User Lesson Plans

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). 

Investigating Ampere's Law for a Long Current Carrying Wire

One of the classes of problems dealing with magnetic fields concerns the production of a magnetic field by a current-carrying conductor or by moving charges.  It was Oersted who discovered back in the early 1800's that currents produce magnetic effects. The quantitative relationship between the magnetic field strength and the current was later embodied in Ampere's Law, an extension of which made by Maxwell is one of the four basic equations of electromagnetism.
 
PocketLab, used in conjunction with a long, straight current carrying wire, offers a great opportunity for students to quantitatively study the relationship between magnetic field strength B and (1) the current i in the wire and (2) the distance r from the wire's center.  Students will be able to confirm the experimental results that
bformula1
The picture below shows the experimental setup used by the author.  A long, straight wire (red) is strung from a ring stand to the floor, and the loose ends of the wire are attached to a DC power supply that allows varying the current as desired.  The current value is shown in amperes in the right-most digital display on the power supply.  The table allowed pulling the sections apart so that the wire could be in the center of the table.  The wire could just as well have been placed along the outer edge of the table.  A ruler is placed on the table zeroed at the center of the wire.  PocketLab can then be placed at the desired distance from the wire.
Experiment%20Setup2
The figure below shows a close up of PocketLab, the NSTA ruler, and the wire.  The ruler is zeroed on the center of the wire, and PocketLab is shown with its left edge at the 3 cm mark on the ruler.  Since PocketLab's magnetic sensor is located about 0.5 cm in from the left edge, shown by the black X drawn on PocketLab, the distance r from the wire in this photo would be 3.5 cm.  PocketLab is set to provide magnetic field magnitude data and is zeroed when there is no current in the wire.
Measuring%20Distance%20from%20Wire3
VARYING CURRENT WHILE KEEPING DISTANCE CONSTANT
 
The video below contains data for magnetic field magnitude while varying the current, but keeping the distance r constant at 1.5 cm throughout.  Data from this video or the attached magnetometer file can be used in Excel to obtain a chart of B vs. i.
The author's Excel chart below clearly shows that the magnetic field B is directly proportional to the current i in amps.  The linear trend/regression fit provides an R-squared value of 0.9999.  It is seen from the linear regression equation that the magnetic field increases by about 7.3 microT for each ampere increase in current.
constantdistance5
VARYING DISTANCE WHILE KEEPING CURRENT CONSTANT
The video below contains data for magnetic field magnitude while varying the distance, but keeping the current constant at about 6 amps throughout.  Data from this video or the attached magnetometer file can be used in Excel to obtain a chart of B vs. r.
The author's Excel chart below clearly shows that the magnetic field B is inversely proportional to the distance r in cm.  The power trend/regression fit provides an R-squared value of 0.9981.  It is seen from the power regression equation that the power is -1.072, very close to the -1 expected for an inverse first-power proportionality.
constantcurrent7

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.
Paddle%20Fan%20with%20Ruler1
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.
timeprogression2
The movie used for all data collection can be seen below.
 

PocketLab Joins Ozobot to Study Position, Velocity and Acceleration Concepts

Ozobot (ozobot.com) is a tiny one inch diameter line-traveling robot that can be used in conjunction with PocketLab to easily study the physics concepts of position, velocity, and acceleration and their time graphs.  PocketLab is simply taped to the top of an Ozobot using double-sided mounting tape.  In other words, Ozobot gives Pocket lab a ride.  The photo below shows this setup, with Ozobot following a 1/4" heavy black line drawn with a chisel tip marking pen.

Ozobot%20Carrying%20PocketLab

A magnetic ruler can be easily constructed to capture position/time information on the Ozobot/PocketLab duo. The photo below shows the magnetic ruler.  Small neodymium magnets are taped 15 cm apart on a stick that can be purchased at hobby shops such a Michaels.  PocketLab is set to record values of magnetic field magnitude.  As the PocketLab/Ozobot pair travel along the line, the magnetic field magnitude rises to a peak when reaching each of the magnets on the ruler.  The data file created by the PocketLab app can then be used to determine the times for each of the peaks.  With position and time known, a graph of position and time can then be constructed, perhaps in Microsoft Excel.

Magnetic%20Ruler2

CONSTANT SPEED EXAMPLE
 
The movie below shows data captured when Ozobot/PocketLab move at a constant speed along the line.
The two charts below were created using Excel with raw data from the PocketLab app magnetometer magnitude.  It is seen that the peaks are very close to being equally spaced in time, roughly 2 seconds apart.  With the distance between magnets fixed at 15 cm, velocity is therefore constant. The slope of the position versus time chart tells us that the velocity is about 7.546 cm/s, as shown by the linear regression data from Excel.  The Excel file is attached for anyone interested in viewing its details.
maggraphconstspeed3
positionvstime4
CONSTANT ACCELERATION EXAMPLE
Ozobot can be programmed using OzoBlockly (ozoblockly.com) in a way that causes Ozobot to accelerate rather than travel at constant velocity.  The OzoBlockly program shown in the figure below was used in this investigation.  Ozobot begins at a speed of 35 mm/s and then increases as it approaches each of line intersections at the magnets by 10 mm/s.  Because Ozobot is carrying the weight of PocketLab, the speeds are actually somewhat less.
ozoblockly%20program5
The movie below shows data captured with Ozobot/PocketLab traveling with  acceleration.  You will note that as its speed increases, the distance between peaks of magnetic field magnitude decrease.
The two charts below were created using Excel with raw data from the PocketLab app magnetometer magnitude.  Again, we note that the peaks are more closely spaced as time progresses.  The thin black straight line on the position vs. time graph clearly shows that the curved blue line implies acceleration.  When a linear fit is done on the velocity vs. time chart, we find that the average acceleration is about 0.2932 cm/s/s.  The Excel file is attached for anyone interested in viewing its details.
magrgaphaccel5positionvelocityvstimel7

The Inverse Cube Law for a Neodymium Dipole Magnet

PocketLab makes is quite easy to investigate and verify the inverse cube law for the magnetic field of a neodymium magnet as a function of distance from the magnet.  All that is needed in addition to The PocketLab is a centimeter ruler, small neodymium magnet, a small block of wood and a little double stick tape.  The photo below shows how the neodymium magnet is taped to the block of wood with the magnet located at the 10 cm mark on the NSTA ruler.  The height of the center of the magnet is at about the height of the circuit board inside of PocketLab.  The X on the front face of PocketLab is very close to the location of the magnetic field sensor inside of PocketLab, 0.5 cm from the left edge of Pocket Lab, in line with the Y-axis of PocketLab.

Setup%20View%202

The photo below shows the set up from above with the left edge of Pocket lab at the 15 cm mark.  The distance between the dipole and the sensor is therefore about 5.5 cm in this photo.

Setup%20View%201

In preparation for data collection, PocketLab is set to display magnetic field magnitude.  It is then moved far from the neodymium magnet and zeroed.  It is then placed at the 12 cm mark on the ruler, making the distance between magnet and sensor 2.5 cm.  After a few seconds, PocketLab is moved to the 13 cm mark, thus increasing the distance by 1 cm to 3.5 cm.  This process is continued through a distance of 8.5 cm.  The magnetic field magnitude can be read directly from the movie, shown below, at each of the known distances.

(Distance, magnetic field magnitude) data pairs are then entered into an Excel spreadsheet, and a chart of Magnetic Field vs. Distance is created.  The chart, shown below, appears to show some sort of an inverse relationship between magnetic field and distance.  The Excel "Add trendline" feature is then used and the "Power" regression fit is applied.  It is found that the power is -2.832, very close to the -3 expected for an inverse cube relationship.

excel%20data%20and%20chart

A copy of the Excel spreadsheet is included in the attachments for anyone interested in viewing it.

PocketLab on a Skier's Edge Machine

The PocketLab is an ideal device for measuring user performance for a variety of exercise equipment.  One example of such equipment is the Skier's Edge, whose company was founded in 1987.  This machine was designed for non-impact lateral conditioning that simulates the experience of downhill skiing.  The photo below shows the skiing machine.  The skier stands on the two black platforms, holding poles and moves the carriage back-and-forth on the curved white tracks.

SkiersEdgeFullCropped

A close-up view of the carriage in the photo below shows that a Pocket Lab has been mounted to the carriage with tabs provide in the PocketLab Maker Kit.  The carriage moves back-and-forth on the curved track in the XZ plane.  Therefore, the Y angular velocity would be a variable of interest to measure.  In addition, the X acceleration would be of interest as it is the major component of the back-and-forth motion provided by the skier's legs.

CloseUp

An iPhone snapshot of the data and video combined is shown below.  The acceleration graph (red) shows that the maximum acceleration is about 4g.  This is a true measure of the skier's power.  The angular velocity graph (blue) shows that the maximum angular velocity is about 75 degrees/sec.  From study of the time axis, both graphs show that the  back-and-forth movements of the skier has a frequency of about 75 per minute.  Increasing this rate while keeping the amplitude of the swings the same would suggest that the maximum g "force" could be increased for a more powerful skier.

ScreenCapture

The action movie shown below includes an overlay of both the acceleration and angular velocity graphs, with maximum acceleration occurring at the ends of the back-and-forth motion, and maximum angular velocity occurring at the center of each swing.



PocketLab Investigation of Fuel Efficiency

With gas prices as high as they are and having a growing concern for the environment, Americans today are becoming conscious about things they can do to improve fuel efficiency.  Many realize that driving at the posted speed limits promotes both safety and reduces the rate at which fuel is consumed.  With these things in mind, some have purchased hybrid vehicles including the Toyota Prius, all-electric vehicles such as the Nissan LEAF, or range-extending vehicles such as the Chevy Volt.  Those with EV's soon realize that they get more miles per charge if they avoid driving at excessively high speeds on the open road.
 
With this background as a base, it would be very instructive if students could perform a laboratory investigation that would provide a quantitative relationship between distance per unit of fuel consumed (a measure of efficiency) and speed.  One way to accomplish this is by investigating these factors using an N-scale electric train set, such as those sold by Bachman for around $100, and commonly available in toy and hobby stores.  In this case, fuel is the unit of electrical energy, i.e. the kilowatt-hour (kWh), or on the scale of a hobby train, more appropriately the watt-second (W-s).  The picture below shows a train set, with a power supply providing the electrical power to run the train,  The power supply provides readings for voltage (V) in volts and current (I) in amps, from which power can be calculated by the product VI.  A meter stick is used to measure the diameter of the circular train track, from which the radius is found to be 0.285 m.  PocketLab is mounted to the top of the engine using tabs that come with the PocketLab Maker Kit.
Setup (2)
Starting with a low voltage, the train was allowed to run for about one lap.  This process was repeated for a sequence of voltages to slightly above the train manufacture's recommended limit of 16 volts.  Voltages can be read from the video.  PocketLab was set to provide angular acceleration.  With PocketLab mounted on the engine with the train moving in the XY plane, Z angular acceleration (shown in green in the video) is the variable of interest, telling us the number of degrees per second that the train revolves on its circular track.  It is important to make sure that angular acceleration is zeroed prior to capturing data and video with the PocketLab app.
The image shown below contains a graph of Z angular velocity vs. time from data produced by the PocketLab app.  The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  The statistical analysis capability of Logger Pro is used to obtain mean angular velocity for each of the voltage steps during the experiment.  This is an extremely useful capability as the angular acceleration does vary a fair amount at each step level.  For example, the graph shows that the mean Z angular velocity for the second voltage step was 32.59 degrees/s. The step nature of the Logger Pro chart is a result of the voltage being increased to a new level after a few laps have been made by the train at a given voltage.
logger
The Microsoft Excel table below summarizes all of the raw data as well as come calculated columns containing power, period, speed, energy per lap, and efficiency.  A copy of the Microsoft Excel file is attached so that you can look at the formulas used in these calcualted columns.  Power is voltage times current (P= VI).  Period is 360/angular velocity.  Speed is 2*Pi*R / period.  The energy per lap in W-s is power multiplied by period.  Efficiency in Laps per W-s is the reciprocal of Energy per lap.  Therefore, efficiency for our N-scale train is measured in laps per W-s.  This is analogous to mpg for a traditional gas consuming car, or to miles per charge or miles per kWh for an EV.
datasheetexcel
The Microsoft Excel table below summarizes all of the raw data as well as come calculated columns containing power, period, speed, energy per lap, and efficiency.  A copy of the Microsoft Excel file is attached so that you can look at the formulas used in these calcualted columns.  Power is voltage times current (P= VI).  Period is 360/angular velocity.  Speed is 2*Pi*R / period.  The energy per lap in W-s is power multiplied by period.  Efficiency in Laps per W-s is the reciprocal of Energy per lap.  Therefore, efficiency for our N-scale train is measured in laps per W-s.  This is analogous to mpg for a traditional gas consuming car, or to miles per charge or miles per kWh for an EV.
graph (1)
Questions for Students:
 
1.  Why is the efficiency of the train so low at low speeds?
 
2.  What are some possible explanations for the train's efficiency dropping after reaching the speed of peak efficiency?
 
3.  Considering a moving automobile as an analogy, (1) What is the efficiency of a car when it is at rest in a traffic jam and why?  (2) What factors reduce the efficiency of a typical car at very high speeds?

Using PocketLab to Investigate Newton's Law of Cooling

In this experiment students will use PocketLab to collect data related to the cooling of a container of hot water as time goes on.  Sir Isaac Newton modeled this process under the assumption that the rate at which heat moves from one object to another is proportional to the difference in temperature between the two objects, i.e., the cooling rate = -k*TempDiff.  In the case of this experiment, the two objects are water and air. Newton showed that TempDiff = To * exp(-kt), where TempDiff is the temperature difference at time t and To is the temperature difference at time zero.
 
In order to do this experiment on Newton's Law of Cooling using PocketLab, we need to wrap PocketLab securely in a plastic sandwich bag, so that water cannot leak into the bag and damage PocketLab.  The figure below shows how this can be done.  PocketLab is wrapped and taped tightly in the plastic bag.  It is immersed in water briefly to make sure that it has no leaks.  A piece of Velcro is attached as shown on the right of the figure.  The purpose of this Velcro is so that PocketLab can then be secured to the bottom of the hot water container without floating.
PLinBaggie
The figure below shows the complete apparatus setup.  PocketLab has been placed and secured to the bottom of a small bottle not much larger than PocketLab.  The bottle needs to be as small as possible, for even a small amount of water in the bottle can take several hours to cool back down to near room temperature again.  The bottle used by the author required only about 70 ml of water, but data was collected for close to two hours.  The room temperature is noted, hot tap water is poured into the bottle until it is full, and data collection is initiated with PocketLab set to record temperature once each second.  The purpose of the optional small block of foam is to insulate the bottle some from the table top, resulting in most heat loss into the air.
PLinWaterBottle
The image shown below contains a graph of temperature from data produced by the PocketLab app.  A red dot is shown once for every two-hundred data points.  The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  The initial rise in temperature is due to the time required for PocketLab to warm up as heat is transferred from the water to PocketLab.  Once equilibrium has been reached, the system begins to show cooling as heat is transferred to the surrounding air at room temperature.  The cooling appears to be negative exponential. The model fit Temperature = A*exp(-Cx)+B was then applied to the region of the graph shown in dark gray.  The correlation coefficient of 1.000 clearly indicates an excellent fit.  x corresponds to time in Newton's equation, A to the initial temperature difference, C to the constant of proportionality k, and B to room temperature.
tempgraph

PocketLab on an Oscillating Cart

An oscillating cart with a PocketLab provides an interesting way to study Newton's Second Law of Motion as well as some principles of damped harmonic motion.  The apparatus setup is shown in the figure below.  The small dynamics cart that can quickly be made from parts included in the PocketLab Maker Kit is shown in its equilibrium position.  Rubber bands are attached to each side of the cart and to two ring stands weighted down with some heavy books.  It is best to use rubber bands that provide as small Newton/meter as possible.  PocketLab is attached to the cart with its x-axis parallel to the rubber bands.

setup (1)

The close-up in the figure below shows that two small pieces of wire are threaded into holes in the cart with the rubber bands attached.  The ends of each wire are twisted together to tighten the rubber band on the cart.

IMG_2190

The movie below shows a typical run, with 20 data points per second and acceleration selected in single-graph mode.  The red trace on the graph is the acceleration of interest, namely acceleration in the X-direction.  The blue and green traces, representing acceleration in the Y and Z directions, are quite erratic due to slight jiggling of the cart, and are not of interest here.  The red curve shows a very regular pattern, in which it can be observed that the magnitude of the acceleration is greatest when the cart is at each end of its swing and zero in the center of the swing.  It is also noted that the magnitude of the acceleration decreases with time in a pattern that suggests exponential decay.

The image shown below contains a graph of x acceleration from data produced by the PocketLab app.  The graph was obtained by importing data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  A model involving the sine function and exponential function was created.  It is seen that the model (the black curve) follows the red acceleration X curve very well.

graph

Students can therefore conclude that this oscillating cart has a negative exponential decay with individual cycles characteristic of the sine function.  There are two constants of particular interest in the model equation shown in the gif image: Accel X = A*exp(-B*x)*sin(Cx+D)+E.  The constant C in the fit is 2*Pi/Period, from which we see that the period is 2*Pi/D = 2*Pi/11.61 = 0.54 s.  This agrees very well with the period obtained by direct observation of the graph.  The constant B in the fit is the reciprocal of the so-called lifetime.  Any exponential decay is characterized by its lifetime, which is the amount of time required for the amplitude to decay to 37% of its initial value.
 
By loading the cart with different masses and collecting PocketLab data on the resulting accelerations, students should be able to verify Newton's Second Law of Motion (Fnet = ma), showing that acceleration is inversely proportion to mass if the net force is held constant.
 
This experiment also provides a nice way to determine the period when the period of the oscillation is quite small, and difficult to measure with a stop watch.

Negative Exponentially Damped Harmonic Motion from a PocketLab Pendulum

This experiment allows one to do a quantitative investigation of the damped harmonic motion of a swinging pendulum.  The pendulum is a piece of wood about a yard long from a Michael's hobby shop one end of which has been attached to a PocketLab by a rubber band.  The other end is taped to the top of a doorway, allowing the resultant pendulum to swing back-and-forth as shown in the image below.

PivotRegion (1)

The y-axis is perpendicular to the XZ plane of the swinging pendulum.  Therefore, the main item of interest is the magnitude of the angular velocity vector in the Y direction (shown as a blue curve in the movie).

The image shown below contains a graph of the Y angular velocity (shown in blue).  The X and Z angular velocities (shown in red and green in the video, respectively) are small and erratic due to slight wobble in the swinging pendulum and are not included in the graph.  The graph was obtained by importing the data from the PocketLab app into Logger Pro, an exceptional educational scientific analysis software from Vernier Software & Technology (vernier.com).  A model involving the sine function and the exponential function was created. It is seen that the model (the black curve) follows the blue angular velocity curve very well.

dhm (1)

Students can therefore conclude that this damped harmonic motion has a negative exponential decay with individual cycles characteristic of the sine function.  There are two constants of particular interest in the model equation shown in the gif image: Angular Velocity = A*exp(-C*x)*sin(Dx+E)+B.  The constant D in the fit is 2*Pi/Period, from which we see that the period is 2*Pi/D = 2*Pi/3.657 = 1.72 s.  This agrees very well with the period obtained by direct observation of the graph.  The constant C in the fit is the reciprocal of the so-called lifetime.  Any exponential decay is characterized by its lifetime, which is the amount of time required for the amplitude to decay to 37% of its initial value.
 
You will note from observing the movie that the angular Y velocity is zero when the pendulum is at the two ends of its swing, and is a maximum at the middle of the swing, both of which are expected.  This is easier to see if you view the movie frame by frame.

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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