SPH4 nit 6.3 Gavitational Potential negy Page of Notes Physics ool box he gavitational potential enegy of a syste of two (spheical) asses is diectly popotional to the poduct of thei asses, and invesely popotional to the distances between thei centes. GM gh (notice the negative sign) Wgavity A gavitational potential enegy of zeo is assigned to an isolated syste of two asses that ae so fa apat (thei sepaation is appoaching infinity) that the foce of gavity between the has dopped to zeo. scape Speed Is the iniu speed needed to poject a ass fo the suface of ass M to just escape the gavitational foce of M. scape negy is the iniu kinetic enegy needed to poject a ass fo the suface of ass M to just escape the gavitational foce of M. Binding negy is the aount of additional kinetic enegy needed by a ass to just escape fo a ass M. A paticle gains o loses kinetic enegy because it inteacts with othe objects that exet foces on it. We ecall that the change in a paticles kinetic enegy is equal to the total wok done on the paticle by the foces acting on it. In any situations it sees as though enegy has been stoed in a syste, to be ecoveed late. Fo exaple, you ust do wok to lift a hae. It sees easonable that in hoisting the hae into the ai you ae stoing enegy in the syste, this enegy is late conveted into kinetic enegy as the hae falls. As the hae is aised (and held) in the ai, to enegy povided is a easue of the potential o possibility fo wok to be done, hat is, thee is a potential fo wok to be done on it by the gavitational foce. Fo this eason, enegy associated with position (whee the hae is held) is called potential enegy. When we apply this idea with a body s weight and height above the gound, we call this gavitational potential enegy. Fo exaple Wok is done by the gavitational foce w duing the vetical otion of a body fo an initial height y to a final height y Now, the gavitational potential enegy deceases if the body oves downwad, and the gavitational potential enegy inceases if it oves upwad.
SPH4 nit 6.3 Gavitational Potential negy Page of Let s deteine the wok done by the weight (foce of gavity) when the body dops fo a height y above the oigin to a lowe height y (see (a) above). he foce o gavity and displaceent ae in the sae diection (down), so the wok gavity Wgavity done on the body by the foce of gavity is positive. W Fs g y y gy gy his expession also gives the coect wok when the body oves upwad and y is geate than y (as in (b) above). In this case y y is negative, and thus Wgavity is negative because the foce of gavity and displaceent ae opposite in diection. his quantity, the poduct of the weight g and the height h above the oigin o coodinates, is called the gavitational potential enegy, gh Now its initial value gy and its final value gy. he change in is the final value inus the initial value, o. We can expess the wok W gavity done by the gavitational foce duing the displaceent fo y to y W gavity he negative sign in font of is essential. When the body oves up, h inceases, the wok done by the gavitational foce is negative, and the gavitational potential enegy inceases ( 0 ). When
SPH4 nit 6.3 Gavitational Potential negy Page 3 of the body oves down, h deceases, the gavitational foce does positive wok, and the gavitational potential enegy deceases ( 0 ). he unit of potential enegy is the joule (), the sae unit as is used in wok. Note: ven though you ight be tepted, it is not coect to call =gh the gavitational potential enegy of the body as g depends upon the ass M and the height h above M that is. xaple A box of ass 56.7 is dopped off the edge of a building and falls 6.00. Deteine the gavitational potential enegy of the box when it is at the edge of the building befoe its fall. Solution: he +y is upwad, and the efeence position (y=0) is the gound. Given: 56.7, g.80, y 6.00, =? s gh s 56.7.80 6.00 3 3.330
SPH4 nit 6.3 Gavitational Potential negy Page 4 of 3 3.33 0 he boxes potential enegy elative to the gound is Note: the answe in the above exaple does not depend upon the path the box took to each the top of the oof. xaple What is the change in gavitational potential enegy of a 73. astonaut, lifted fo the ath s suface and placed into a cicula obit at an altitude of 5.50 0 k? Solution: G N 6.67 0, 73., M 4 5.8 0, 6 6.38 0,? Note: ust change 5.50 0 k to 5 5.50 0 N 6.67 0 5.8 0 73. 4 GM 6 4.580 6.380 N 6.67 0 5.8 0 73. 4 GM 6 5 4.0 6.380 5.500 4.0 4.580 3.63 0 8
SPH4 nit 6.3 Gavitational Potential negy Page 5 of he change in gavitational potential enegy is 8 3.63 0 scape fo a Gavitational Field We have seen that any two asses have gavitational potential enegy of GM at a sepaation of distance. he negative value of this potential enegy is chaacteistic of a potential well. See potential enegy diaga below. If the Space Shuttle was sitting on the gound at A on the diaga, then it would have value of and a kinetic enegy of zeo, thus a total enegy of Afte blastoff, the ocket is now is at an inceasing altitude and the inceases along the ed cuve. Suppose that the Shuttle engines give the Shuttle a fixed speed such that the kinetic enegy is epesented by the vetical distance AB on the ed cuve. Now its total enegy is
SPH4 nit 6.3 Gavitational Potential negy Page 6 of As its altitude inceases, inceases along the ed cuve AC, yet eains constant along the hoizontal segent BC. he kinetic enegy deceases and, at any point, is given by the length of the vetical line fo the cuve to the hoizontal line BC. When the shuttle eaches an altitude coesponding to point C, has deceased to zeo, and the ocket can go no highe. Instead, it falls back down, with as on the upwad tip. and govened by the sae constaints Now the lage the speed, the highe point B is on the blue line. It is inteesting to deteine the iniu speed the Shuttle would need to escape the potential well. hat is the distance fo the cuve to the hoizontal line ust be coveed by the. his also eans that the Shuttle would each an infinite distance whee 0 est. v 0 GM v GM N 6.670 5.8 0 6 6.380 4 s.0 k. s 4 his speed is called the escape speed Recalling fo a pevious chapte Now v v GM v GM GM GM GM GM befoe coing to a his tells us that the total enegy of anything in obit is negative and is equal to one-half
SPH4 nit 6.3 Gavitational Potential negy Page 7 of the value of the gavitational potential enegy at its adius of obit. xaple A 8.00 0 satellite is placed in a geosynchonous obit aound the ath a) Deteine the adius of the satellite s obit b) Deteine the gavitational potential enegy of the satellite when it is still on the launch pad. c) Deteine the total enegy of the satellite when it is in obit. d) Deteine how uch wok the launch ocket ust do on the satellite to place it in obit. e) Once in obit, deteine how uch additional enegy would the satellite equie to escape fo ath s potential well. Solution: a) N 4 4 G 6.670, 4.0h 8.640 s, M 5.8 0 Fo any satellite (independent of ass) 4 F F 3 3 G GM GM 4 N 6.670 5.80 8.640 4 7 4.0 s 4 4 his adius epesents an altitude of 7 6 7 4. 0 6.380 3.58 0 b) 6 6.38 0 8.00 0 At the ath s suface GM N 6.670 5.80 8.000 6 6.380 0 5.000 4
SPH4 nit 6.3 Gavitational Potential negy Page 8 of he gavitational potential enegy is 0 5.00 0. c) 7 4. 0 he total enegy of a satellite in cicula obit, bound to the ath, is given by: GM v GM N 6.670 5.80 8.000 7 4.0 3.780 4 he total enegy of the satellite in geosynchonous obit is 3.78 0 d) W ( obit) 3.780 5.000 4.60 0 ath 0 e) o escape ath s potential enegy well, the total enegy of the satellite ust be zeo o geate. Since in obit the enegy was 3.78 0 on enegy. 3.78 0, then the satellite ust gain at least
SPH4 nit 6.3 Gavitational Potential negy Page of xta Notes and Coents