# Lecture 22: Potential Energy

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1 Lecture : Potental Energy We have already studed the work-energy theorem, whch relates the total work done on an object to the change n knetc energy: Wtot = KE For a conservatve orce, the work done by that orce as an object travels rom pont 1 to pont s dened. We dene a new knd o energy, potental energy, as: W = U = U U F 1 Note that here we are talkng about the work done by a sngle orce

2 In general, One-dmensonal Case ( ) ( ) d U = U r U r = W = F r Note that only a derence n potental energy s meanngul, snce we can always redene what we call r = 0 I the orce only acts n one dmenson, and only depends on one dmenson, (call t ), ths smples to: ( ) ( ) = ( )d U U F r r

3 Eample 1: Gravty Gravty s a constant orce that acts n only one drecton (at least near the Earth s surace). The change n gravtatonal potental energy as an object moves rom one heght to another s: ( ) ( ) d ( ) U = U y U y = F y = mg dy ( ) = mg y y = mg y y y y y As we epected, the change n potental energy only depends on the change n heght o the object

4 Eample : Sprng We start wth a sprng at ts natural length, and compress t a dstance d The work done by the sprng s: 1 W = kd = kd Thus the change n potental energy s: d 0 1 U = W = kd We would obtan the same result or a sprng stretched a dstance d

5 Potental Energy and Force For, one-dmensonal problems, we have dened: ( ) ( ) = ( )d U U F Takng the dervatve o each sde gves: du ( ) du ( ) d Snce and are completely arbtrary, we have: Any conservatve orce can be dened as the dervatve o a potental energy uncton ( ) F ( ) = F + d ( ) F ( ) du = d

6 Mechancal Energy Let s say we have a system o partcles, or whch: 1. No orces outsde the system do any work. All nternal orces are conservatve Now magne that the partcles n the system move. We know that the total work done on partcle s: Summng all these ndvdual works, we nd W tot, = KE Wtot = Wtot, = KE = KE But snce we assumed that no outsde orce does any work, the total work s due to the (conservatve) nternal orces

7 Thereore we can also denty the total work wth a change n potental energy o the system: Wtot = U So we have: KE KE = U + U I we dene the sum o the knetc and potental energes to be the mechancal energy E, ths tells us that: E = In other words, energy s conserved or ths system 0 = 0

8 Analogy: Islands o the coast Imagne a set o resort slands, located at varous dstances rom the coast: The dashed lnes are brdges, and the numbers are the toll you must pay to cross each brdge But these tolls are strange, n that you pay the money gong away rom the coast, but get t back when you move toward the coast

9 Note that reachng sland 3 rom the coast wll always cost you \$1, no matter whch set o brdges you take In act, any round-trp costs a total o \$0 Now magne there are our people, all wth \$10 n ther pocket One s standng on the coast, and one s on each o the slands Who s the rchest? The person on sland 3 s, snce he can earn \$1 just by travelng back to the coast In contrast, the one standng on the coast doesn t have enough money to even reach sland 3 We see that n addton, to n-pocket money, one has potental money, the amount o whch depends on your locaton

10 Now magne that the tolls were arranged as ollows: I one starts rom the coast, travels drectly to sland 3, then to sland, then to sland 1, then back to the coast, one earns \$ or the round trp By contnung on ths cycle, one can make an nnte amount o money Or, by gong n the opposte drecton, lose an nnte amount There s no well-dened potental money n ths case

11 Eamnng the Analogy The brdge tolls represent the work done by a orce I the orce s conservatve, the work done by that orce n any round-trp s zero 1. In ths case, we can dene a potental energy (the potental money n the analogy). Each object also has a knetc energy ( money n pocket ) 3. Snce all the work s done by orces (tolls) nternal to the system, total energy (total wealth) s constant I the orce s not conservatve, there s no way to dene a potental energy In ths case none o the above concepts make sense

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