Potential Energy and Conservation of Energy

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1 Potential Enegy and Consevation of Enegy

2 Consevative Foces Definition: Consevative Foce If the wok done by a foce in moving an object fom an initial point to a final point is independent of the path (A o B), B W c! # F " d c A (path independent) then the foce is called a consevative foce which we denote by F c

3 Example: Gavitational Foce Conside the motion of an object unde the influence of a gavitational foce nea the suface of the eath The wok done by gavity depends only on the change in the vetical position W g = F g!y = "mg!y

4 Change in Potential Enegy Definition: Change in Potential Enegy The change in potential enegy of a body associated with a consevative foce F c is the negative of the wok done by the consevative foce in moving the body along any path connecting the initial and the final positions. B!U " #% F $ d = # W A c c

5 Wok-Enegy Theoem: Consevative Foces The wok done by the total foce in moving an object fom A to B is equal to the change in kinetic enegy z total f total 1 mv 2 # 1 2 W! % F " d = f mv 0! $ K z When the only foces acting on an object ae consevative foces F total = F then the change in potential c enegy is Theefoe!U = "W = "W total c!"u = "K

6 Consevation of Enegy fo Consevative Foces When the only foces acting on an object ae consevative!k +!U = 0 Definition: Mechanical Enegy The mechanical enegy is the sum of the kinetic and potential enegies E mechanical! K + U Equivalently, the mechanical enegy emains constant in time E f mechanical = K f + U f = K i + U i = E i mechanical

7 Checkpoint Poblem: Enegy and Choice of System You lift a ball at constant velocity fom a height h i to a geate height h f. Consideing the ball and the eath togethe as the system, which of the following statements is tue? 1. The potential enegy of the system inceases. 2. The kinetic enegy of the system deceases. 3. The eath does negative wok on the system. 4. You do negative wok on the system. 5. Two of the above. 6. None of the above.

8 Change in PE: Constant Gavity Foce: F gav = mg = Fgav, ĵ =!mg ĵ y Wok: W gav = F gav, y!y = "mg!y Potential Enegy:!U = "W gav = mg!y = mg (y f " y 0 ) Choice of Zeo Point: Whateve gound is convenient

9 Woked Example: Change in Potential Enegy fo Invese Squae Gavitational Foce Conside an object of mass m 1 moving towads the sun (mass m 2 ). Initially the object is at a distance 0 fom the cente of the sun. The object moves to a final distance f fom the cente of the sun. Fo the object-sun system, what is the change in potential duing this motion?

10 Woked Example Solution: Invese Squae Gavity Gm 1m 2 Foce: F m 1,m 2 =! ˆ 2 Wok done: W = f f Gm m Gm m f! ) F # d = ) % ' $ 1 2 " &( d = = Gm m! 1 1 " 1 2 $ % & ' f 0 ( Potential Enegy! 1 1 " #U = $W = $Gm m % Change: gav gav 1 2 % ' f $ 0 ( & & Zeo Point: U gav ( 0 =!) = 0 Potential Enegy U gav ( ) = Function! Gm m 1 2

11 Checkpoint Poblem: Change in Potential Enegy Sping Foce Connect one end of a sping of length l 0 with sping constant k to an object esting on a smooth table and fix the othe end of the sping to a wall. Stetch the sping until it has length l and elease the object. Conside the objectsping as the system. When the sping etuns to its equilibium length what is the change in potential enegy of the system?

12 Change in PE: Sping Foce Foce: F = F î =!kx î x Wok done: W sping = x= x f " ( 2 2!kx)dx =! 1 k (x! x ) f 0 2 x=x 0 Potential Enegy 1 2 2!U Change: sping = "W sping = k (x f " x ) 0 2 Zeo Point: U sping (x = 0) = 0 Potential Enegy U ( x ) = 1 kx 2 sping Function 2

13 Summay: Change in Mechanical Enegy Total foce: Total wok: W total = Change in potential enegy: Total wok done is change in kinetic enegy: Mechanical Enegy Change: Conclusion: final F total = F total + F c total nc final F total! d = total total (F + F )! d " " initial initial c nc final!u total = " $ F total # d initial W total =!" U total +W = "K nc!e mechanical "! K +!U total c W =!K +!U total nc

14 Demo slide: potential to kinetic enegy B97 ge=demo.php?letnum=b 97&show=0 This demonstation consists of dopping a ball and a pendulum eleased fom the same height. Both balls ae identical. The vetical velocity of the ball is shown to be equal to the hoizontal velocity of the pendulum when they both pass though the same height.

15 Stategy: Using Multiple Ideas Mechanical Enegy Consevation Newton s Second Law fo adial diection

16 Checkpoint Poblem: Loop-the- Loop An object of mass m is eleased fom est at a height h above the suface of a table. The object slides along the inside of the loop-the-loop tack consisting of a amp and a cicula loop of adius R shown in the figue. Assume that the tack is fictionless. When the object is at the top of the tack (point a) it pushes against the tack with a foce equal to thee times it s weight. What height was the object dopped fom?

17 Demo slide: Loop-the-Loop B95 e=demo.php?letnum=b 95&show=0 A ball olls down an inclined tack and aound a vetical cicle. This demonstation offes oppotunity fo the discussion of dynamic equilibium and the minimum speed fo safe passage of the top point of the cicle.

18 Potential Enegy and Foce In one dimension, the potential diffeence is B U (x)! U (x 0 ) =!" F x dx A Foce is the deivative of the potential enegy du F =! x dx 1 x kx 2 Examples: (1) Sping Potential Enegy: U sping ( ) = 2 F x sping = # du d! 1 kx 2 ", = # % = #kx dx dx & $ 2 ' Gm m (2) Gavitational Potential Enegy: 1 2 U gav ( ) =! F gavity = # du d! Gm m " Gm m, d = # d & $ # 1 2 % ' = # 1 2 2

19 Enegy Diagam Choose zeo point fo potential enegy: U (x = 0) = 0 Potential enegy function: 1 U (x) = kx 2, U (x = 0) = 0 2 Mechanical enegy is epesented by a hoizontal line since it is a constant E mechanical = K(x) + U(x) = 1 2 mv 2 x kx2 Kinetic enegy is diffeence between mechanical enegy and potential enegy (independent of choice of zeo point) K = E mechanical! U Gaph of Potential enegy function U(x) vs. x

20 Checkpoint Poblem: Enegy Diagams The figue above shows a gaph of potential enegy veses position fo a paticle executing one dimensional motion along the axis. The total mechanical enegy of the system is indicated by the dashed line. At the paticle is somewhee between points A and G. Fo late times, answe the following questions. At which point will the magnitude of the foce be a maximum? At which point will the kinetic enegy be a maximum? At how many of the labeled points will the velocity be zeo? At how many of the labeled points will the foce be zeo?

21 Checkpoint Poblem: Foce and Potential Enegy A paticle of mass, moving in the x-diection, is acting on by a potential, " U (x) =!U 1 $ " # $ x % 3 " x % 2 % ' $ x '! $ 1 & # x1 ' # & & ' whee U 1 and x 1 ae positive constants and U(0)=0. Sketch U(x)/U 1 as a function of x/x 1. Find the points whee the foce on the paticle is zeo. Classify them as stable o unstable. Calculate the value of U(x)/U 1 at these equilibium points. Fo enegies that lies in the ange 0<E<(4/27) U 1 find an equation whose solution yields points along the x-axis about which the paticle will undego peiodic motion. Suppose E= (4/27) U 1 and that the paticle stats at x = 0 with speed v 0. Find v 0.

22 MIT OpenCouseWae SC Physics I: Classical Mechanics Fo infomation about citing these mateials o ou Tems of Use, visit:

13.10 Worked Examples

13.10 Worked Examples 13.10 Woked Examples Example 13.11 Wok Done in a Constant Gavitation Field The wok done in a unifom gavitation field is a faily staightfowad calculation when the body moves in the diection of the field.