Lab #7: Energy Conservation

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1 Lab #7: Enegy Consevation Photo by Kallin Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual and exhilaating eceational activities is the bungee jump. In this spot, a seies of intetwined elastic cods ae attached to a haness that is eithe fastened about the peson's body o attached to his/he ankles. The othe end of the cod typically is attached to a high place like a special bungee jumping stuctue, a bidge, o even a hot ai balloon. The peson leaps fom this stuctue and falls feely until the bungee cod begins to go tense. The jumpe's speed deceases until he/she momentaily comes to est and then acceleates upwads. This motion cycle continues fo a numbe of oscillations until the jumpe is bought to est. While the spot of bungee jumping in its pesent fom is quite ecent, its oigin dates back seveal centuies to the itual of "land diving in the Pacific Achipelago. Thee, the pupose of the itual was to demonstate couage and offe injuies to the gods fo a plentiful havest of yams. It wasn't until the late 1970's that this was tansfomed into a eceational activity. (See The Physics of Bungee Jumping by P.G. Menz in The Physics Teache, Nov. 1993). In this expeiment we shall conside a system, simila to the bungee jumping scenaio, in which two foces act upon a given mass. Ou goal will be to make an oveall judgment egading the consevation of enegy pinciple as it applies to this situation. The system will be compised of a mass suppoted by a sping. The mass will be made to oscillate in the vetical plane, as shown below. Notice that, while oscillating, the mass expeiences the following two foces: the (constant) foce of gavity and the (non-constant) foce povided by the sping. Dissipative foces that ae pesent, such as ai esistance and fiction, ae small, so they will be ignoed. It is impotant to undestand the diffeences between the natue of constant and non-constant foces. In this expeiment, the foce of gavity is consideed to be constant in both magnitude and diection, egadless of the position of the mass that is oscillating. F gavity The diection of the foce of gavity is vetically downwad. = mg The foce exeted on the mass by the sping, howeve, is not constant. Its magnitude vaies depending upon the

2 position of the oscillating mass. Hooke s Law explains that the foce exeted by the sping is popotional to the distance that the sping is stetched fom its equilibium position. In othe wods, as the stetch of the sping inceases, so does the foce with which it pulls on the mass. F sping = kx The negative sign indicates that the diection of the foce is opposite the diection of the stetch. Theefoe, since the sping is stetched downwad thoughout the oscillation, the diection of the foce exeted by the sping is always upwad in this expeiment. Note the following positions of the sping and mass system defined in the following figue: Figue A: Sping at Rest. System is consideed to be un-stetched. Stating efeence point, x=0. Motion Senso eads h o. Figue B: Sping System with Mass, m, added at Rest. System is at equilibium. Refeence point of oscillation, y=0 when x= x eq. Motion Senso eads h eq. Figue C: Sping System in Oscillation at some position below equilibium. Motion Detecto eads h, x (the stetch) of the sping is lage than in Figue B, and y is downwad. Figue D: Sping System in Oscillation at some position above equilibium. Motion Detecto eads h, x (the stetch) of the sping is smalle than in Figue B, and y is upwad. Figue A Figue B Figue C Figue D x = 0 x y = 0 x eq y x y h o h eq h h h = 0

3 Notice that thee ae thee displacements defined in the above Figues: x, h, and y. Each is impotant and all ae elated to one anothe. The diections of each vecto ae also impotant: x (the stetch) is always downwad; h (the height measued by the motion detecto) is always upwad; and y (the displacement fom the equilibium position duing oscillation), is eithe upwad o downwad depending upon the moment consideed. Positive and negative values can be assigned to indicate the diection of each of these vaiables. Notice, also, in Figue B, the significance of the foces acting upon mass, m. Since this position is the equilibium position, the net foce on the mass at this moment equals zeo. Figue B x = 0 x eq F sping = kx eq y = 0 F gavity = mg h eq h = 0 Theefoe, F gavity F = 0 + sping Which can be estated as: mg kx = 0 Eq. 1 + eq Thee ae also thee types of mechanical enegy contained in the sping-mass system duing oscillation: Kinetic Enegy, Gavitational Potential Enegy, and Sping (Elastic) Potential Enegy. Each ae defined as follows: Kinetic Enegy: KE = 1 mv Gavitation Potential Enegy: Sping (Elastic) Potential Enegy: GPE = mgh SPE = 1 kx If the system is ideal, then m is the mass placed upon the hange plus the mass of the hange, k is the sping constant, v is the speed of the mass, and x and h ae defined as in the figues above. An ideal system is one in which the sping is mass-less.

4 Consideing these thee types of mechanical enegy, the Total Mechanical Enegy of the sping-mass system at any time, theefoe, can be given by: Total Mechanical Enegy: E total + 1 = 1 kx mv + mgh Eq. This expession is tue as long as thee is no extenal wok added to the system by the peson who sets it into motion. Theefoe, when stating the oscillation, it is impotant to do the following pocedue: 1. Lift the mass hange so that it is at est at position x=0. (Since it is at est, v=0 and E total = mgh 0.). Release the system by quickly dopping you hand down and out of the way. In ode to make a meaningful analysis of the above Total Enegy expession, it is helpful to ewite it in tems of just one displacement vecto, y. By looking at the Figues A, B, C, and D, one can veify that the displacement vectos x and h can be defined as follows: (When veifying, emembe to assign positive and negative values to each tem.) x = x + eq y Eq. 3 h = h + eq y Eq. 4 These expessions can be substituted into Equation and then eaanged using algeba: E total = mv + mgh kx E 1 1 ) total = mv + mg( heq + y) + k( xeq + y E total eq eq eq + = 1 mv + mgh + mgy + 1 k( x + x y y ) E 1 1 total = mv + mgheq + mgy + kxeq + kxeqy + 1 ky Reaanging and combining tems esults in the following: total 1 1 mv + ky + ( mg + kxeq ) y + mgheq 1 kxeq E = + Fo analysis, conside the following thee goups of tems: total [ 1 mv + ] [ ] [ ] 1 ky + mg + kx ) y + mgh 1 kx E + = Eq.5 ( eq eq eq The left-most backet descibes the kinetic enegy and gavitational potential enegy (as measued fom the pespective of y) of mass, m. Both of these quantities ae known to change thoughout the oscillation. The coefficient of the cente tem is descibed in Equation 1 as being equal to zeo. Howeve, this is only an appoximation because ou system is not ideal and the total mass of the system is not actually m. The behavio of this tem is detemined by the behavio of y. Since the vaiable, y, is know to oscillate, this tem descibes an oscillation whose amplitude is elatively small. (Ideally, the amplitude should be equal to zeo.) All tems contained in the ight-most backet ae constants. Theefoe, the total value contained in this backet is also a constant.

5 Consideing all of these tems togethe, the ideal case pedicts that the Total Enegy of the sping-mass system should be descibed as follows: whee C is a constant. E total = mv + 1 ky + C 1 Eq. 6 The Consevation of Enegy pinciple states that, if all foms of enegy ae consideed, then E = constant Eq. 7 total If this is tue, then, ideally, the fist two tems in Equation 6 (the Kinetic Enegy of the system and the Sping Potential Enegy of the System as measued fom the pespective of y) should also add to a constant value. Check out: Lab # Enegy Consevation

6 Lab #7: Enegy Consevation Goals: Detemine the sping constant, k, of you paticula sping using a gaphical method. Compae the oscillating values of the kinetic enegy and sping potential enegy of a sping-mass system. Veify the Consevation of Enegy pinciple as it applies to a sping-mass system. Analyze the data fo evidence of non-ideal effects and othe foms of enegy. Equipment List: Science Wokshop Motion Senso Foce Senso Sping with 50 g mass hange attached Table clamp with vetical and hoizontal posts Slotted Masses of 50 g, 100 g, & 00 g Activity 1: The Sping Constant The pupose of this activity is to detemine the sping constant, k, of you paticula sping. The equation suggests that in ode to detemine k we must measue the foce exeted on the sping and how fa the sping stetches as a esult of this foce. 1. Set up the equipment as shown in the pictue. Place the motion detecto on the floo diectly beneath the mass hange. Please, do not allow any masses to fall onto the detecto!. Using Science Wokshop set up the Motion Senso to display a gaph of Position vs. Time. Be sue that the motion detecto can see the bottom of the mass hange. Note: Since the mass will be at est fo each data ecoding, the gaph should be a hoizontal line. 3. Display the mean y-value (Position) fo each un. (This value will effectively cancel the effects of any slight motion of the mass.) 4. Note: if desied, Steps and 3 can be eplaced by taking Position data with a Digits window. 5. While the mass hange (without any additional masses on it) is hanging feely and at est, ecod position data. This initial position (measued fom the motion detecto) will seve as a efeence point thoughout the lab. Recod this value. 6. Connect the Foce Senso to the analog channel of the Science Wokshop inteface and set it up to ead a Digits display window of the foce value. Remembe that the foce ead by the Foce Senso is equivalent in magnitude to the foce exeted by the sping. (Note: This Foce data can be taken, instead using a Foce vs. Time gaph and displaying the mean y-value (Foce) of each data un.) h o = Lab # Enegy Consevation

7 7. Push the TARE button on the side of the Foce Senso to eset the pobe to zeo while just the sping is suspended fom it. This will calibate the senso to only ead additional foce ceated by adding moe mass. Note: Remembe to ecalibate the foce senso often thoughout the lab. 8. Stating with 50 gams, hang successively highe masses (not to exceed 350 gm) to the hange while ecoding the magnitude of the stetch of the sping. Remembe: the motion detecto does not measue the stetch, x, of the sping. With 50 gams added With 100 gams added With 150 gams added With 00 gams added With 50 gams added With 300 gams added With 350 gams added h (metes) X (metes) Foce (Newtons)n Using Excel, ceate a gaph of F vs. x (Foce vs. stetch). (It is pemissible to gaph only the magnitudes.) 10. Based upon you gaph, what is the value of k that you obtain fo you sping? (Including units.) Explain how this value was obtained. (Also ecod it fo use in the Post Lab.) k = 11. Theoetical Question: (Do not take any data.) How much would you sping stetch if a mass of 15 kg was attached? Show you wok. Activity : Compaing Sping Potential Enegy and Kinetic Enegy of a Sping-Mass System The pupose of this activity is to measue and compae the sping potential enegy and the kinetic enegy of the sping-mass system. 1. It is acceptable to delete all data uns, gaphing windows, and digital window displays fom Activity 1.. Choose one of the mass amounts used in Activity 1 and place it upon the hange. Please do not allow the mass to fall onto the motion detecto. Recod this mass value. m = 3. While the mass is hanging feely and at est, begin ecoding position data. This initial position, h eq, will seve as ou equilibium efeence point (y=0) thoughout this lab. Also detemine the value fo x eq. (Note: This data was aleady ecoded in Activity 1. Theefoe it is not necessay to etake it.) h eq = x eq = Lab #6 Enegy Consevation

8 4. Use the Science Wokshop Expeiment Calculato to calculate the position, y. (See Equation 4 in the Intoduction of the Lab. This effectively shifts ou coodinate system to the at est position of the hanging mass.) [Hint: You can check the coectness of you calculation by gaphing y vs. Time. When the mass is at equilibium, y = 0. ] 5. Use the Expeiment Calculato to define a calculation called Sping Potential Enegy. This calculation should be equal to (0.5*k*y*y). (See Equation 5 in the Intoduction of the Lab.) 6. Use the Expeiment Calculato to define a calculation called Kinetic Enegy. This calculation should be equal to (0.5*m*v*v). (See Equation 5 in the Intoduction of the Lab.) Remembe that m should be expessed in units of kilogams. 7. Ceate a gaphing window of Sping Potential Enegy vs. y and then use the Add Plot button to add a gaph of Kinetic Enegy vs. y to the same window. 8. Ceate a new gaphing window of y vs. Time, and then use the Add Plot button to add gaphs of Sping Potential Enegy vs. Time and Kinetic Enegy vs. Time to the same window. 9. Lift the mass upwad until the sping-mass system is appoximately at position x = 0. Pess Recod and elease the mass. Recod data fo 5 to 10 complete oscillations. 10. Resize and impot these gaphing windows into you template. 11. Answe the following questions: Sping Potential Enegy a. At what position(s) in the oscillation does the sping have the geatest sping potential enegy? b. At what position(s) in the oscillation does the sping have the geatest sping potential enegy? c. What is the maximum sping potential enegy value ecoded in this pat of the lab? SPE max = d. What is the minimum sping potential enegy value ecoded in this pat of the lab? SPE min = Kinetic Enegy a. At what position(s) is the Kinetic Enegy of the mass the geatest? b. At what position(s) is the Kinetic Enegy of the mass the least? c. What is the maximum amount of Kinetic Enegy ecoded in this pat of the lab? KE max = d. What is the minimum amount of Kinetic Enegy ecoded in this pat of the lab? KE min =

9 Compaison of SPE and KE a. Compae the positions of the maximum and minimum enegy values. Descibe what appeas to be happening to the enegy ove the couse of one oscillation. Does this obsevation suppot the Consevation of Enegy pinciple stated in Equations 6 and 7 in the Intoduction to the Lab? Explain. b. Compae the minimum enegy values obtained fom you data. What conclusions can you daw fom this? c. Compae the maximum enegy values obtained fom you data. What conclusions can you daw fom this? (Why ae they not the same?) Peiod of Oscillation a. Fom you gaphs, detemine the time it took the mass-sping system to complete one oscillation. T measued = Activity 3: Total Enegy 1. Use the Expeiment Calculato to define a calculation called Total Enegy (Kinetic Enegy + Sping Potential Enegy).. Ceate a new gaphing window of Total Enegy vs. Time. 3. Lift the mass to position x = 0, as befoe, and ecod data fo 0 to 5 oscillations. Impot this gaph to you template. Detemine the slope and y-intecept of the Total Enegy vs. Time gaph. Slope = Y-intecept =

10 7. Answe the following questions: (Hint: Refe to Equations 5, 6, & 7 in the Intoduction of the Lab.) a. Does the Total Enegy vs. Time gaph suppot the Consevation of Enegy pinciple? Why o why not? b. Compae the Total Enegy vs. Time gaph to Equation 5 in the Intoduction of the lab. Why does you data appea to oscillate slightly? Explain. c. Is the initial (when eleased) Total Enegy value of the system (calculated on you gaph) actually equal to mgh 0 as claimed in the Intoduction of the Lab? Veify by calculation. Explain you findings. Lab # Enegy Consevation

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