Conservation of Energy 1 of 8

Transcription

1 Conservation of Energy 1 of 8 Conservation of Energy Te important conclusions of tis capter are: If a system is isolated and tere is no friction (no non-conservative forces), ten KE + PE = constant (Our text uses te notation K +U = constant) If tere is friction, ten KE + PE + E term = constant. (E term = termal energy) Two examples of PE (potential energy) PE grav = mg PE elastic = (1/2)kx 2 Questions you sould ave: Wy is KE+PE = constant, wen system isolated and no friction? Wat is te definition of potential energy, PE, and wy PE grav = mg, PE elastic = (1/2)kx 2? It is not enoug to know formulas. You sould know were te formulas come from. Potential Energy So, ow do we define potential energy, PE, and get PE grav = mg? If a force involves no dissipation (no friction), ten it can be a special type of force called a conservative force. Te defining property of a conservative force is tat te work done by te force depends only te initial and final positions, not on te pat taken. We sowed in a previous concept test tat gravity is a conservative force. Te force of friction is not a conservative force, because te work done depends on te pat taken: te longer te pat te more work is done by friction. i f

2 Conservation of Energy 2 of 8 Conservative forces include: gravity (F = mg) te spring force, or elastic force (F = kx) te normal force Te normal force is someting of a special case. Te work done by te normal force is always zero, so te normal force is "trivially" pat-independent: te work is zero, regardless of pat, and regardless of initial and final positions. Associated wit every conservative force is a kind of energy called potential energy (PE or U). PE is a kind of stored energy. If a configuration of objects as PE, ten tere is te potential to cange tat PE into oter kinds of energy (KE, termal, ligt, etc ). Te definition of te PE associated wit a conservative force involves te work done by tat force. Let s first review te concept of work. Recall: If I lift a mass m, a distance, at constant velocity (v = constant), wit an external force F ext, suc as my and, ten te work done by gravity is te negative of te work done by te external force. f i F ext = mg F grav = mg same magnitudes, opposite directions So W ext = +mg and W grav = mg. Tis is true for te special case v = constant, but it turns out tat it is always true tat W ext = W grav, regardless of te motion, so long as te KE at te final position is te same as te KE at te intial position. So, W ext = W grav, if te mass starts and finises at rest: v i = v f = 0. Wit tis example in mind, we are ready to define PE. If a force F (suc as gravity) is a conservative force, ten we define te PE associated wit tat force by PE F WF Wext

3 Conservation of Energy 3 of 8 In words: te cange in potential energy is te negative of te work done by te conservative force and it is terefore te positive of te work done by an external force opposing te conservative force. Only canges in PE are pysically meaningful. We are free to set te zero of potential energy werever we want. PE W mg. In tis formula, te eigt is te eigt above (+) or below () grav ext te =0 level. So is really f i f. So I sould really write te formula as 0 PEgrav mg. If I coose to set PE i = 0 at i = 0, ten te formula becomes PEPE i mg( i) or simply, PEgrav 0 0 PEgrav mg mg In te previous capter, we sowed tat te work done by an external force to stretc or compress a spring by an amount x is W ext = +(1/2)kx 2. We terefore ave tat te elastic potential energy contained in a spring is PE elas Wext PE 1 elas 2 k x 2 In writing tis formula, we ave set PE elas = 0 at x = 0 (te unstretced position). (Te normal force never does work, so PE normal Wnormal 0. We can set te PE associated wit te normal force equal to zero and forget about it.) Were is potential energy located? I lift a book of mass m a eigt and say tat te book as PE grav = mg. But it is not correct to say tat te PE in te book. Te gravitational PE is associated wit te system of (book + eart + gravitational attraction between book and eart). Te PE is not "in te book" or "in te eart"; it is in te book-eart system wic includes te gravitational field surrounding te book and te eart. For te case of elastic potential energy, te PE elas actually is inside te spring. It is located in te increased electrostatic potential energy in te cemical bonds joining te atoms of te spring. Conservation of mecanical energy. Definition: mecanical energy E mec = KE + PE. We are now in a position to sow tat

4 Conservation of Energy 4 of 8 E mec = KE + PE = constant (if no friction and system isolated). Recall te Work-KE Teorem: W net = KE. Now if tere is no friction, te net force involves conservative forces only, so W net = W c PE W c, so we ave Wnet Wc KE PE or (c for conservative force). But we just defined KE PE 0 KE PE constant (if no friction) Example of Conservation of Energy (no friction). A pendulum consists of a mass m attaced to a massless string of lengt L. Te pendulum is released from rest a eigt above its lowest point. Wat is te speed of te pendulum mass wen it is at eigt /2 from te lowest point? Assume no dissipation (no friction). L v o = 0 In all Conservation of Energy problems, begin by writing (initial energy) = (final energy) : E i = E f KE i + PE i = KE f + PE f /2 v =? 0 + mg = (1/2) mv 2 + mg(/2) (cancel m's and multiply troug by 2) 2g = v 2 + g v 2 = g v g Notice: Using Conservation of Energy, we didn't need to know anyting about te details of te forces involved and we didn't need to use F net = ma. Te Conservation of Energy strategy allows us to relate conditions at te beginning to conditions at te end; we don't need to know anyting about te details of wat goes on in between. Suppose tere are two conservative forces acting on a system, and no non-conservative forces. Ten we ave F net F c1 F c2 (For instance, tere may be gravity and a spring force, but no friction.) Ten we ave Wnet Fnet dr Fc1 Fc2 dr Fc1 dr Fc2 dr Wc1 Wc2 Te Work-KE Teorem ten gives Wnet Wc1 Wc2 PE 1 PE 2 KE or KE PE 1PE 2 0 KE PE1 PE2 constant (if no friction)

5 Conservation of Energy 5 of 8 Anoter example of Conservation of Energy: A spring-loaded gun fires a dart at an angle from te orizontal. Te dart gun as a spring wit spring constant k tat compresses a distance x. Assume no air resistance. Wat is te speed of te dart wen it is at a eigt above te initial position? y o = 0 v f =? E i = E f KE i + PE grav,i + PE elas,i = KE f + PE grav,f + PE elas,f (1/2)kx 2 = (1/2)mv 2 + mg m v 2 m g m 2 x v 2 g Notice tat te angle never entered into te solution. Wat if tere is friction? k k Up till now, we ave assumed tat tere is no sliding friction and no eat transferred in any of tese problems. (Having static friction in a problem usually causes no difficulties, because static friction does not generate termal energy.) How do we andle sliding friction and eat generated? If a system is isolated from external forces so tat no external work is done, and if no eat is transferred, and if tere is no sliding friction so tat no termal energy is generated (tat's a lot of "if's"), ten we can assert tat KE + PE = constant (isolated system, no termal energy involved) If, owever, tere is sliding friction, ten some of te mecanical energy (KE+PE) can be transformed into termal energy (E term ). In tis case, we ave KE + PE + E term = constant (isolated system) We now sow tat te amount of termal energy generated is te negative of te work done by friction: E term = W fric Notice tat te work done by sliding friction is always negative, since sliding friction always exerts a force in te direction opposite te motion. Consequently, W fric is a positive quantity. F fric x

6 Conservation of Energy 6 of 8 Wen tere is sliding friction, te net force consists of bot conservative forces (like gravity) and non-conservative forces (te friction), so W net = W c + W nc. But W net = KE (by Work-KE teorem) and W c = PE, so we ave KE + PE W nc = 0 (isolated system, wit friction) Since te total energy must remain constant in an isolated system, we identify te term W nc as E term = W fric, since we want KE + PE + E term = 0 (wic is te same as KE + PE + E term = constant). KE+PE=total energy graps Suppose a roller coaster of mass m rolls along a track saped like so: m x track Te sape of te track is a grap of eigt vs. orizontal position x. Since te gravitational potential energy of te coaster is PE = mg, were mg is a constant, a grap of PE vs. x looks te same as te grap of vs. x, but wit te vertical axis measuring energy (joules) rater tan eigt (meters). Assuming no friction, te total mecanical energy E tot = KE+PE of te roller coaster remains constant as it rolls along te track. We can represent tis constant energy wit a orizontal line on our grap of energy vs. x. From tis "energy grap", we can read te KE and te PE of te coaster at any point. PE KE E tot = KE+PE turning point PE KE max, PE min ere x

7 Conservation of Energy 7 of 8 Relation between force and PE. For tis discussion, I'll use te standard notation for potential energy: U = PE. I want to sow tat (in 1D) du F dx (Force is te negative "gradient" of te potential energy.) Consider a small cange in U, U, tat occurs wen a small cange in x, x, is made. By te definition of potential energy PE F = U = W F = Fx (I can write W F = Fx even if F varies wit x, because x is small and so F constant.) If I divide troug by x, we get, U d U F x d x Here F is te conservative force associated wit te potential energy U. Tis F is not, in general, te net force in te problem. lim x0 Power Power = rate at wic work is done = rate at wic energy is converted from one form to anoter: P W t E t units of power = [P] = joules/second = J/s = watts (W) Every second, a 100 W ligt bulb converts 100 joules of electric potential energy into eat and ligt. Te power company sells potential energy in units of kilowatt ours. 1 kwr = 1000 J/s 3600 s = J Anoter popular unit of energy is te Calorie (spelled wit a capital C). A typical candy bar as 300 Calories of stored cemical energy. Tere are two kinds of calories, spelled wit a little "c" or a big "C": 1 calorie (cal) = "little calorie" = J 1 Calorie = 1000 cal = "big Calorie" = "food calorie" = 1 kcal = 4186 J

8 Conservation of Energy 8 of 8 Te "food Calorie" is te "big Calorie" and it sould be spelled wit a big C. (Cemists like to use te little calorie, wic is defined as te amount of eat required to raise te temp of a gram of water by 1 o centigrade.) Calorie example: Dr. D, wo as mass m = 68 kg, eats a 300 Cal candy bar and ten climbs 10 stories ( = 35 meters) to is office on te 10 t floor of Gamow Tower. How many Calories as e burned? Work done = PE = mg = (68 kg) (9.8 m/s 2 ) (35 m) = J (1Cal/ 4186 J) = 5.6 Cal A measly 5.6 Cal!?!? Well, it's not quite tat bad. He was also doing a lot of ineffective work turning around in te stairwell, flailing is limbs, etc as e climbed, so te total mecanical work was more, maybe 10 Cal total. Also, te uman body is not a very efficient macine: only about 25% of te food Calories burned come out of te body as mecanical work; te rest goes into eat. (Dr. D was flused and panting after is 10-story climb.) So to produce 10 Cal of work, is body burned about 40 Cal still not very muc. Moral: You can't burn many Calories instantly by exercising. However, by exercising regularly, you build muscles wic increases your resting metabolic rate (RMR). A typical out of-sape male as a RMR of about 70 watts, meaning 70 joules per second burned by just breating, digesting, tinking. (70 W is about 1400 Cal/day). By exercising regularly, tat RMR can be raised to 90 watts (1860 Cal/day). So by exercising regularly, you burn about an extra 500 Cal per day just from your increased resting metabolic rate. "Lose weigt wile you sleep!" Wit te increased RMB, you can eat about 1 candy bar per day more tan normal and still not gain weigt. Power example: Te same Dr. D can climb to te 10 t floor in 60 seconds (if e puses!). Wat is te mecanical power e generates (due to increased PE only, not including eat generated)? P = PE / t = mg / t = (68 kg) (9.8 m/s 2 ) (35 m) / (60 s) = 390 W almost enoug to ligt four 100W ligt bulbs for tat 1 minute. A orse can generate a power of 1 orsepower for several ours. 1 p = 746 W. So Dr. D can generate about ½ p for 1 minute (and ten e is needs to take a nap). Horses are pretty powerful! Currently (2009) te power company sells energy at a rate of $0.10 per kilowattour. One kwr is enoug to ligt ten 100 W bulbs for 1 our and you get tat for 10 cents!