October 20, 2016: Collisions in two dimensions

Anthony Betancourt
Lab performed: Oct. 20, 2016
Professor Wolf

Lab #15: Collisions in Two Dimensions

Purpose:

The main objective of this lab is to determine if momentum and energy are conserved between two colliding objects.  The first collision is a steel ball with another steel ball bearing, the second collision is a steel ball with a glass marble.

Procedure:

1. Set up the glass table and a large lab stand with a 90 degree arm that is directly over the center if the glass table.
2. Setup up a camera or use a smartphone on the end of the 90 degree arm.
3. Grab two identical balls and one made of different material and mass.  Make sure they are the same in size.
4. For the first collision, place a ball in the center of the glass. Press record on the camera and roll the second ball toward the ball in the center.  
5. Once a clear video showing the collision between the two balls has been captured, repeat for the second collision with two different mass balls.  
6. When both videos have been captured, upload the videos into quicktime to shorten the length to just before and after the collision.
7. Once the videos are shortened, upload them into logger pro software and analyze each video and trace them.  
8. Logger pro can use the traced dots and place them on a graph of position vs. time
9. After the graphs are created, find the velocities for each ball.
10. Afterwards, you must derive a formula for the x-center of mass and y-center of mass.  Then make a graph of both center of mass vs. time graphs.
11. Determine if momentum is conserved.

Measured Data:

(Video analysis in logger pro software)

(video analysis of same balls; velocities before collision)

(video analysis; velocities after collision)

(glass and steel balls; velocity before collision)

(glass and steel balls; velocity after collisions)

(Sample calculations showing momentum and energy; x direction)

Analysis:

The experiment was fairly easy to perform.  The difficulty comes from the video analysis.  The best solution would be to zoom in on the glass table top with the camera used.  Aside from the technical difficulties, the data gathered was sufficient enough to give information on momentum and energy.  From the graphs we can conclude the velocity of the balls from the slope of the linear fit line in both the x and y direction.  The velocity we achieve can be used in the momentum and energy calculations to see if the experiment did turn out as expected.

Conclusion:

The experiment went as expected beside the difficulty with tracking the ball in logger pro video analysis.  Once the graphs were finished, it was fairly simple to calculate the moment and energy for the balls.  The uncertainties that were ran into included: the logger pro software crashing unexpectedly, aiming the balls correctly to get a good angle after the collisions, the camera set up could have been zoomed in to provide better close up footage, and difficulty in setting up the glass table to be completely.  

October 12, 2016: Ballistic Pendulum

Anthony Betancourt
Lab Partners: Josh Fofrich, Christine, Jarrod , Alex Reyes
Lab Performed: Oct. 12, 2016

Lab Activity: Ballistic Pendulum

Purpose:

The main objective of this lab activity is to provide a real world example of the conservation of energy and also conservation of momentum.  These concepts are used in order to relate the max height the ball and nylon block reach to the initial speed of the ball before the collision.  

Theory:

In order to determine the balls velocity before the collision we must relate the balls kinetic energy to the momentum transferred into the block that leads it to rise upward a certain angle.  

Procedure:

1. Measure and record the mass of the ball and block.  Tilt apparatus on its side over the scale in order to weight block.
2. Make sure the block is in line and level with the ball exiting from the spring loaded canon.  
3. Pull back on the plunger until its in the "locked and loaded" position. Be sure to take note on the locking position and continue using the same for the remainder of the lab.
4. Place the angle indicator at the 0 position.
5. Fire ball into the block.  Record the angle the block rises up to.
6. Repeat four or five times and get the average.  
7. Next, Setup ballistic apparatus on the edge of the lab table.  
8. Measure the distance from the barrel to the top of the lab table.  Measure the height of the lab table, Record measurements.
9. Fire the ball out away from the table's edge and note where it lands.  Place a carbon paper and white sheet in the area where it lands.  
10. Fire the ball several times until a cluster forms where the ball has landed.  Measure the distance from the table's edge to the cluster area where the ball landed. Record measurement.   

Measures Data:

(Measured Data)

(Ballistic Pendulum Apparatus)

(Calclations for initial velocity of ball)

(distance ball traveled from table top; 2.65m)

(cluster formed from impact of the steel ball; 2 near bottom are insignificant, practice shots)

(calculations for verifying launch speed)

(Calculation for uncertainty)

(Updated uncertainties calculations including the length of the string)

Analysis:

In order to find the initial launch speed of the ball we must relate the movement of the ball and block swing upward to some height, h, and the collision that occurs with the ball and block after the ball is fired.  We first use the conservation of momentum to derive an equation for the balls initial velocity. After we can use the energy of the system to transfer of kinetic energy into gravitational potential energy through some angle the block and ball rise.  The balls initial kinetic velocity derived and solved for, then this value input into the first equation for the momentum and finally solved for the initial speed the ball was launched at. In order to verify our initial speed, the ball is shot out from the table edge and measured the distance the ball traveled.  We can use kinematics to find the time for the balls decent in the y-direction. Once we have time, we use it for finding the velocity of the ball in the x-direction dividing the distance over the time.  Lastly, we find the uncertainties in the lab to account for any discrepancies in our values for the initial velocity.   

Conclusion:

The launch speed calculated using conservation of momentum and energy was slightly higher than the value using kinematics.  If we take into account for the uncertainties in measurements, the values are closer to the same.  The measurements from lab could be off from the length of the string, the apparatus being completely level on the table top, and accuracy of the measurements on the pendulum itself.  There are also uncertainties in the verification of the speed of the ball from firing off the lab table.  The measurement for the distance traveled could be off due to using two measuring sticks, the cannon itself being completely horizontal before firing, and the measurement for the cluster where the ball lands.  

October 10, 2016: Magnetic Potential Lab

Anthony Betancourt
Lab Partner: Josh Fofrich
Lab Performed: Oct. 10, 2016
Professor Wolf

Lab #13: Magnetic Potential Energy

Purpose:

The main objective of this lab is to utilize the lab equipment provided and establish a method for representing the force of magnets with an equation.  Through the use of two magnets, with repelling force, on an incline that is elevated on one end, and measurement between the separation of both magnets can lead to this equation.

Procedure:

1. Set up air track and glider on a level surface. be sure to have a stack on books or wooden block available to place under track to create incline.  
2. Make sure magnet on end of glider faces the magnet at the end of the track and that they repel each other. 
3. Measure the mass of the glider, record data.
4. Turn on the air pump to the track.
5. Place one book or block under the track to create a small inline. the glider will start to fall toward the end with the magnet.  
6. Using a smartphone, measure the angle of elevation on the track.
7. Allow time for the glider to stabilize. Once glider has stopped moving turn off air pump. 
8. The two opposing magnets have a distance between them, measure this distance and record data. 
9. Repeat steps 1 through 8 with varying inclines and record the data for each angle.  Be sure to include a range of angles both large and small.  

Measured Data:

(Air track set up)

(Angle of incline and separation distance between magnets)

(calculations for mgsin𝜃; Force of the magnet)

(power curve fit; x-axis is separation distance, y-axis is mgsin𝜃)

(Calculations for the function of Magnetic Potential with uncertainties, Umag)

(verifying conservation of energy)

Analysis:

Over the course of the experiment, the one issue that was most prevalent was getting the calipers in between the magnets to get an accurate measurement for the separation.  Another smaller issue was getting a variety of inclines to result in better range of separation distances.  One way we figured out to alleviate this problem was using the chairs in lab to position the track end on.  Care must be take to properly secure the track with a foot or weight so it doesn't slide off the chair.  Once enough data had been collected, we calculated the force of the magnets from the separation and mgsin𝜃.  The magnetic force and separation distance can be plotted on x and y-axis respectively to achieve a power fit curve.  The power fit curve gave us an equation with a correlation value of 0.8332 with uncertainties as shown above in picture 5.  The final picture verifies the conservation of energy as the glider moves toward the motion detector and being repelled by the opposing magnets and traveling back to its initial position.  

Conclusion:

This experiment was straightforward in procedure and the data that needed to be collected.  The main issues had were due to issues of achieving various angles, the measurement between magnets, and possibly plotting data into logger pro.  Uncertainties associated with the power fit curve are included above in the analysis.  Other uncertainties might be from the measurements taken to find the separation between the magnets, the data that was strike through in the graphing in order to achieve higher correlation value, and possibly the strength of the magnets used in the lab might differ from other lab groups magnets.  The more trials conducted will result in higher correlation value and also provide better data to choose from for graphing power fit curve.   

October 5, 2016: Conservation of energy for an oscillating mass/spring system

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #12: Oscillating Mass/Spring System

Purpose:

The purpose of this experiment is to provide a representation for the conservation of energy in a oscillating mass/spring system.  The experiment can show how a spring conserves energy in the form of elastic potential energy and kinetic energy while the mass oscillates.   The logger pro software can visually represent the energy graph as the mass moves up and down.

Procedure:

1. Mount a able clamp and long metal rod to the lab table.
2. Place 90 degree clamp and short metal rod near the top of the vertical metal rod.
3. Place force sensor on the end of the horizontal rod. Be sure to zero force sensor after placing spring on it.  
4. Place motion sensor on floor directly underneath the spring facing up.
5. Hang 50 gram mass hanger on the spring and zero the motion sensor.
6. Verify operation of motion sensor by collecting data and pulling hanging mass down ward and letting it oscillate and making sure graph shows appropriate shape.  

Theory:

We must be able to relate the ideas of kinetic and potential energy to the use of a oscillating mass/spring system.  This experiment is a simple and effective way of combing both concepts into  easily obtainable data.  The use of the hanging mass on the end of the spring can give us the spring constant, change in position from unstretched with the stretched length after hanging the 50 gram mass.  The kinetic energy of the system can be collected from using the motion sensor to track the oscillating mass and getting a graphical representation of its movement.  This graph can be further analyzed using logger pro's features.  

Measured Data:

(Diagram depicting set up for oscillating mass/ spring system)

(Graph showing position vs. time, velocity vs. time, KE vs. time)

(graphs of Kinetic, Elastic, and Gravitational Energies, all vs. time)

(graph depicting GPE, EPE, KE vs velocity)

(graphs of GPE, KE, EPE vs positions and vs. time; Also included is Energy Sum)

Analysis:

The first screen shot is of the mass oscillating back and forth in front of the motion sensor. The graph must be nice and smooth in order to be best utilize to derive energy graphs.  The third graph on that slide is a derived calculation of kinetic energy using "1/2 MV^2".  Second screen shot is of the three energies derived in logger pro: Elastic Potential Energy, Kinetic Energy, and Gravitational Potential Energy.  Same technique used to derive the kinetic energy graph was utilized for elastic and gravitational potential energy.  The graphs of EPE and GPE are opposite of one another since the EPE of the system is transferred into GPE when the spring is unstreched and vice versa when GPE is less and EPE is greater due to the stretch in the spring.  Third screen shot shows the graphs of each energy vs velocity.  The graphs look as such due to the oscillation of the mass and alternate from positive to negative.  The last screenshot has all three energy graphs vs. position and vs. time including the Energy sum.  The energies vs. position graph is to visually represent the conservation of energy in each one over the entirety of the system.  The energy sum graph should be a straight line with little to no slope depicting all the energies canceling each other out.  

Conclusion:

This experiment has a simple yet effective out come to derive the various energies and utilizing the logger pro software to analyze the graphs.  Where someone might have some difficulty is in the setup of the experiment.  It takes several trial runs to position the motion detector in the correct spot under the mass in order for the data collection to result a smooth graph.  Any other difficulty might be due to certain uncertainties in the lab equipment and procedural faults.  The spring utilized in this experiment was not a uniformed shape, possibly effecting the EPE due to skewed change in position from stretching.  In order to achieve a smooth graph, we had to place a larger mass on the end of the spring to help alleviate the shakiness and bouncing around from the spring oscillating.  Also with the heavier mass, the bar bent at 90 degrees near the top of the long metal bar starts to bend when oscillating possibly effecting the change in position for the spring.  Lastly, not properly securing the force sensor on the 90 degree bar can result in faulty force reading.  

October 3, 2016: Work-Kinetic Energy Theorem Acitivity

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #11: Work-Kinetic Energy Theorem

Purpose:

The purpose of this lab activity is to introduce to the student a real world representation of the work-kinetic energy theorem.  The lab utilizes dynamics carts attached to a spring at one end and to a force sensor on the other.  Logger Pro will be utilized to get a visual representation of the kinetic energy and calculate the work done.

Procedure:  

Experiment 1

1. Set up a dynamics track and cart on your lab table along with the logger pro box and laptop designated to your lab group.
2. Attach a motion sensor on one end of the track, pointed toward the cart, and attach the force sensor on the other end utilizing a track bumper.
3. Attach a spring between the cart and the force sensor.
4. be sure to zero the force probe and motion detector with spring un-stretched.  
5. open experiment file "L11E2-2 (Stretching Spring)" to display force vs. position graph. 
6. to begin graphing, move the cart toward the motion sensor until the spring is stretched to 0.6m.
7. Find work done by using integration routine in the software. 

Experiment 2

1. Use same set up as procedure above. 
2. Under calculated column, enter a formula that will allow you to calculate kinetic energy of the cart.
3. Make sure x-axis is "position".   Zero force probe and motion detector and set position and force at the starting position.
4. To begin graphing, pull back the cart and release so that spring returns back to it natural position.  you might have to strike through data cells that don't showcase a smooth curved graph.

Theory:

In order to find the work done by stretching the spring we have to use the logger pro software to analyze the force vs. position graph and integrate the area under the graph which will tell us the work done.  In the second experiment, in order to find the change in kinetic energy after the cart is released the graph displaying KE vs. position needs to be under the Force vs. position graph in order to track the kinetic energy at a certain position.

Measured Data:

(Experiment 1; top graph shows force of spring on cart)

(Experiment 2; Top graph shows work done by the spring; bottom shows kinetic energy at 0.236J)

(Experiment 2; Larger area under graph for work done; bottom is KE at 0.557J)

(Experiment 2; larger area of work done; KE is 0.461 J)

Analysis:

The first graph shows the force of the spring on the cart while being stretched approximately 0.6m.  The slope of this graph is the force constant of the spring.  The work done is the area underneath the graph.  The second, third, and fourth graph all show the different amounts of work done and the amount of kinetic energy at the different positions.  

Conclusion:

The work done on the cart by the spring increases as the kinetic energy of the cart decreases.  This shows us that work done by the spring on the cart is inversely proportional to the kinetic energy in the system.  The work energy theorem states that all the work done on a system is equal to the change in its kinetic energy.  As the cart gains kinetic energy from the spring force, the change in that kinetic energy becomes more apparent than when it was at rest, thus, affecting the total amount of net work done on the cart.  

October 3, 2016: Centripetal Force with a motor

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #9: Centripetal Acceleration with a Motor

Purpose:

The purpose of this lab is to familiarize the student with the acceleration forces involved with circular motion.  The apparatus used in this lab can replicate a centripetal acceleration acting upon an object suspended by a length of string (L) above the ground of a height (H) at a calculated angular velocity (omega) that is derived from counting rotations over a timed interval.  From the derived data, an angle (theta) can be derived along with a radius from the axis of rotation (R+X).

Procedure: 

1. Check to make sure the apparatus is setup for the lab.  Take measurements of the height of the swinging arm down to the floor (H).
2. Take the measurement of the length of string the rubber stopper is hanging from (L).  Take the measurement from the center of rotation of the swinging arm to the attached string (R).  
3. Turn on the motor to the apparatus and wait for it to reach a constant velocity. Once the velocity stabilizes, use a piece of paper attached horizontally to a lab stand and slowly move the paper up until the rubber stopper make contact, then move slightly back down. This measurement will be h.  
4. As the rubber stopper spins around, start timing how long it takes to complete 10 revolutions.  Take the number of revolutions times 2pi and divide by the time taken for all 10 revolutions and this will give you omega.  
5. Repeat steps 1-4 at different voltages to increase the speed of the motor, thus giving different height (h) measurements, different theta angles (angle created by string from attached point on swing arm), and different omega values.  Complete 6 different trials.  

Theory:  

As the rubber stopper moves in a circular motion, the voltage on the motor will dictate how fast the rubber stopper spins.  As the angular velocity increases, so does the angle (theta) at the point where the string is attached changes.  To establish the relationship between omega (⍵) and theta, a free body diagram can show the forces acting in the y and x direction.  The equations for sum of forces can be re-arranged to show the value of omega.  

Measured Data:

(Apparatus diagram with measurements)

(Derivation for omega)

(Spreadsheet table with measured data and 𝜔 calculations)

(𝜔 calculations from timing 10 rotations of the apparatus; Trial #1 is calculated)

(Correlation between timed omega and derived omega)

Analysis: 

Once the timed rotations are calculated for omega (10 rotations multiplied by 2𝞹 and divided by the time it took) those values are then compared to the values from the derived equation of 𝜔.  An Excel spreadsheet can easily correlate the two data sets into a graph with a linear trend line.  Overall, both data sets were very close in values but tend to differ as the speed of the motor was turned up.  The faster the rubber stopper went around, greater the angle that was created from the the attached string to the swinging arm.  The derived values for omega did not increase as rapidly as the timed values for omega.  Once the data values were compared, a correlation of .90589 resulted between the Derived vs. Timed omega values.  This correlation shows the relative accuracy for the timed portion of the experiment but with some added uncertainty in the end calculation. 

Conclusion:

In order to calculate a relatively accurate value for an objects angular speed, these two methods can be utilized.  The timed is accurate up to a certain extent.  As the object increases in angular speed, the method of counting the revolutions and timing them becomes less accurate as the speed increases.  This is due to the fact of having to watch the object rotating and count how many revolutions while simultaneously using a stop watch to timed the accurately. One way to increase the accuracy of this method for higher speeds, would be to allow the object to complete more revolutions while allowing more elapsed time and result in greater accuracy for average angular speed.  The derived equation for omega also has certain uncertainties associated with it.  One being the forces acting on the spinning object, such as air resistance, preventing the omega value to be as accurate as the calculated result.  Other uncertainties in this experiment could be from the swinging arm (the meter stick used) was not completely level from the start of the experiment causing the height of the actual swinging arm to be slightly off.  With the inherent air resistance acting on the swinging rubber stopper, the string might not be  exactly in line with the swinging arm as the apparatus is moving in a circular motion.   

September 28, 2016: Centripetal Acceleration vs. Angular Frequency

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #8: Centripetal Acceleration vs. Angular Frequency

Purpose:

The main objective is this lab is to establish a relationship with the centripetal acceleration creating from a large turntable and the angular frequency that it experiences .  Gathering enough data from the time it takes to complete 10 rotations, measurements from power supply source, and calculating various rotational speeds are all methods involved in this lab.

Procedure:

1. Turn on power supply to the turntable to 6.4v.  Once turntable is on allow it to achieve a constant speed until taking measurements. Use 200 g mass on turn table
2. Once table speed has stabilized, allow lab pro to gather data from photo gate and accelerometer.  
3. After timing 10 rotations, move accelerometer to a different distance from the center of rotation.  
4. Gather data from new radius and repeat until 4 trials have been completed at this voltage.  
5. Apply a voltage of 7 volts to turntable using the same 200 g mass at a radius of 58 cm. Record  time for 10 rotations and record mean value on graph.  
6. Apply 7.3 volts to table and record time for 10 rotations as well as mean value on graph.  Use same radius as step 5.
7. Apply 7.6 volts to turn table and record time for 10 rotations as well as mean value on graph.  Use same radius as in step 5.  
8. Next three trials will use a radius of 58cm, 7.3 volts, and only change the mass used. Use masses of 200g, 100g, and 50g.  Record time for 10 rotations and mean value from graph.  

Theory:

To find the relationship between the centripetal acceleration and angular frequency, we make sure the table is spinning at a constant speed by reading a value of zero in the x direction from the accelerometer. Then we vary the masses attached at the end of the accelerometer in various radiuses from the center.  This is done to record any changes in the angular speed as the mass is moved out further from the center as well as observing the changing of different masses effecting the angular speed and centripetal acceleration.  If the apparatus is a perfect representation of the relationship between angular speed and centripetal acceleration, the outcome should be a correlation of one on a angular speed squared vs. acceleration graph.

Measured Data:

(turntable apparatus)

(close up of accelerometer and mass)

(calculations for omega)

(table of gathered data)

(graph showing Force vs. Radius*w^2)

(graph showing Force vs. Mass*Radius)

(graph showing Acceleration vs. Angular speed^2)     

Analysis:

The first two graphs have a correlation value that is within the accepted value.  The data collected from the lab demonstration proved to be reliable.  The graph depicting Force vs. Radius*w^2 has a correlation value of 0.9996 which can almost accurately predict the amount of mass used in the system.  The graph showing Force vs. Mass*Radius has a correlation value of 0.8909 which can still be used to calculate the angular speed of the system given a mass and radius of the circle, but unfortunately only to a certain degree of accuracy.  The last graph shows the the centripetal acceleration vs. w^2 with a slope of 0.8932 which can be used to predict, to a certain degree of accuracy, the value for the force acting in the system.  Some uncertainties may be a result of the unstable turn table apparatus and possibly the power supply to the motor turning the table being able to provide a steady rate. 

Conclusion:

The experiment provided some exceptional data for student evaluation.  Although some kinks need to be sorted out with the apparatus, the overall experiment went better than expected.  With some adjustment to the turntable, the experiment will be a great learning tool in this subject of angular speed and centripetal acceleration.   The data is accurate enough to reproduce any future experiments with very little uncertainty.