Chapter Seven - Potental Energy and Conservaton o Energy 7 1 Potental Energy Potental energy. e wll nd that the potental energy o a system can only be assocated wth specc types o orces actng between members o a system. The am ount o potental energy n the system s determned by the conguraton o the system. Movng members o the system to derent postons or rotatng them may change the conguraton o the system and thereore ts potental energy as shown n gure (7-1). Fgure (7-1), representng external agent lts a boo slowly rom a heght y to a heght y. In mechancs, the specc orms o potental energy are: (a) Gravtatonal potental energy.(b) Elastc potental energy. Potental energy s a scalar quantty and ts SI unt s. Chapter Seven - Potental Energy and Conservaton o Energy 7 Gravtatonal Potental Energy: The gravtatonal potental energy U g s the energy that an object o mass m has by vrtue o ts poston relatve to the surace o the earth. That poston s measured by the heght y o the object relatve to an arbtrary zero level. The wor done by the external agent on the system (object and the Earth) as the object undergoes ths upward dsplacement s gven by the product o the upward led orce F and the upward dsplacement o ths orce, Δr = Δy Ug mgy ( F ). y ( mg)( y y ) mgy mgy ext App Gravtatonal potental energy s the potental energy o the object - Earth system. As an object alls ts energy s transerred rom gravtatonal energy to netc energy by the gravtatonal orce.as shown n g (7-). Fgure (7-), representng transerred rom gravtatonal energy to netc energy. 10
From ths result, we see that the wor done on any object by the gravtatonal orce s equal to the negatve o the change n the system s gravtatonal potental energy. 7 Elastc Potental Energy The elastc potental energy o the system can be thought o as the energy stored n the deormed sprng (one that s ether compressed or stretched rom ts equlbrum poston). The wor done by an external led orce F on a system consstng o a bloc connected t o the sprng s gven by Equaton: ext 1 x 1 x ext : The wor done by sprng and s the orce constant. In ths stuaton, the ntal and nal x coordnates o the bloc are measured rom ts equlbrum poston, x = 0. Thereore the elastc potental energy stored n a sprng o orce constant s gven by: U S 1 x Notes 1- The elastc potental energy stored n a sprng s zero whenever the sprng s unreormed (x =0) as shown n (a) n the ollowng gure. Chapter Seven - Potental Energy and Conservaton o Energy - Energy s stored n the sprng only when the sprng s ether stretched or compressed. - hen an object o mass m pushed aganst the sprng compressng t a dstance x, as shown n (b) n the ollowng, the elastc potental energy stored n the sprng s (1/)x. 4- hen the object s released rom rest, the sprng snaps bac to ts orgnal length and the stored elastc potental energy s transormed nto netc energy o the object, as shown n (c) n the ollowng gure. 5- The elast c potental energy s a maxmum when the sprng has reached ts maxmum compresson or extenson (that s, when x s a maxmum). 6- Because the elastc potental s proportonal to x, U s s always postve n deormed sprng. Fgure (7-), representng the elastc potental energy stored n a sprng. 7-4 Conservatve and Non-conservatve Forces 104
7 4 1 Conservatve Force: Conservatve orces have two mportant propertes: 1- A orce s sad to be conservatve the wor done by the orce actng on a partcle movng between two ponts s ndependent o the path the partcle taes between the ponts. - The wor done by a conservatve orce movng through any closed path s zero. (A clo sed path s one n whch the begnnng and end ponts are dented). Fgure (7-4), representng the wor done by a conservatve orce movng through any closed path. Chapter Seven - Potental Energy and Conservaton o Energy For example o a conservat ve orce: (1) the gravtatonal orce : The wor done by the gravtatonal orce on an object s ndependent on the path o the object s movements. It only depends on the derence o the object s ntal and nal poston n the drecton o the orce. Fgure (7-5), representng the gravtatonal orce. mgy mgy g g ) The orce that a sprng exerts on any object attached to the sprng : The wor done on the object by the sprng depends only on the ntal and nal poston o the dstorted sprng. 1 1 g x x 7 4 Non-conservatve Forces: A orce or whch the wor depends on the path s called a non-conservatve orce. For example o a non- conservatve orce s the rctonal orce. 105
The wor done by rcton along that path 1 s gven by: 1 The wor done by the rcton orce along path 14 s gven by: d d d d 14 d Notes Fgure (7-6), representng the wor done aganst the orce o netc rcton depends on the path taen as the boo s moved rom to. 7 5 Relatonshp between Conservatve Forces and Potental Energy e can dene a potental energy uncton U, such that the wor done by a conservatve orce equals the decrease n the potental energy o the system. The wor done by a conservatve orce F as a partcle moves along the x-axs s: Chapter Seven - Potental x Energy and Conservaton o Energy c Fx dx Uؤ x here F x s the component o F n the drecton o the dsplacement. That s, the wor done by a conservatve orce equals the negatve o the change n the potental energy assocated wth that orce. The above equaton can also express as: In general, U = U(x, y, z) c ÄU U F dr ÄU U U U F dx 1- U s negatve when F x and dx are n the same drecton, as when an object s lowered n a gravtatonal eld or when a sprng pushes an object toward equlbrum. - I a conservatve orce does postve wor on a system, potental energy s lost. x x x - I a conservatve orce does negatve wor, potental energy s ganed. 4- I the conservatve orce s nown as a uncton o poston, we can use the equaton: 106
U x x x F dx U 5- The above equaton s used to calculate the change n potental energy o a system as an object wthn the system moves rom x to x. x 7 6 How to nd F x the potental energy o the system s nown I the pont o lcaton o the orce undergoes an nntesmal dsplacement dx, we can express the nntesmal change n the potental energy o the system du as: du F dx Thereore, the conservatve orce s related to the potental energy uncton through the relatonshp: du F x dx That s, any conservatve orce actng on an object wthn a system equals the negatve dervatve o the potental energy o the system wth respect to x. In three dmensons, the expresson s: U U U F î ĵ ˆ U x y z Snce U=U(x, y, z) and U x Chapter Seven - Potental F x Energy ; F y and ; F z Conservaton o Energy Let's very ths expresson correctly gves the gravtatonal orce and the elastc orce when usng the gravtatonal potental energy and the elastc potental energy: x U y (1) The gravtatonal potental energy uncton s U g = mgy From the equaton: (x) F y du(y) dy U z It ollows that: (x) F y du(y) dy d dy mgy mg () The elastc potental energy o the deormed sprng s U s = (1/)x From the equaton: (x) F x du(x) dx It ollows that: du(x) Fx (x) dx 7 7 Conservaton o Mechancal Energy d dx 1 x x 107
The total mechancal energy o a system s dened as: the sum o the netc energy K and the potental energy U. The prncple o mechancal energy conservaton states that: the total mechancal energy o a system remans constant n any solated system o objects that nteracts only through conservatve orces. An solated system: s a system that there s no net wor s done on the system by external orces. You can wrte the conservaton o energy statement n many derent mathematcal orms. Here are some o them: E E E ÄE ÄK ÄU K K U U K E The above equaton s vald only when no energy s added to or removed rom the system. Furthermore, there must be no non-conservatve orces dong wor wthn the system. 0 U I more than one conservatve orce acts on an object wthn a system, a potental energy uncton s assocated wth each orce. In such a case, we can ly the prncple o mechancal energy conservaton or the system as: K U K U Chapter Seven - Potental Energy and Conservaton o Energy here the number o terms n the sums equals the numbers o conservaton orces present. For example, an object connected to a sprng oscllates vertcally, two conservatve orces act on the object; the sprng orce and the gravtatonal orce. So that we can express the above equaton as: K U E K U g, U s, I some o the orces actng on objects wthn the system are not conservatve, then the mechancal energy o the system does not reman constant. Let us examne two types o non-conservatve orces, (a) An led orce; and (b) The orce o netc rcton. 0 K K (a) or Done by an Appled Force I s the wor done by an led orce on an object and c s the wor done by the conservatve orce on the object, then the net wor done on the object s related to the change n ts netc energy accordng to the wor-netc energy theorem and s gven by: ÄK net net c 0 U U Kؤ c g, U s, 0 Potental energy s related to conservatve orces only, so we can use the expresson: c Uؤ From substtuton: 108
Eؤ Uؤ Kؤ The rght sde o the above equaton represents the change n the mechancal energy o the system. e conclude that, an object s part o a system, then an led orce can transer energy nto or out o the system. (b) Force o Knetc Frcton Knetc rcton s an example o a non-conservatve orce. hen an object moves through a dstance d on a horzontal surace that s not rctonless, the only orce that does wor s the orce o netc rcton. Ths orce causes a decrease n the netc energy o the object, so that: ÄK d rcton I an object moves on an nclne that s not rctonless, a change n the gravtatonal potental energy o the object-earth system also occurs, and d s the amount by whch the mechancal energy o the system changes because o the orce o the netc rcton. In such cases: ÄE ÄK ÄU d here: E ÄE E Example 7.1: An Esmo returnng rom a successul shng trp pulls a sled loaded wth salmon. The total mass o the sled and salmon s 50.0 g, and the Esmo exerts a orce o 1.0 10 Chapter Seven - Potental Energy and Conservaton o Energy N on the sled by pullng on the rope. (a) How much wor does he do on the sled the rope s horzontal to the ground (θ = 0 n Fgure) and he pulls the sled 5.00 m? (b) How much wor does he do on the sled θ = 0 and he pulls the sled the same dstance? (Treat the sled as a pont partcle, so detals Such as the pont o attachment o the rope mae no derence.) Soluton: (a) Fnd the wor done when the orce s horzontal. FÄx (1.0 10 N)(5.00) 6.00 10 (b) Fnd the wor done when the orce s exerted at a 0 angle. ( F cos )Äx (1.0 10 N)(cos0)(5.00) 5.010 Example 7.: Suppose that n Example 7.1 the coecent o netc rcton between the loaded 50.0g sled and snow s 0.00. (a) The Esmo agan pulls the sled 5.00 m, exertng a orce o 1.0 10 N at an angle o 0. Fnd the wor done on the sled by rcton, and the net wor. (b) Repeat the calculaton the led orce s exerted at an angle o 0.0 wth the horzontal Soluton: (a) Fnd the wor done by rcton on the sled and the net wor, the led orce s horzontal. F y n mg 0 n mg rc x nx mgx ( 0.)(50)(9.8)(5) 4.9 10 109
110 Sum the rctonal wor wth the wor done by the led orce rom Example 7.1 to get the net wor (The normal and gravty orces are perpendcular to the dsplacement, so they don t contrbute): 610 ( 4.9 10 ) 0 0 1.110 net rc n g (b) Recalculate the rctonal wor and net wor the led orce s exerted at a 0.0 angle. F n mg F sn 0 n mg F sn y rc x nx ( mg F sn ) x (0.)(50 9.8 1. 10 sn 0)(5) 4.10 net rc n g 5.10 ( 4.10 ) 0 0 90 10 Example 7.: A 60.0-g ser s at the top o a slope, as shown n Fgure. At the ntal pont A, she s 10.0 m vertcally above pont B.(a) Settng the zero level or gravtatonal potental energy at B, nd the gravtatonal potental energy o ths system when the ser s at A and then at B. Fnally, nd the change n potental energy o the ser Earth system as the ser goes rom pont A to pont B. (b) Repeat ths problem wth the zero level at pont A. (c) Repeat agan, wth the zero level.00 m hgher than pont B. Soluton: (a) Let y = 0 at B. Calculate the potental energy at A, at B, and the change n potental energy. Chapter Seven - Potental Energy and Conservaton o Energy PE PE mgy PE 60 9.810 5.8810 0 5.8810 5.8810 (b) Repeat the problem y = 0 at A, the new reerence pont, so that PE = 0 at A. PE mgy 60 9.8 ( 10) 5.8810 PE PE 5.8810 0 5.8810 (c) Repeat the problem, y = 0 two meters above B. PE PE mgy mgy 609.88 4.7010 60 9.8 ( ) 1.1810 PE PE 1.1810 4.7010 5.8810
111 H. [1] A ball o mass m s dropped rom a heght h above the ground. (a) Neglectng ar resstance, determne the speed o the ball when t s at a heght y above the ground. (b) Determne the speed o the ball at y at the nstant o relea se t already has an ntal speed v at the ntal alttude h. [] A 4g partcle moves rom the orgn to poston C whch has coordnates x=5m and y=5m as shown n the gure. One orce on t s the orce o gravty n the negatve y drecton. Calcu late the wor done by gravty as the partcle moves rom O to C along (a) OAC; (b) OC. Is the orce conservatve? Explan. Chapter Seven - Potental Energy and Conservaton o Energy [] A pendulum conssts o a sphere o mass m attached to a lght cord o length l, as shown n the ollowng gure. The sphere s released rom rest when the cord maes an angle θ 0 wth the vertcal, and the pvot at P s rctonless. (a) Fnd the speed o the sphere when t s at the lowest pont B. (b) hat s the tenson T B n the cord at B?
[4] An electrcally charged partcle s held at rest at the pont x = 0, whle a second partcle wth equal charge s ree to move along the postve x-axs. The potental energy o the system s U(x) = c / x, where c s a postve constant that depends on the magntude o the charges. Derve an expresson or the x-component o orce actng on the movable charge, as a uncton o ts poston. [5] A ser starts rom rest at the top o a rctonless nclne o heght 0m. At the bottom o the nclne, she encounters a horzontal surace where the coecent o netc rcton betwe en the ss and the snow s 0.1. (a ) How ar does travel on the horzontal surace beore comng to rest? (b) Fnd the horzontal dstance the ser travels beore comng to rest the nclne also has a coecent o netc rcton equal to 0.1. [6] A sngle conservatve orce acts on a 5g partcle. The equaton F x = (x + 4) N, where x s n m, descrbes ths orce. As the partcle moves along the x-axs rom x=1m to x = 5m, calculate Chapter Seven - Potental Energy and Conservaton o Energy (a) The wor done by ths orce; (b) The change n the potental energy o the system; (c) The netc energy o the partcle at x = 5m ts speed at 1m s m/s. [7 Two blocs are connected by a lght strng that passes over a rctonless pulley. The bloc o mass m 1 les on a horzontal surace and s connect to a sprng o orce constant. The system s released rom rest when the sprng s unstretched. I the hangng bloc o mass m alls a dstance h beore comng to rest. Calculate the coecent o netc rcton between the bloc o mass m 1 and the surace. [8] A 5g bloc s set nto moton up an nclned plane wth an ntal speed o 8m/s. The bloc comes to rest ater travelng m along the plane, whch s nclned an angle o 0 º to the horzontal. For ths moton determne: (a) The change n the blocۥs netc energy. (b) The change n the potental energy. (c) The rctonal orce exerted on t (assumed to be constant). 11
[9] The coecent o rcton between the g bloc and surace n the ollowng gure s 0.4. The system starts rom rest. hat s the speed o the 5g ball when t has allen 1.5m, usng energy methods? [10] Suppose the ntal netc and potental energes o a system are 75 and 50 respectvely, and that the nal netc and potental energes o the same system are 00 a nd - 5 respectvely. How much wor was done on the system by non-conservatve orces? [11] A dver o mass m drops rom a board 10m above the water surace, as n the gure. Fnd hs speed 5m above the water surace. Neglect ar resstance Chapter Seven - Potental Energy and Conservaton o Energy [1] A 0.5g bloc rests on a horzontal, rctonless surace as n the gure; t s pressed aganst a lght sprng havng a sprng constant o = 800N/m, wth an ntal compresson o cm. To what heght h does the bloc rse when movng up the nclne? [1] An 8.0 g mountan clmber s n the nal stage o the ascent o Pe s Pea, whch 401m above sea level. 11
114 (a) hat s the change n gravtatonal potental energy as the clmber gans the last 100m o alttude? Use U=0 at sea level. (b) Do the same calculaton wth U=0 at the top o the pea. [14] A g mass starts rom rest and sldes a dstance d down a rctonless 0 º nclne. hle sldng, t comes nto contact wth an unstressed sprng o neglgble mass, as shown n the ollowng gur e. The mass sldes an addtonal 0.m as t brought momentarly to rest by compresson o the sprng (=400N/m). Fnd the ntal separaton d between the mass and the sprng. Chapter Seven - Potental Energy and Conservaton o Energy [15] A g bloc s attached to a sprng o a orce constant 500N/m on a horzontal table. The bloc s pulled 5cm to the rght o equlbrum and released rom rest. Fnd the speed o the bloc as t passes through equlbrum (a) The horzontal surace s rctonless and (b) The coecent o rcton between bloc and surace s 0.50.