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Magnetic Field on the Axis of a Current Loop

In this lesson students will find that a current-carrying loop can be regarded as a magnetic dipole, as it generates a magnetic field for points on its axis.  The figure below shows a diagram and the equation for the magnetic field B.  Derivation of this equation requries knowledge of the Biot-Savart Law, calculus and trigonometry.  But in this lesson we are interested only in comparing experimental results from PocketLab's magnetometer to the theoretical equation in the figure below.  More advanced students can consider derivation of the equation, if they wish.
 
Diagram and Theory
 
There are many ways that you can make a current loop.  The author used a plastic ribbon spool approximately 3" in diameter and 3/4" wide, and then wrapped 10 turns of insulated wire around the spool.  The ends of the wire were connected to a DC power supply that supplied constant current for the current loops.  The photo below shows PocketLab with its magnetometer centered in the middle of the spool on the axis of the spool.  PocketLab's magnetic sensor is located about 0.5 cm in from the its edge, shown by the black X drawn on PocketLab.  PocketLab is set to provide magnetic field magnitude data and is zeroed when there is no current in the wire loops.
 
PocketLab at x=0 Position
 
The two photos below provide two more views of the apparatus setup.  It is important to keep the magnetic sensor on the loop axis, as it is moved to known distances from the center of the loop.  The author used some small blocks of wood for this purpose.  A meter stick with its zero point at the center of the loops allows moving PocketLab gradually outward, increasing the value of x by one cm for each move of PocketLab.  The author's setup used a current of i = 5.12 amp, R = 0.0361 m, with x varying from 0.00 m to 0.10 meters by steps of 0.01 m each.  The number of loops N = 10.
 
Pocket Lab along Axis
 
Axis View
 
The graph shown below, constructed in Excel, was obtained from data in the magnetometer.csv file produced by the PocketLab app.  The horizontal plateaus are labeled with the average value of the magnetic field for each plateau.  These averages could be obtained using Excel, but are much easier and quicker to obtain using Logger Pro, an exceptional educational data analysis software package produced by Vernier Software & Technology (vernier.com).  The highest plateau is where PocketLab was at x=0.  The next plateau is for x=0.01 m, the next for x=0.02 m, and so on, through x=0.10 m.
 
Magnetic Field vs. Time
 
The graph below, produced in Excel, summarizes the experimental results for magnetic field vs. distance along the axis, as they compare to the expected results from the theoretical equation.  A good discussion would be for students to suggest reasons for discrepancies between theory and experiment.
 
Experiment vs. Theory

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. 
 
  

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.
 
Setup
 
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.
 
Angular Velocity Graph
 
 

A Quantitative PocketLab Study of Momentum, Impulse, and Force in the Collision of Two Carts

You don't need an expensive air track to do a quantitative study of momentum, impulse, and force involved in the collision of two carts.  You can get very good results by the use of two PocketLabs, two iPhones, and a pair of carts from the PocketLab Maker Kit.
 
As shown in the picture below, each cart has a PocketLab mounted on one of its wheels, so that the z-axis is the axis of interest when the carts are moving.  You can only connect one PocketLab to an iPhone at time, thus the need for two iPhones, each running the VelocityLab app.  Since we are interested only in velocity, it really makes no difference where on the track the zero position is for each of the carts.  The track, made from a pair of rails with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This helps to keep the carts on "the straight and narrow" as they move. Styrofoam bumpers at the ends of the tracks keep the carts from falling off of the table.  The collision is buffered by an 8" cable tie that is stuck (using thick double stick mounting tape) to one of the wood blocks that have been taped to each cart.  Cart B has a larger extra block of wood taped to it to give it higher mass.  The carts are given initial pushes in any direction desired.  The VelocityLab apps will display movement to the right as positive velocity and movement to the left as negative velocity.  The velocity graphs will likely not be synced in time, as it is quite difficult to start the VelocityLab apps on both iPhones at exactly the same time. But this is OK, as we only need to know the initial and final velocities as well as the difference between end and start times for the collisions.  Knowing these things will allow us to compute impulse and momentum changes as well as the average force during the collision.
 
Setup
 
The Excel table in the figure below provides all of the detailed calculations for four different scenarios for cart A and cart B.  All of the items in blue are raw data that are quickly obtained from a balance and the VelocityLab apps.  The impulse time should be the same for both carts, so the white column shows the average impulse time which is used in computing the force, shown in the red columns.  The green columns show the initial and final system (cart A and cart B combined) momenta.  The yellow columns show the change in momenta for each of the carts.  We note that for each of the four trials:
 
(1) The initial and final system momenta are reasonably close to being equal, as we would expect from the law of conservation of momentum.
(2) The change in the momenta for each of the carts are similar in magnitude but opposite in sign.
(3) The average force acting on each of the carts is also similar in magnitude but opposite in sign, as we would expect from Newton's Third Law of Motion.
 
Example Collision Scenarios
 
As an example, the velocity vs. time graphs below show the details associated with trial 2 in the above table.  From the graphs and the red markup shown on the graphs, it should be clear how initial and final velocities as well as impulse times were obtained for each of the two carts.
 
Velocity Graphs
 
As can be seen from the equations in the table showing the four trial results, the calculations are straight-forward but a bit tedious.  Therefore a copy of the Excel spreadsheet is provided as an attachment.  Your students can simply do scenarios of their own choosing and enter raw data into the blue columns.  Excel will then do all the calculations for the remaining columns.     

Conservation of Momentum When Two Carts "Explode"

Do you have two carts from the PocketLab Maker Kits?  Do you have two PocketLabs?  You probably have at least two students in your physics class with iPhones.  Do they have the VelocityLap app installed on their iPhones?  Then you have the major components for your students to investigate conservation of momentum when two carts on a track "explode".
 
As shown in the picture below, each cart has a PocketLab mounted on one of its wheels, so that the z-axis is the axis of interest when the carts are moving.  You can only connect one PocketLab to an iPhone at time, thus the need for two iPhones, each running the VelocityLab app.  The carts are shown in their zero positions at the center of a track on a table.
 
Setup
 
The track, made from a pair of rails with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This helps to keep the carts on "the straight and narrow" as they move.  The "explosion" is provided by a cable tie that is stuck (using thick double stick mounting tape) to one of the wood blocks that have been taped to each cart.  The carts are pushed together, compressing the cable tie, and then released. The stored elastic potential energy of the cable tie provides the energy for the
"explosion", causing the two carts to move off in opposite directions.

The photo below shows a closeup of the cable ties.  As can be seen from the NSTA ruler, they are about 8" long before looping the tie.  The looped cable tie that was used in the momentum experiments is also shown in the photo.
 
Cable Ties
 
The total momentum of the system is zero before the explosion.  If momentum is conserved, we would expect it to be zero after the collision as well.  The figure below shows the velocity graphs obtained from data from the VelocityLab app for both carts A (on the left) and B (on the right).  These two graphs are really the only ones needed as they provide the velocity of the carts after the explosion.  These two graphs will likely not be synced in time as it is quite difficult to start the VelocityLab apps on both iPhones at exactly the same time. But this is OK, as we only need to know the final velocities and not the time at which they occurred.
 
 Velocity Graphs
 
The mass of cart A is 0.159 kg.  The mass of cart B, which has an extra wood block taped to the cart, is 0.223 kg.  The most important data on the charts have been highlighted with a star.  The graph shows that the velocity of cart A just after the explosion is -0.323 m/s.  Therefore, its momentum  = mv = -0.051 kg-m/s.  Similarly, the graph shows that the velocity of cart B just after the explosion is 0.244 m/s, and has a momentum = pv = 0.054 kg-m/s.  The sum of the final momenta is 0.003 kg-m/sec, very close to the initial zero momentum of the system before the explosion.  We have good evidence that the law of conservation of momentum is conserved even in explosions!  In explosions the vector sum of the exploded pieces is the same as the initial momentum of the system.  It is certainly worth while to ask students to think of real world examples of such explosions.
 
Below is a video of a typical run of the experiment:
 
 
 
 

A Classic Conservation of Momentum Experiment with PocketLab

A well-known conservation of momentum experiment that has been around for many years involves dropping a brick onto a horizontally moving cart.  With PocketLab and the VelocityLab app, your students can perform this experiment easily with the cart from the PocketLab Maker Kit and a small block of wood.  The snapshot below shows the setup with the author about to drop the block of wood onto the cart coming from the left.  A pair of rails, with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This is optional but does help to keep the cart on "the straight and narrow". 
 
Ready to Drop Wood Block on Cart
 
The video below shows what a typical run of the experiment looks like.  The cart is given an initial push at the far left end of the track, receives the wood block near the middle of the track, and then the combined cart and wood block hit the styrofome bumper at the right end of the track.
 
 
The figure below shows position, velocity, and accelertion vs. time graphs that were constructed in Excel from data obtained from a .csv file created by the VelocityLab app.  The graphs are marked up in red, noting how the motion of the cart applies to the graphical interpretation of the data.

The two most important pieces of information needed to verify the law of conservation of momentum are identified with stars in the velocity graph: (1) the speed of the cart just before receiving the dropped wood block, and (2) the speed of the combined cart and wood block just after the receiving the wood block.  (Note that the mass of cart is 0.144 kg and the mass of the wood block is 0.096 kg.)
 
Position, Velocity, and Acceleration Graphs
 
With all of the above information, it is quite easy to compute the momentum of the system right before the cart receives the wood block [p = mv = 0.144 kg x 0.841 m/s = 0.121 kg-m/s], and right after the wood block is dropped [p = mv = (0.144 kg + 0.096 kg) x 0.524 m/s = 0.126 kg-m/s].  The momentum before and after agree within about 4%, providing good verification of the law of conservation of momentum.
 
(A VelocityLab.csv file for a typical run of this experiment is attached for those who may be interested in viewing it.)

A Momentum Conservation Experiment for an Inelastic Collision Between Two Carts

Do you have two PocketLab Maker Kit carts, and do you have the free VelocityLab app?  Then you are all set to do some experiments in conservation of momentum with PocketLab!  This lab discusses how to setup and perform an inelastic collision in which one cart (A) is moving toward another cart (B) that is at rest.  When cart A hits cart B, they stick and move off together.  The photo below shows the two carts shortly before the collision would occur.  PocketLab is mounted on a front wheel of cart A.  Small pieces of wood are stuck to the carts and protrude further than the wheels.  Some thick double-stick mounting tape is attached to the ends of the wood that overhang the wheels.  When the carts collide, the tapes stick together, providing an inelastic collision.  In such a collision some of the kinetic energy of the carts is converted into other forms of energy, including heat and sound.
 
Closeup of the Two Carts
 
The figure below shows the setup used by the author to study conservation of momentum.  A pair of rails, with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This is optional but does help to keep the carts on "the straight and narrow".  Cart A is given an initial push at the far left end of the track, while cart B is at rest near the middle of the track.  After sticking together, the carts hit the white Styrofoam bumper at the right end, recoil some and then stop.
 
Setup
 
The figure below shows position, velocity, and acceleration vs. time graphs that were constructed from data obtained from a .csv file created by the VelocityLab app.  The graphs are marked up in red, noting how the motion of the carts applies to the graphical interpretation of the data.

The two most important pieces of information needed to verify the law of conservation of momentum are identified with stars in the velocity graph: (1) the speed of cart A just before colliding and sticking to cart B, and (2) the speed of the combined carts just after sticking together.  (Note that the mass of cart A is 0.159 kg and the mass of cart B is 0.102 kg.)
 
VelocityLab Graphs
 
With all of the above information, it is quite easy to compute the momentum of the system right before the collision [p = mv = 0.159 kg x 0.802 m/s = 0.128 kg-m/s], and right after the collision [p = mv = (0.159 kg + 0.102 kg) x 0.453 m/s = 0.118 kg-m/s].  The momnetum before and after agree within about 8%.  Calculations of the kinetic energy of translation reveal that about half of this kinetic energy is lost:  0.051 J just before the collision, and 0.027 J just after the collision.
 
The VelocityLab.csv file has been attached for those that may be interested in it.  The video below is a combined video and data, showing graphs of position, velocity, and acceleration in sync with the motion of the carts.
 

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.
 
Close-up of Yo-yo
 
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.
 
Setup
 
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.)
 
Yo-yo Angular Velocity Graph
 
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.
 
Extrapolating with Logger Pro

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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.  
 
    
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