Chapter 7: Conservation of Energy

Transcription

1 Lecture 7: Conservaton o nergy Chapter 7: Conservaton o nergy Introucton I the quantty o a subject oes not change wth tme, t means that the quantty s conserve. The quantty o that subject remans constant even ts change between two ponts s erent. The best way to eplan that quantty s the energy. I the energy s conserve n a system, then t s known that the total amount o ths energy remans constant even ts shape changes. e know that rom prevous Chapter the energy s the capacty to o work. In act, the work s a kn o energy. The work s relate to the orce that s apple to a boy to change ts poston. So, there s relaton between energy an the orce. Conservatve/non-conservatve orces The orces that are apple on boes to o work are calle conservatve or ervatve orces. On a straght lne, a boy moves orwar an backwar, then the orce eerte on ths boy s calle conservatve. In one menson, the gravtatonal orce an the sprng orce can be gven as eamples or conservatve orces. I a boy moves upwar, the gravtaton orce oes a work whch s negatve: 1 1 mg y y 7.1 Then, the boy alls reely, an the work one by the gravtaton orce becomes postve: mg y y 7. The total work one by the gravtatonal orce s 0 snce we know that the ntal an nal ponts or the object s same: 1 1 y y, y y For the sprng orce: F k 7.4 The total work one by the sprng s agan 0 snce. Then 1 k I a orce F apple on a boy changes ts poston uner a rcton orce, then ths orce can not be conservatve. ven though the object changes ts poston orwar an backwar, there wll be always a rcton orce whch s n the opposte recton. Snce the work one by the rcton orce s negatve, the total work one by the orce F wll not be 0. So, the rctonal orce s a non-conservatve orce. In summary, a conservatve orce s a orce or whch the work one on a close path s equal to zero. 1

2 Lecture 7: Conservaton o nergy Conservatve systems an mechancal energy The system means an object an ts envronment whch nteracts wth that object. A car movng on an nclne plane s n the system o the car tsel, the nclne plane, an the earth ue to gravtaton. So; the system s not only the object but ts envronment that has eects on the moton o that object. I orces n a system are conservatve than ths system s calle conservatve system. An eample or a conservatve system s reely-allng object: I the ar resstance s neglecte, then the work one on the object s: mg y y 7.6 sng energy-work theorem, an then rom ths equaton we see that each se s equal to a constant. Then we say where s the potental energy or the object at heght y an s the kc energy. Then t s seen that cons tan t total Ths s the Mechancal nergy or the object an t s constant. Potental energy an the conservaton o mechancal energy The amount o change n the potental energy s gven by: here, the negatve sgn n the ntegraton shows us that the ecrease n the potental s conserve by the ncrease n the kc energy. 7.1 then 7.13 a we get 7.10 F 7.11

3 Lecture 7: Conservaton o nergy 7.14 The energy s conservatve n a conservatve system. Gravtatonal potental energy The work one by the gravtatonal orce s mg y y 7.15 I I y 0 0, y 0 0, There s no erence or the sgn o the potental energy or the object. You can take whether y 0 or y 0. The erence between two ponts wll be always equal to each other. Potental energy or a sprng The Hooke law states that F k 7.16 an the work one by the sprng s gven by k Snce the potental energy s k I 0 k As seen n the last quaton, the potental energy or a sprng can not be negatve. Non-conservatve orces e know that the mechancal energy s wrtten as Now, we know that the work one by an object may not be conservatve all tme. Then we wrte the work one by the orce as an we know, so an we sa 1 cons cons

4 Lecture 7: Conservaton o nergy here, t s seen obvously that 0 an can not be calculate easly. To n we must know the ntal an nal contons o the object. amples an Problems Queston 7.1: As t s known the ar resstance s a kn o orce wth a magntue proportonal to v, an t always acts n the opposte recton o the velocty o the partcle. Is t a conservatve orce? plan your reason. Soluton 7.1: No! Let us conser an object thrown nto the ar: t rst reaches a mamum heght then returns to the groun. It thus completes a roun trp close path. By our rst prncple o conservatve orces, the total work one by ar resstance over ths close loop must be zero. However, snce the ar resstance always opposes the moton o objects movng wth a velocty, t acts n the opposte recton as the splacement o the object or the entre trp. Thus the total work over ths close loop must be negatve, an ar resstance, much lke rcton, s a non-conservatve orce. Queston 7.: The orce o a mass on a sprng s gven by F k. Calculate the total work one by the sprng over one complete oscllaton. To o ths calculaton, assume that the ntal splacement o the mass s rom, to -, an then t returns back to ts orgnal splacement rom, to In ths way conrm the act that the sprng orce s conservatve!. Soluton 7.: To calculate the total work one urng the trp, we must evaluate the ntegral F. Snce the mass changes ts rectons n the moton, we must actually evaluate two ntegrals: one rom, to -, an one rom, to : F k an nally 0 k 1 1 k k 0 0 Fgure 7.1: a ntal poston o mass, b poston o mass halway through oscllaton, c nal poston o mass. 4

5 Lecture 7: Conservaton o nergy Thus the total work one over a complete oscllaton a close loop s zero, conrmng that the sprng orce s nee conservatve. 5