shows how VelocityLab allows for a quick and easy demonstration of damped harmonic motion.  The photo below shows the experiment setup as performed by the author.  A jellied cranberry sauce can was selected as there is virtually no sloshing of the cranberry sauce as the can oscillates back-and-forth on a curved piece of laminate flooring.  The center of the flooring is clamped down to the table with an adjustable wrench.  The ends of the laminate flooring are raised a little with some small wood blocks.  The cranberry sauce can is shown at rest at the VelocityLab zero position.  PocketLab has been mounted to  one end of the can with some Velcro.
 
Setup
 
The video below shows a VelocityLab video combined with data for a typical run.  Note at the start of the video how the author carefully rolls the can to the top left side of the curved surface in a way that preserves the zero position and makes positions to the left of center negative while those to the right are positive.
 
 
The figure below shows a combined graph of position, speed, and acceleration vs. time. The graph was obtained by using Excel to massage the data in the pos_vel_acc.csv file produced by VelocityLab.  A number of questions about the video and graph are worth discussing with the students including:
 
1. What can be said about speed and acceleration when the cylinder is at maximum amplitude on either side of the curved surface?
 
2. What can be said about the speed and acceleration when the cylinder is rolling while at the center of the curved surface?
 
3. What happens to the magnitude of the maximum amplitude of position, velocity, and acceleration with each successive oscillation of the cylinder?
 
Graph of Position, Velocity, and Acceleration

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.
 
Diagram
 
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.
 
Setup
 
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.
 
Graphs
 
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.  
 
    

Investigating Gay-Lussac's Law and Absolute Zero of Temperature with PocketLab and a Mason Jar

Gay-Lussac's Law states that when the volume of a container of gas is held constant, while the temperature of the gas is increased, then the pressure of the gas will also increase.  In other words, pressure is directly proportional to the absolute temperature for a given mass of gas at constant volume.  Although this is, strictly speaking, true only for an ideal gas, most gases that surround us behave much like an ideal gas.  Even ordinary air, which is a mixture of gases, can behave like an ideal gas.
 
In this experiment, a PocketLab that is sealed inside a Mason jar can be used to verify Gay-Lussac's Law as well as extrapolate a value for the absolute zero of temperature. The PocketLab is set to "Two-Graph" mode, recording pressure in mBar and temperature in celsius degrees.  Considering the PocketLab specifications  for the temperature sensor, it is seen that the allowed range is from -20C to 85C.  It would be perfect if we could measure the pressure of the air in the Mason jar for three different temperatures covering the allowed range.  The photos in the figure below show three such possibilities.
 
Three Temperatures
 
The photo on the left shows PocketLab sealed in a Mason jar on a table at room temperature, about the middle of the allowed range.  The photo in the middle shows the PocketLab Mason jar in a freezer, which will give us a temperature near the low end of the allowed range.  The photo on the right shows the PocketLab Mason jar in an oven set to a maximum temperature of 170F (77C), just a little below the high end of the allowed range.  It took about an hour for the PocketLab temperature sensor to reach the desired values in the freezer and in the oven, so patience is required.
 
For safety, protective goggles should be worn.  In addition, gloves should be worn when removing the jar from the freezer, as it is cold enough to cost frost bite if handled too long.  Gloves should also be worn when removing the jar from the oven, as it will be at a temperature that is not too far from that of boiling water.  It is also essential to monitor the temperature on the iPhone to make sure that it doesn't exceed the high end of the allowed range.  It should be removed from the oven and the oven turned off a little before reaching the high end of the allowed range.  For the author, the stainless steel freezer and oven did not stop PocketLab from communicating data with the iPhone setting on a nearby counter.  The author also kept the iPhone charging cord attached during the experiment to avoid running out of charge on the iPhone battery.
 
The Excel graph shown below summarizes the experimental results.  The three data points fall very close to a straight line obtained by doing a linear trend/regression.  The line is extended to the left until it reaches the temperature axis.  At that temperature, -233C, the pressure would be zero.  The value -233C can be obtained from the regression equation by setting y to 0 and solving for x. With the absolute zero of temperature at -273.15C by international agreement, our value of -233C represents an error of about 14.7%.  It would likely promote a good classroom conversation to discuss possible causes for this error.
 
Gay-Lussac Law and Absolute Zero         
 
 

Quantitative Experiment to Determine the Relationship Between a Pendulum's Length and Period

PocketLab is a perfect device for determining the quantitative relationship between the length of a pendulum and its period of oscillation.  Pendulums of known lengths were made from balsa wood strips such as those available from Michaels and other hobby stores.  The photo below shows six such pendulums of lengths 15, 30, 45, 60, 75, and 90 cm alongside a meter stick.  The picture shows that PocketLab was taped with double-stick mounting tape to the pendulum whose length is 45 cm.
 
Balsa Wood Pendulums
 
The photo below shows the apparatus setup. The balsa wood pendulum with PocketLab attached is hung with masking tape from a ring stand supported by the weight of several books to keep it stable.  The orientation of PocketLab shows that it will be swinging in the XZ plane.  Therefore, the Y angular velocity data will provide information necessary to compute the period of the pendulum.
 
Setup
 
The video below shows the PocketLab graphs superimposed on the actual moving pendulums.  The Y angular velocity, shown in blue, contains the data of interest in the analysis.
 
 
The Excel graph below shows the Y angular velocity in deg/s obtained from the gyroscope data file for the pendulum of length 15 cm.  (Note that half the length (3.2 cm) of PocketLab is subtracted from 15 cm, giving a length of 11.8 cm to the approximate center of mass of PocketLab.)   As shown in red in the graph, the period is calculated by averaging the time for ten complete oscillations.  This process is repeated for all six pendulum lengths.  All gyroscope data for these six pendulums can be found in the attached gyroscope Excel file.
 
Angular Velocity vs. Time
 
Below is shown an Excel chart the summarizes the results of the experiment.  When a power regression type is applied to the data, it is seen that the power turned out to be 0.4773, very close to the theoretical 0.5 expected for such a pendulum.  It can be concluded that the period of a simple pendulum is proportional to the square root of the length of the pendulum.
 
Pendulum Period vs. Length
 
    

Determining the Radius of Curvature of a Gradual Street Turn

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.
 
Setup
 
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.
 
Aerial View of Route on Bing Map
 
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.
 
Angular Velocity Chart
 
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.
 
Centripetal Acceleration 
 
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. 

Investigating Boyle's Law with PocketLab

With a pressure sensor built into PocketLab, there must surely be some way to investigate Boyle's Law.  This law states that pressure and volume of an ideal gas are inversely proportional to one another provided that the temperature and amount of gas are kept constant within a closed system.  What is needed is a closed system that is large enough to hold PocketLab in a way that pressure can be sensed while changing the volume of the enclosed gas (in our case, air).
 
Educational Innovations, Inc. (teachersource.com) currently sells a Microscale Vacuum Apparatus for $35.95 (as of 8/9/2016) into which PocketLab easily fits.  The photo below shows this apparatus with PocketLab inside the polycarbonate bell jar.
 
Vacuum Pump Setup
 
The student can determine the volume of the bell jar by a variety of methods.  It turns out to be about 220 cc.  An estimate of the volume of the air within the attached hose is about 2 cc.  Estimating the volume of the solid parts of PocketLab that displace air in the bell jar to be about 12 cc, the volume of the trapped air is then approximately 210 cc when the syringe is at the zero mark.
 
Before collecting data with PocketLab's pressure sensor, the piston in the syringe is pumped enough times to bring the pressure down to approximately 300 mBar.  Then the piston is pulled out and held at the 5 cc (or ml) mark for a few seconds.  Then it is pulled further to the 10 cc mark and held for a few seconds.  This process is continued by steps of 5 cc through the 30 cc mark.  Meanwhile, PocketLab records the pressure of the trapped air, as shown in the video below.  We therefore know the pressure readings and the corresponding volume readings (210 cc + 5 cc = 215 cc; 210 cc + 10 cc = 220 cc, etc., through 210 cc + 30 cc = 240 cc).
 
 
An Excel chart of the data is shown in the figure below.  It clearly shows that increasing pressure results in decreasing volume--some kind of an inverse relationship.  But what is the best trend/regression type fit for this data?
 
Collected Data
 
When linear, exponential, and power trend/regression types are applied in Excel, all there types provide R-squared value that are very close to one, ordinarily indicative of a good fit.  See the three figure below for these charts.
 
Linear Trend
Exponential TrendPower Trend
 
As we can see in all three charts, our data is but a tiny portion of the spectrum of (volume, pressure) pairs possible.  This is because of the large space required to hold PocketLab and its pressure sensor.  The collected data would likely show more of a curve if this space was small compared to the volume of the syringe.  Students can be encouraged to provide intuitive arguments against the linear and exponential trends (questions about what happens if the pressure is zero or the volume is zero--can any finite quantity of gas occupy zero volume?)
 
This leaves us with the power trend/regression as the only reasonable choice.  The exponent is shown as -1.157, close to the value -1 that we would expect if volume and pressure were inversely proportional (i.e., if PV = constant). 
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