Potential Energy and Conservative Force Fields
Purpose:
· To analyze a body in motion in a conservative force field and develop a mathematical model describing the relationship between potential energy of the body and the distance travelled.
· Apply the mathematical model to calculate the force acting on the body throughout its journey.
· Analyze the work done and power generated by the body.
Materials:
· Pasco track and cart with two 250-gram masses
· Motion detector and CBL system
· Triple beam balance
· Metric ruler
· Level
· Ring stand and mounting clamp for the motion detector
Background:
When an upward applied force raises an object in the Earth’s gravitational field, carrying it from rest at some initial point Pi to rest at some final point Pf , positive work is done and the body’s potential energy, U, increases. The force exerted on the body by the field acts downward and does an equal amount of negative work, therefore the change in potential energy DU equals negative work due to gravity (DU = WG). In general, “the change in potential energy of an object when it moves from one point to another, while interacting via a conservative force field, equals the negative of the work done on the object by the field.” (Hecht, 2000, pg 243).
In the case of gravity,
12PEGf-PEGi= -PiPfF ds'>
In one dimension (force and displacement in the same direction):
12∆U = -xixfF(x) dx'>
The above equation allows us to find the change in potential energy for a given force. What if we are given the potential energy change and are required to determine the force? We take the derivative of the above equation (Blazey, 2006):
12dUdx= d-Fxdxdx=- dFxdxdx'>
12dUdx= -F(x) '>
12Fx= -dUdx'>
The graph of potential energy U as a function of x yields force as the slope of the line tangent to the curve at any given value of x. This force points downhill. Hilly terrain or roller coasters, where the potential energy is given by the relationship U = mgh, exhibit the property of potential energy changing linearly with height, just like on a graph of U as a function of x. The net force is zero where the derivative dU/dx = 0, at the tops of hills or the bottom of valleys.
In this lab, you will explore a one-dimensional system with a body moving linearly in a conservative force field, gravity. The body, a dynamics cart, moves down an inclined track with “down the track” considered as the positive x-direction. The cart begins with all potential energy and reaches the bottom of the track with all kinetic energy (the height of the lower end of the track will be established as the reference level for h=0). You will collect potential energy vs. position data while the cart is accelerating down the track and utilize a graph of this data to develop a mathematical expression for the potential energy as a function of position. This equation will allow you to determine the force acting on the cart throughout its journey.
Procedure:
1. Assemble the materials as indicated in the diagram below. The motion detector should be mounted to a ring stand.
2. Measure the mass of the cart, and two 250 gram masses, and record as mass of the cart, m (these items will have to be massed separately due to the range limitation of the triple beam balance).
3. Determine the measurements for h2 and h1 and record the difference between the two as the total height. (The diagram exaggerates the height difference for ease of interpretation.)
4. Assemble the motion detector and CBL system and initiate the physics program for a motion detector. Use a sample interval of 0.1 seconds and collect 50 samples.
5. Begin data collection and release the cart with the two masses on board.
6. Review the distance vs. time graph to ensure that the graph is continuous with no perturbations in data. If the results seem reasonable, proceed to Data Analysis.
Purpose:
· To analyze a body in motion in a conservative force field and develop a mathematical model describing the relationship between potential energy of the body and the distance travelled.
· Apply the mathematical model to calculate the force acting on the body throughout its journey.
· Analyze the work done and power generated by the body.
Materials:
· Pasco track and cart with two 250-gram masses
· Motion detector and CBL system
· Triple beam balance
· Metric ruler
· Level
· Ring stand and mounting clamp for the motion detector
Background:
When an upward applied force raises an object in the Earth’s gravitational field, carrying it from rest at some initial point Pi to rest at some final point Pf , positive work is done and the body’s potential energy, U, increases. The force exerted on the body by the field acts downward and does an equal amount of negative work, therefore the change in potential energy DU equals negative work due to gravity (DU = WG). In general, “the change in potential energy of an object when it moves from one point to another, while interacting via a conservative force field, equals the negative of the work done on the object by the field.” (Hecht, 2000, pg 243).
In the case of gravity,
12PEGf-PEGi= -PiPfF ds'>
In one dimension (force and displacement in the same direction):
12∆U = -xixfF(x) dx'>
The above equation allows us to find the change in potential energy for a given force. What if we are given the potential energy change and are required to determine the force? We take the derivative of the above equation (Blazey, 2006):
12dUdx= d-Fxdxdx=- dFxdxdx'>
12dUdx= -F(x) '>
12Fx= -dUdx'>
The graph of potential energy U as a function of x yields force as the slope of the line tangent to the curve at any given value of x. This force points downhill. Hilly terrain or roller coasters, where the potential energy is given by the relationship U = mgh, exhibit the property of potential energy changing linearly with height, just like on a graph of U as a function of x. The net force is zero where the derivative dU/dx = 0, at the tops of hills or the bottom of valleys.
In this lab, you will explore a one-dimensional system with a body moving linearly in a conservative force field, gravity. The body, a dynamics cart, moves down an inclined track with “down the track” considered as the positive x-direction. The cart begins with all potential energy and reaches the bottom of the track with all kinetic energy (the height of the lower end of the track will be established as the reference level for h=0). You will collect potential energy vs. position data while the cart is accelerating down the track and utilize a graph of this data to develop a mathematical expression for the potential energy as a function of position. This equation will allow you to determine the force acting on the cart throughout its journey.
Procedure:
1. Assemble the materials as indicated in the diagram below. The motion detector should be mounted to a ring stand.
2. Measure the mass of the cart, and two 250 gram masses, and record as mass of the cart, m (these items will have to be massed separately due to the range limitation of the triple beam balance).
3. Determine the measurements for h2 and h1 and record the difference between the two as the total height. (The diagram exaggerates the height difference for ease of interpretation.)
4. Assemble the motion detector and CBL system and initiate the physics program for a motion detector. Use a sample interval of 0.1 seconds and collect 50 samples.
5. Begin data collection and release the cart with the two masses on board.
6. Review the distance vs. time graph to ensure that the graph is continuous with no perturbations in data. If the results seem reasonable, proceed to Data Analysis.
Data:
Paragraph.
U (PE)
0.13315231743
0.133864807107
0.133689424719
0.133864807107
0.133899883235
0.13389403721
0.133900614
0.133899883235
0.133841422986
0.13126989177
0.124067538556
0.12266452881
0.123201649995
0.121740049565
0.120582493885
0.117677011194
0.116569987044
0.118536509943
0.117458560585
0.113029538778
0.111264772484
0.108072734765
0.111521213817
0.116117013839
0.110486494656
0.105251996805
0.103190540786
0.101674321952
0.0998064539005
0.0982096256362
0.0962452898732
0.0914576386004
0.0900368348896
0.0907501606471
0.0889562732411
0.087125586508
0.0848806340575
0.0806284177845
0.0829698316292
0.0814148295113
0.0757877372865
0.077424878162
0.0724441977736
0.0703065708131
0.0724438924786
0.0685703467784
0.0676933077757
0.0694412253128
0.0694412253128
0.0698744732325
x L4
0.106066
0.110508
0.108009
0.105788
0.106066
0.10551
0.105233
0.10551
0.104955
0.108009
0.121614
0.140218
0.156044
0.173814
0.191862
0.211298
0.233233
0.254057
0.273494
0.295706
0.320418
0.344574
0.372618
0.393164
0.415932
0.442865
0.13315231743
0.133864807107
0.133689424719
0.133864807107
0.133899883235
0.13389403721
0.133900614
0.133899883235
0.133841422986
0.13126989177
0.124067538556
0.12266452881
0.123201649995
0.121740049565
0.120582493885
0.117677011194
0.116569987044
0.118536509943
0.117458560585
0.113029538778
0.111264772484
0.108072734765
0.111521213817
0.116117013839
0.110486494656
0.105251996805
0.103190540786
0.101674321952
0.0998064539005
0.0982096256362
0.0962452898732
0.0914576386004
0.0900368348896
0.0907501606471
0.0889562732411
0.087125586508
0.0848806340575
0.0806284177845
0.0829698316292
0.0814148295113
0.0757877372865
0.077424878162
0.0724441977736
0.0703065708131
0.0724438924786
0.0685703467784
0.0676933077757
0.0694412253128
0.0694412253128
0.0698744732325
x L4
0.106066
0.110508
0.108009
0.105788
0.106066
0.10551
0.105233
0.10551
0.104955
0.108009
0.121614
0.140218
0.156044
0.173814
0.191862
0.211298
0.233233
0.254057
0.273494
0.295706
0.320418
0.344574
0.372618
0.393164
0.415932
0.442865
Data Analysis:
1. Time data are stored in List 1 of the TI-84. Distance data are stored in List 4, and velocity data are stored in List 5. You may proceed with data analysis using LoggerPro following these instructions:
i. Before you can upload your data to LoggerPro, you must calculate potential and kinetic energies using the data collected. Using the TI-84, enter the following command: 120.5*m L52 STO L2'> (m is the mass of the cart with the two masses on board). This command stores the values for kinetic energy in List 2.
ii. Next, enter the following command: 12m*9.81*h- L2 STO L3'>. This command stores the values for potential energy in List 3.
iii. Connect the calculator to a desktop computer via the USB port and open the LoggerPro software. Click on the calculator icon so that the computer can detect the calculator. Import Lists 3 (potential energy) and 4 (distance). Delete any data points that resulted before the cart was released and after the cart hit the bumper.
iv. Graph U vs. x. Use the linear regression function to determine a mathematical expression for the resulting graph.
2. Using the relationship between potential energy, distance and force, determine the force acting on the cart during its journey down the track.
0.058 m/s
Extensions (yes, you must do these).
1. Using the basic principles of F = -dU/dx and U = mgh, (and ignoring friction) prove that the force experienced by the cart moving down the track is equal to the x-component of the cart’s weight vector.
Force is equal to the mass times gravity times the sin of theta. the sin of theta is equal to h over x
(right triangle) F=d(mgh)/dx = d(mgxsin(theta))/dx
F=mgsin(theta)*d(x)/dx = mgsin(theta)
2. Determine the work done by gravity from x0 to xf.
0.057*1.31-0.057*0.106 = 0.068628
3. Determine the power generated when work is done from x0 to xf.
Power is Work over Time
P=fd/t
Fd=0.068
t=5 seconds
0.068/5= 0.0136
Conclusion:
Our purpose was to analyze a body in motion in a conservative force field and develop a mathematical model describing the relationship between potential energy of the body and the distance travelled. We applied the mathematical model to calculate the force acting on the body throughout its journey. We analyzed the work done and power generated by the body. The possibility of error was existant. The motion sensor may have easily malfunctioned or not give correct information. Even with these errors, the data was used and applied.
Works Cited 1. Blazey, G. (2006). Unit 5 Conservation of Energy. Retrieved November 16, 2007, from bama.ua.edu/~andreas/phys253_2007/phys253_2007.html: nicadd.niu.edu/people/blazey/Physics_253_2006/Lessons/Unit5_ConservationofEnergy/CE5.ppt
2. Harr, R. (2006). PHY5200 Lecture 20. Retrieved November 16, 2007, from http://hep.physics.wayne.edu/~harr/courses/5200/f06/ : hep.physics.wayne.edu/~harr/courses/5200/f06/lecture20.htm
3. Hecht, E. (2000). Physics: Calculus. Pacific Grove, CA: Brooks/Cole (pg. 243).
1. Time data are stored in List 1 of the TI-84. Distance data are stored in List 4, and velocity data are stored in List 5. You may proceed with data analysis using LoggerPro following these instructions:
i. Before you can upload your data to LoggerPro, you must calculate potential and kinetic energies using the data collected. Using the TI-84, enter the following command: 120.5*m L52 STO L2'> (m is the mass of the cart with the two masses on board). This command stores the values for kinetic energy in List 2.
ii. Next, enter the following command: 12m*9.81*h- L2 STO L3'>. This command stores the values for potential energy in List 3.
iii. Connect the calculator to a desktop computer via the USB port and open the LoggerPro software. Click on the calculator icon so that the computer can detect the calculator. Import Lists 3 (potential energy) and 4 (distance). Delete any data points that resulted before the cart was released and after the cart hit the bumper.
iv. Graph U vs. x. Use the linear regression function to determine a mathematical expression for the resulting graph.
2. Using the relationship between potential energy, distance and force, determine the force acting on the cart during its journey down the track.
0.058 m/s
Extensions (yes, you must do these).
1. Using the basic principles of F = -dU/dx and U = mgh, (and ignoring friction) prove that the force experienced by the cart moving down the track is equal to the x-component of the cart’s weight vector.
Force is equal to the mass times gravity times the sin of theta. the sin of theta is equal to h over x
(right triangle) F=d(mgh)/dx = d(mgxsin(theta))/dx
F=mgsin(theta)*d(x)/dx = mgsin(theta)
2. Determine the work done by gravity from x0 to xf.
0.057*1.31-0.057*0.106 = 0.068628
3. Determine the power generated when work is done from x0 to xf.
Power is Work over Time
P=fd/t
Fd=0.068
t=5 seconds
0.068/5= 0.0136
Conclusion:
Our purpose was to analyze a body in motion in a conservative force field and develop a mathematical model describing the relationship between potential energy of the body and the distance travelled. We applied the mathematical model to calculate the force acting on the body throughout its journey. We analyzed the work done and power generated by the body. The possibility of error was existant. The motion sensor may have easily malfunctioned or not give correct information. Even with these errors, the data was used and applied.
Works Cited 1. Blazey, G. (2006). Unit 5 Conservation of Energy. Retrieved November 16, 2007, from bama.ua.edu/~andreas/phys253_2007/phys253_2007.html: nicadd.niu.edu/people/blazey/Physics_253_2006/Lessons/Unit5_ConservationofEnergy/CE5.ppt
2. Harr, R. (2006). PHY5200 Lecture 20. Retrieved November 16, 2007, from http://hep.physics.wayne.edu/~harr/courses/5200/f06/ : hep.physics.wayne.edu/~harr/courses/5200/f06/lecture20.htm
3. Hecht, E. (2000). Physics: Calculus. Pacific Grove, CA: Brooks/Cole (pg. 243).
