Lab #7: Energy Conservation

Transcription

1 Lab #7: Energy Conservation Photo by Kallin Reading Assignment: Chapter 7 Sections,, 3, 5, 6 Chapter 8 Sections - 4 Introduction: Perhaps one of the most unusual and exhilarating recreational activities is the bungee jump. In this sport, a series of intertwined elastic cords are attached to a harness that is either fastened about the person's body or attached to his/her ankles. The other end of the cord typically is attached to a high place like a special bungee jumping structure, a bridge, or even a hot air balloon. The person leaps from this structure and falls freely until the bungee cord begins to go tense. The jumper's speed decreases until he/she momentarily comes to rest and then accelerates upwards. This motion cycle continues for a number of oscillations until the jumper is brought to rest. While the sport of bungee jumping in its present form is quite recent, its origin dates back several centuries to the ritual of "land diving in the Pacific Archipelago. There, the purpose of the ritual was to demonstrate courage and offer injuries to the gods for a plentiful harvest of yams. It wasn't until the late 970's that this was transformed into a recreational activity. (See The Physics of Bungee Jumping by P.G. Menz in The Physics Teacher, Nov. 993). In this experiment we shall consider a system, similar to the bungee jumping scenario, in which two forces act upon a given mass. Our goal will be to make an overall judgment regarding the conservation of energy principle as it applies to this situation. The system will be comprised of a mass supported by a spring. The mass will be made to oscillate in the vertical plane, as shown below. Notice that, while oscillating, the mass experiences the following two forces: the (constant) force of gravity and the (non-constant) force provided by the spring. Dissipative forces that are present, such as air resistance and friction, are small, so they will be ignored. It is important to understand the differences between the nature of constant and non-constant forces. In this experiment, the force of gravity is considered to be constant in both magnitude and direction, regardless of the position of the mass that is oscillating. F gravity The direction of the force of gravity is vertically downward. r r = mg The force exerted on the mass by the spring, however, is not constant. Its magnitude varies depending upon the

2 position of the oscillating mass. Hooke s Law explains that the force exerted by the spring is proportional to the distance that the spring is stretched from its equilibrium position. In other words, as the stretch of the spring increases, so does the force with which it pulls on the mass. r F spring r = kx The negative sign indicates that the direction of the force is opposite the direction of the stretch. Therefore, since the spring is stretched downward throughout the oscillation, the direction of the force exerted by the spring is always upward in this experiment. Note the following positions of the spring and mass system defined in the following figure: Figure A: Spring and Mass Hanger at Rest. System is considered to be un-stretched. Starting reference point, x=0. Motion Sensor reads h o. Figure B: Spring System with Mass, m, added at Rest. System is at equilibrium. Reference point of oscillation, y=0 when x= x eq. Motion Sensor reads h eq. Figure C: Spring System in Oscillation at some position below equilibrium. Motion Detector reads h, x (the stretch) of the spring is larger than in Figure B, and y is downward. Figure D: Spring System in Oscillation at some position above equilibrium. Motion Detector reads h, x (the stretch) of the spring is smaller than in Figure B, and y is upward. Figure A Figure B Figure C Figure D x = 0 y = 0 x eq y x y x h o h eq h h h = 0

3 Notice that there are three displacements defined in the above Figures: x, h, and y. Each is important and all are related to one another. The directions of each vector are also important: x (the stretch) is always downward; h (the height measured by the motion detector) is always upward; and y (the displacement from the equilibrium position during oscillation), is either upward or downward depending upon the moment considered. Positive and negative values can be assigned to indicate the direction of each of these variables. Notice, also, in Figure B, the significance of the forces acting upon mass, m. Since this position is the equilibrium position, the net force on the mass at this moment equals zero. Figure B x = 0 x eq F spring = kx eq y = 0 F gravity = mg h eq h = 0 Therefore, F gravity F = 0 r r spring Which can be restated as: mg kx = 0 Eq. eq There are also three types of mechanical energy contained in the spring-mass system during oscillation: Kinetic Energy, Gravitational Potential Energy, and Spring (Elastic) Potential Energy. Each are defined as follows: Kinetic Energy: KE = mv Gravitation Potential Energy: Spring (Elastic) Potential Energy: GPE = mgh SPE = kx If the system is ideal, then m is the mass placed upon the hanger, k is the spring constant of our ideal spring, v is the speed of the mass, and x and h are defined as in the figures above. An ideal system is one in which the spring and mass hanger are mass-less. Since this experiment is not ideal, a correction factor will be derived and then verified in the Post Lab.

4 Considering these three types of mechanical energy, the Total Mechanical Energy of the spring-mass system at any time, therefore, can be given by: Total Mechanical Energy: E total = kx mv mgh Eq. This expression is true as long as there is no external work added to the system by the person who sets it into motion. Therefore, when starting the oscillation, it is important to do the following procedure:. Lift the mass hanger so that it is at rest at position x=0. (Since it is at rest, v=0 and E total = mgh 0.). Release the system by quickly dropping your hand down and out of the way. In order to make a meaningful analysis of the above Total Energy expression, it is helpful to rewrite it in terms of just one displacement vector, y. By looking at the Figures A, B, C, and D, one can verify that the displacement vectors x and h can be defined as follows: (When verifying, remember to assign positive and negative values to each term.) x = x eq y Eq. 3 h = h eq y Eq. 4 These expressions can be substituted into Equation and then rearranged using algebra: E total = mv mgh kx E ) total = mv mg( heq y) k( xeq y E total eq eq eq = mv mgh mgy k( x x y y ) E total = mv mgheq mgy kxeq kxeq y ky Rearranging and combining terms results in the following: total mv ky ( mg kxeq ) y mgheq kxeq E = For analysis, consider the following three groups of terms: total [ mv ] [ ] [ ] ky mg kx ) y mgh kx E = Eq.5 ( eq eq eq The left-most bracket describes the kinetic energy and gravitational potential energy (as measured from the perspective of y) of mass, m. Both of these quantities are known to change throughout the oscillation. The coefficient of the center term is described in Equation as being equal to zero. However, this is only an approximation because our system is not ideal and the total mass of the system is not actually m. The behavior of this term is determined by the behavior of y. Since the variable, y, is know to oscillate, this term describes an oscillation whose amplitude is relatively small. (Ideally, the amplitude should be equal to zero.) All terms contained in the right-most bracket are constants. Therefore, the total value contained in this bracket is also a constant.

5 Considering all of these terms together, the ideal case predicts that the Total Energy of the spring-mass system should be described as follows: where C is a constant. E total = mv ky C Eq. 6 The Conservation of Energy principle states that, if all forms of energy are considered, then E = constant Eq. 7 total If this is true, then, ideally, the first two terms in Equation 6 (the Kinetic Energy of the system and the Spring Potential Energy of the System as measured from the perspective of y) should also add to a constant value.

6 Lab #7: Energy Conservation Goals: Determine the spring constant, k, of your particular spring using a graphical method. Compare the oscillating values of the kinetic energy and spring potential energy of a spring-mass system. Verify the Conservation of Energy principle as it applies to a spring-mass system. Analyze the data for evidence of non-ideal effects and other forms of energy. Equipment List: Science Workshop Motion Sensor Force Sensor Spring with 50 g mass hanger attached Table clamp with vertical and horizontal posts Slotted Masses of 50 g, 00 g, & 00 g Activity : The Spring Constant The purpose of this activity is to determine the spring constant, k, of your particular spring. The equation suggests that in order to determine k we must measure the force exerted on the spring and how far the spring stretches as a result of this force.. Set up the equipment as shown in the picture. Place the motion detector on the floor directly beneath the mass hanger. Please, do not allow any masses to fall onto the detector!. Using Science Workshop set up the Motion Sensor to display a graph of Position vs. Time. Be sure that the motion detector can see the bottom of the mass hanger. Note: Since the mass will be at rest for each data recording, the graph should be a horizontal line. 3. Display the mean y-value (Position) for each run. (This value will effectively cancel the effects of any slight motion of the mass.) 4. Note: if desired, Steps and 3 can be replaced by taking Position data with a Digits window. 5. While the mass hanger (without any additional masses on it) is hanging freely and at rest, record position data. This initial position (measured from the motion detector) will serve as a reference point throughout the lab. Record this value. 6. Connect the Force Sensor to the analog channel of the Science Workshop interface and set it up to read a Digits display window of the force value. Remember that the force read by the Force Sensor is equivalent in magnitude to the force exerted by the spring. (Note: This Force data can be taken, instead using a Force vs. Time graph and displaying the mean y-value (Force) of each data run.) h o =

7 7. Push the TARE button on the side of the Force Sensor to reset the probe to zero while just the spring and mass hanger are suspended from it. This will calibrate the sensor to only read additional force created by adding more mass. Note: Remember to recalibrate the force sensor often throughout the lab. 8. Starting with 50 grams, hang successively higher masses (not to exceed 350 gm) to the hanger while recording the magnitude of the stretch of the spring. Remember: the motion detector does not measure the stretch, x, of the spring. h (meters) X (meters) Force (Newtons) Empty mass hanger 0 0 With 50 grams added With 00 grams added With 50 grams added With 00 grams added With 50 grams added With 300 grams added With 350 grams added 9. Using Excel, create a graph of F vs. x (Force vs. stretch). (It is permissible to graph only the magnitudes.) 0. Based upon your graph, what is the value of k that you obtain for your spring? (Including units.) Explain how this value was obtained. (Also record it for use in the Post Lab.) k =. Theoretical Question: (Do not take any data.) How much would your spring stretch if a mass of 5 kg was attached? Show your work. Activity : Comparing Spring Potential Energy and Kinetic Energy of a Spring-Mass System The purpose of this activity is to measure and compare the spring potential energy and the kinetic energy of the spring-mass system.. It is acceptable to delete all data runs, graphing windows, and digital window displays from Activity.. Choose one of the mass amounts used in Activity and place it upon the hanger. Please do not allow the mass to fall onto the motion detector. Record this mass value. (Also record it for use in the Post Lab.) m = 3. While the mass is hanging freely and at rest, begin recording position data. This initial position, h eq, will serve as our equilibrium reference point (y=0) throughout this lab. Also determine the value for x eq. (Note: This data was already recorded in Activity. Therefore it is not necessary to retake it.) h eq = x eq =

8 4. Use the Science Workshop Experiment Calculator to calculate the position, y. (See Equation 4 in the Introduction of the Lab. This effectively shifts our coordinate system to the at rest position of the hanging mass.) [Hint: You can check the correctness of your calculation by graphing y vs. Time. When the mass is at equilibrium, y = 0. Positions above this should have a positive y-value. Positions below this value should have a negative y-value.] 5. Use the Experiment Calculator to define a calculation called Spring Potential Energy. This calculation should be equal to (0.5*k*y*y). (See Equation 5 in the Introduction of the Lab.) 6. Use the Experiment Calculator to define a calculation called Kinetic Energy. This calculation should be equal to (0.5*m*v*v). (See Equation 5 in the Introduction of the Lab.) Remember that m should be expressed in units of kilograms. 7. Create a graphing window of Spring Potential Energy vs. y and then use the Add Plot button to add a graph of Kinetic Energy vs. y to the same window. 8. Create a new graphing window of y vs. Time, and then use the Add Plot button to add graphs of Spring Potential Energy vs. Time and Kinetic Energy vs. Time to the same window. 9. Lift the mass upward until the spring-mass system is approximately at position x = 0. Press Record and release the mass. Record data for 5 to 0 complete oscillations. 0. Resize and import these graphing windows into your template.. Answer the following questions: Spring Potential Energy a. At what position(s) in the oscillation does the spring have the greatest spring potential energy? b. At what position(s) in the oscillation does the spring have the greatest spring potential energy? c. What is the maximum spring potential energy value recorded in this part of the lab? SPE max = d. What is the minimum spring potential energy value recorded in this part of the lab? SPE min = Kinetic Energy a. At what position(s) is the Kinetic Energy of the mass the greatest? b. At what position(s) is the Kinetic Energy of the mass the least? c. What is the maximum amount of Kinetic Energy recorded in this part of the lab? KE max = d. What is the minimum amount of Kinetic Energy recorded in this part of the lab? KE min =

9 Comparison of SPE and KE a. Compare the positions of the maximum and minimum energy values. Describe what appears to be happening to the energy over the course of one oscillation. Does this observation support the Conservation of Energy principle stated in Equations 6 and 7 in the Introduction to the Lab? Explain. b. Compare the minimum energy values obtained from your data. What conclusions can you draw from this? c. Compare the maximum energy values obtained from your data. What conclusions can you draw from this? (Why are they not the same?) d. Correction to the Mass: In the Post Lab, you will study this in more detail. However, for now, make the following change to the mass value in your Kinetic Energy Calculation in the Experiment Calculator: KE (corrected) = 0.5 * (m 0.05 (0.85/3)) * v * v Look at your KE vs. y graph again and record the corrected maximum value of Kinetic Energy. (corrected) KE max = Does this value agree more closely with the value obtained for SPE max? Explain why this might occur? Period of Oscillation a. From your graphs, determine the time it took the mass-spring system to complete one oscillation. (Also record this value for use in the Post Lab.) T measured = b. Explain why the periods of the two Energy vs. Time graphs differ from the period of y vs. Time graph. How are the periods of the graphs related to one another? Activity 3: Total Energy. Use the Experiment Calculator to define a calculation called Total Energy (Kinetic Energy Spring Potential Energy).. Create a new graphing window of Total Energy vs. Time. 3. Lift the mass to position x = 0, as before, and record data for 0 to 5 oscillations. 4. Make an accurate sketch or printout of this graph for later use in the Post Lab. 5. Import this graph to your template. 6. Determine the slope and y-intercept of the Total Energy vs. Time graph. Record these values for use in the Post Lab. Slope = Y-intercept =

10 7. Answer the following questions: (Hint: Refer to Equations 5, 6, & 7 in the Introduction of the Lab.) a. Does the Total Energy vs. Time graph support the Conservation of Energy principle? Why or why not? b. Compare the Total Energy vs. Time graph to Equation 5 in the Introduction of the lab. Why does your data appear to oscillate slightly? Explain. c. Is the initial (when released) Total Energy value of the system (calculated on your graph) actually equal to mgh 0 as claimed in the Introduction of the Lab? Verify by calculation. Explain your findings.

11 Name: Section #: Post Lab #7: Energy Conservation The purpose of the Lab was to verify the Conservation of Energy principle as it applied to a spring-mass system. However, as stated in the Introduction, the derivation of this principle made two very distinct assumptions:. The sources and/or consumers of non-mechanical energy were negligible. (Example: outside work done by a person, friction, and air resistance).. The system was ideal (the spring and mass hanger were mass-less). Evidence from your data should support the fact that both of these assumptions are false. The purpose of the Post Lab is to analyze the data for this evidence. Attach the following data (determined in Activities & 3) to the Post Lab:. The spring constant, k, of the spring. k =. The amount of mass placed upon the mass hanger: Mass (added) = 3. The estimated time between successive oscillations of the system. T measured = 4. The Total Energy vs. Time graph. 5. The slope and y-intercept of the Total Energy vs. Time graph. Slope = Y-intercept = Energy Dissipation: Notice that the Total Energy vs. Time graph appears to dissipate, or decrease over time. In other words, it does not appear to be constant and, therefore, conserved. a. Given the slope and Y-intercept of your Total Energy vs. Time graph, calculate how much time will it take for the system to come to rest? Show your work. b. Estimate the number of oscillations that it will take for your system to come completely to rest.

12 Show your calculation. c. Why does the Total (Mechanical) Energy of the spring-mass system decrease over time? Does this fact prove the Conservation of Energy principle to be incorrect? Support your answer. The Non-ideal Situation: Since the spring and mass hanger were not mass-less, (the mass of the spring was approximately 85 grams and the mass of the hanger was 50 grams), the kinetic energy that was lost and gained by these parts of the system should have been considered. However, in what way should they have been incorporated into the analysis? Consider the following figure: At Rest In Motion l L dm dm x M X M Assume that the spring has mass m. Consider a small section of the mass of the spring, dm, of length d l. While at rest, the distance from the support to this section of spring is labeled l. The distance to the attached mass, M, is labeled L. As the spring oscillates the mass at the end of the spring travels with speed V while the segment of the spring (dm) travels with a somewhat different speed v. The distance that dm stretches from its equilibrium position is labeled x. The distance that M stretches from its equilibrium position is labeled X. (Note: The true value of mass M equals the sum of the mass hanger and the added mass.) The following ratios are equivalent: x X v = l = L V

13 The total kinetic energy of the system is actually the sum of the kinetic energies of the spring parts and mass M: K. E. = MV v dm K. E. = MV or L 0 v m ( ) dl Substituting for the speed, v, of the spring segment using the ratios above yields: K. E. = MV Evaluating the integral and simplifying results in the following: L L m V ( )( ) L L l dl m K. E. = MV V 3 Combining these two terms into one term shows that the Total Kinetic Energy of the system can be given by: m K. E. = M V 3 In other words, the total mass of the non-ideal system is given by: Total Mass = M 3 m 0 Analysis: Total Mass = mass of the hanger added masses m spring 3. Calculate the corrected value for the mass of the system. m total =. The following equation describes the theoretical period (time to complete one oscillation) of a massspring system: T = π where m is the mass of the oscillating system and k is the spring constant of the spring. a. Calculate the theoretical period of the mass-spring system if the system had been ideal. m k b. Determine the theoretical period of the mass-spring system given that it was not ideal.

14 3. By what % does each of the theoretical periods differ from the actual period that was measured during data collection. Do your results support the correction made to the mass of the system? Explain. 4. In your opinion, why is it useful to analyze physics concepts from an ideal perspective? What would you caution your fellow classmates to keep in mind whenever they analyze physics data and solve physics problems?