TAMPINES JUNIOR COLLEGE 2009 JC1 H2 PHYSICS GRAVITATIONAL FIELD

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1 TAMPINES JUNIOR COLLEGE 009 JC1 H PHYSICS GRAVITATIONAL FIELD OBJECTIVES Candidates should be able to: (a) show an undestanding of the concept of a gavitational field as an example of field of foce and define gavitational field stength as foce pe unit mass. (b) ecall and use Newton s law of gavitation in the fom F Gm1m. (c) deive, fom Newton s law of gavitation and the definition of gavitational field stength, the equation g GM fo the gavitational field stength of a point mass. (d) ecall and apply the equation g GM fo the gavitational field stength of a point mass to new situations o to solve elated poblems. (e) show an appeciation that on the suface of the Eath g is appoximately constant and is equal to the acceleation of fee fall. (f) define potential at a point as the wok done in binging unit mass fom infinity to the point. (g) solve poblem using the equation GM fo the potential in the field of a point mass. (h) ecognise the analogy between cetain qualitative and quantitative aspects of gavitational and electic fields. (i) analyse cicula obits in invese squae law fields by elating the gavitational foce to the centipetal acceleation it causes. (j) show an undestanding of geostationay obits and thei application. REFERENCES 1. Physics, Robet Hutchings. Advanced Level Physics, Nelkon & Pake 3. Fundamentals of Physics, Halliday and Resnick 009 Tampines Junio College

2 Tampines Junio College Gavitational Field 1 GRAVITATIONAL FORCE Impotant 1.1 What is gavitational foce? Evey paticle of matte in the univese attacts evey othe paticle. This attactive foce is called the gavitational foce. The moe massive and close the paticles ae, the geate is the gavitational foce between them. Gavitational foce is one of the fundamental foces that exist in natue. 1. Newton s law of gavitation The Newton s law of gavitation states that evey paticle in the univese attacts evey othe paticle with a foce which is diectly popotional to the poduct of thei masses and invesely popotional to the squae of thei distance apat. Conside two paticles of masses m 1 and m, and distance apat. F F m 1 m The gavitational foce between the two paticles of mass m 1 and m is given by Gm1m F whee G is the gavitational constant which has a value of N m kg and is the distance between the two paticles. Note: 1. The two paticles exet equal and opposite attactive foce on each othe. The foce is diected along the line joining the two paticles.. Fo a sphee of unifom density, the foce that it exets on othe objects can be obtained by teating the sphee as a point mass at the cente of the sphee. Question: Can Newton s law of gavitation be applied to non-point masses o non-spheical masses? Answe: If the objects involved ae neithe point masses no spheical masses, the fomula is still applicable if the objects ae placed sufficiently fa apat such that thei sizes become negligible compaed to the distance sepaating them. Ms Lim YH (009)

3 Tampines Junio College Gavitational Field Quick Check 1 Eath and the Moon ae gavitationally attacted to each othe. Does the moe massive Eath attact the moon with, (a) a geate foce (b) the same foce (c) o less foce, than the moon attacts Eath? Impotant Quick Check Compaed to the weight of that object at sea level, the weight of an object on peak of Himalayas (about 8848 m above sea level) will be (a) the same. (b) slightly smalle. (c) much smalle. Quick Check 3 On the gound the gavitational foce on an object is W. What is the gavitational foce at a height R, whee R is the adius of the eath? A) 0.5W B) 0.5W C) W D) W * See woked example 1 in tutoial. GRAVITATIONAL FIELD Gavitational field is a egion in which a mass expeiences a gavitational foce..1 Field of a mass The egion aound a mass is a gavitational field. The following diagams show the field lines aound a point mass and aound a unifom sphee (such as the Eath). The lines ae always adial and diected towads the cente of mass. Note that the field lines ae imaginay lines. The field aound the mass is not unifom. The field is stonge at points nea the mass (lines ae close) and becomes weake futhe away (lines ae futhe apat). This can be deduced fom the density of the field lines. The quantity that indicates the stength of the field at a point is known as the gavitational field stength. Ms Lim YH (009) 3

4 Tampines Junio College Gavitational Field. Gavitational field stength The stength of the gavitational field at seveal positions can be compaed by measuing the gavitational foce acting on a standadised mass placed at these positions. Impotant The gavitational field stength at a point is defined as the gavitational foce pe unit mass acting at that point. The usual symbol fo field stength is g and its units ae N kg 1..3 Deiving the field stength equation G M g Conside a point mass M. The point mass sets up a gavitational field aound it. Let us deive an expession fo the field stength at point P at a distance fom the point mass. M P G M m If we place a small mass m s at P, the foce expeienced by it is F G M ms F Foce pe unit mass acting at P is then = G M = m s Field stength at P (which is foce pe unit mass) is theefoe given by m s s G M g.4 Repesentation of the Field Stength The diagam shows the aow epesenting the field stength g at point P due to a point mass M. Field stength is a vecto, its diection is towads the point mass poducing the field. M g P Note: Field stength g is invesely popotional to the squae of the distance (g 1/ ). Such a field is known as an invese squae law field. A minus sign actually appeas on the ight hand side of the fomulae fo the gavitational foce and gavitational field stength, so that they look like Ms Lim YH (009) 4

5 Tampines Junio College F Gm1m Gavitational Field Impotant and G M g The minus sign is used to indicate the attactive natue of the foce (athe than epulsive). It NEED NOT be included when we ae calculating a value fo the foce o the field stength, except duing gaph sketching. Quick Check 4 The gavitational field geneated by (o due to) the moe massive Eath will be (a) geate (b) the same (c) o less than the gavitational field geneated by the moon? Quick Check 5 The gavitational field (due to the Eath) expeienced by an object on the peak of Himalayas (about 8848 m above sea level) is (a) the same as (b) slightly smalle than (c) much smalle than the gavitational field (due to the Eath) expeienced by that object at sea level. Quick Check 6 Kaen s mass is 50 kg and Kenny s mass is 75 kg. They ae both sitting in the TPJC Auditoium (consideing only thei inteaction with Eath only) They will expeience the same a) Gavitational foce b) Gavitational field stength * See woked example in tutoial. Quick Check 7 If a thid object sits between two massive objects like (say when the moon obits ound the eath, it will expeience the Sun s gavitational field as well). What will be the total gavitational field? How should they be added? (A) as a scala (B) as a vecto *Ssee woked example 3 in tutoial Ms Lim YH (009) 5

6 Tampines Junio College Gavitational Field.5 Gaphical epesentation of Eath s field stength The gaph shows how the field stength g vaies with distance fom the cente of the Eath. The field stength follows the invese squae law outside the Eath. The negative sign fo field stength indicates that the diection is towads the Eath s cente. Impotant Eath M field stength g G M R G M g distance ** (see appendix 6).6 Acceleation of fee fall Conside a mass m at the suface of the Eath. Let F be the gavitational foce exeted on the mass. By definition, field stength at the Eath suface would be g = F / m If the mass is allowed to fall feely, its acceleation would be a = F / m = g We thus see that the field stength g at the Eath suface is numeically equal to the acceleation of fee fall. The field stength g at the Eath suface is 9.81N kg 1 (o 9.81 m s - ) and is appoximately constant nea the Eath suface because of the lage size of the Eath. 3 GRAVITATIONAL POTENTIAL ENERGY The potential enegy of a body is the enegy it possesses due to its position in a field of foce. Gavitational potential enegy is theefoe the enegy a mass possesses when it is placed in a gavitational field. Ms Lim YH (009) 6

7 Tampines Junio College 3.1 Gavitational Potential Enegy Nea the Suface of the Eath Conside the diagam on the ight. We have aleady leant that when lifting a mass m though a vetical height h (whee h << ), the potential enegy gained by the mass is Gavitational Field m Impotant Incease in gavitational p.e. = mgh h o U = mgh m Stictly speaking, the quantity mgh epesents the change in potential enegy when the mass is lifted though a vetical height h athe than the absolute amount it possesses at a height h. We conveniently choose the lowe level (usually the gound level) as the zeo potential enegy level so that mgh is the exta enegy a mass has when it is at height h compaed to when it is at the zeo potential enegy level. At this point we would also like to ecall fom the ealie topic on Wok, Enegy and Powe that incease in potential enegy is actually wok done against gavity. It is based on this undestanding that the fomula U = mgh is deived. 3. Fomal Definition of Gavitational Potential Enegy The gavitational potential enegy of a mass at a point is defined as the wok done by an extenal foce in binging the mass fom infinity to that point. Hee, the wok is done by an extenal foce acting in opposite diection to the gavitational attaction. Altenatively, the gavitational potential enegy of a mass at a point can also be undestood as the wok done against gavity in binging the mass fom infinity to that point. To illustate the meaning of the gavitational potential enegy, conside the following diagam of a point mass M. The egion aound the point mass M is the gavitational field set up by the mass. The small mass m will be acceleated to P automatically by the gavitational attaction acting on it. The function of the extenal foce is theefoe to pevent it fom gaining speed (o kinetic enegy) as it moves fom infinity to P. M P gavitation al foce m at infinity extenal foce The symbol used is U and the unit is the Joule (J). Ms Lim YH (009) 7

8 Tampines Junio College Mathematically, the definition can be witten as Gavitational Field Impotant Hence U U F G d d M and m ae point o spheical masses So will the two equations give the same answe if we do some calculations? Lets find out. A 3-kg mass is pojected to a height h above the Eath s suface. Taking the adius of the Eath is 6400 km and mass of Eath = 6 x 10 4 kg. Use U = mgh and U to find the change in gavitational potential enegy if (i) h = 1000 m (ii) h = 1000 km (i) Using U = mgh, U = 3 x 9.8 x 1000 =.94 x 10 4 J Using U, U J (ii) Using U = mgh U = 3 x 9.8 x =.94 x 10 7 J Using U, U J Ms Lim YH (009) 8

9 Tampines Junio College Gavitational Field Conclusion: Both equations gave close answes in (i) but gave quite diffeent answes in (ii). Why? Impotant Note: Stictly speaking, the tem U gives the total potential enegy of the whole system of two masses M and m. When two masses M and m ae placed at a distance apat, it is not ight to say that each mass possesses this amount of potential enegy. Rathe, we should egad this as the enegy shaed by the two masses. PE f Consistency between PE = mgh and PE = h PE i Now, using PE PE i- = PE f = h mass m aised fom suface of Eath though height h U R E h RE 1 1 RE RE h R E h RE RE RE h h R R h E E Now if h is small compaed to than, then R E 1 1 R h E R E Theefoe ΔU h RE GM m h R E mgh Hence U = mgh is a special case of the moe geneal PE. It is valid only if the change in height h is small compaed to the adius of the Eath, when g emains constant. Ms Lim YH (009) 9

10 Tampines Junio College Gavitational Field 3.3 Pojection of a Body fom the Suface of a Planet When a body of mass m is pojected fom the suface of the Eath with an amount of kinetic enegy E K, the futhest distance away fom the cente of the Eath,, that it will each is obtained by using consevation of enegy, i.e. assuming no enegy loss. Impotant The total enegy of the body is conseved, i.e. sum of kinetic enegy and potential enegy emains constant thoughout. In such a case, the kinetic enegy of the body is conveted to potential enegy as it moves away fom the Eath: M = mass of Eath, = adius of Eath KE + PE at suface of Eath = KE + PE at futhest distance away E K + ( ) = 0 + ( ) Note: Cannot use E K = mgh as the value of h may not be small compaed to adius of Eath. This poblem may also be epesented gaphically by the enegy-distance gaph: Potential Enegy E K 0 dist. fom cente of Eath E T = E p + E k E K E p R E The total enegy E T is obtained by adding E K to. The futhest distance is obtained fom the intesection of the line epesenting E T and the potential enegy cuve because at this point, all the enegy is in the fom of potential enegy. Ms Lim YH (009) 10

11 Tampines Junio College Gavitational Field Condition Fo Escape to Infinity (Escape Speed) v Impotant Eath ocket to infinity KE 0 PE = 0 By law of consevation of enegy, since total enegy (KE + PE) at infinity is geate o equal to zeo, theefoe the condition fo escape is KE + PE 0 i.e. ½ mv + (-/ ) 0 v So the escape speed is G M v = G M 4 GRAVITATIONAL POTENTIAL The quantity gavitational potential enegy depends on both the size of the mass as well as the position of the mass. We would like to define a enegy quantity that is only dependent on the position (just like gavitational field stength). This quantity is the gavitational potential. The gavitational potential at a point is defined as the wok done by an extenal agent in binging a unit mass fom infinity to that point. The usual symbol fo gavitational potential is and its SI unit is J kg -1. Suppose we want to find the gavitational potential at a paticula point P. Theoetically, to detemine the potential at P, we would bing in a unit mass (1 kg) all the way fom infinity to P. The wok done in binging the unit mass fom infinity to P is the gavitational potential at P. Fo example, if - J of wok is done in binging a unit mass fom infinity to P, the gavitational potential at P would be - J kg 1. Hee, the wok is done by an extenal foce and not the gavitational attaction. The value can be negative if the extenal foce is opposite to the displacement vecto. Ms Lim YH (009) 11

12 Tampines Junio College Gavitational Field 4.1 Fomula fo Gavitational Potential Fom the definition of gavitational potential, we see that its elation with gavitational potential enegy is given by Impotant U m and hence GM We see that the gavitational potential at a point only depends on its distance fom the souce of gavitation (the mass M). If a mass m is placed at a point with a gavitational potential, its gavitational potential enegy is given by U = m. Note: Both the gavitational potential and the gavitational potential enegy ae scala quantities. The negative sign pesent in thei fomulae is pat of thei numeical values and cannot be left out. * see woked example 4 and 5 in tutoial. 5 RELATIONSHIP BETWEEN FIELD STRENGTH AND POTENTIAL The elationship between gavitational field stength and gavitational potential is d g d You can easily veify that this elation fo the point mass case. The gavitational field stength at a point is equal to the negative of the potential gadient at that point. The minus sign indicates that the potential falls when moving in the diection of the field. The field stength is numeically equal to the potential gadient. Ms Lim YH (009) 1

13 Tampines Junio College Gavitational Field Conside the following gaph that shows how the gavitational potential vaies with the distance fom the cente of a planet. Impotant gavitational potential gadient gives field stength distance The gadient at any point on the cuve epesents the field stength at that point. Similaly, by multiplying mass on both sides of the above equation, we obtain the elation between the gavitational foce F and the gavitational potential enegy U: F du d * See appendix 7 fo moe details and wok example 6 in tutoial fo calculations. 6 ORBITS What keeps a satellite obiting aound the Eath? It is the Eath s gavitational attaction that holds a satellite in its obit. Without gavitational attaction, the satellite would move in a staight line athe than in a cicula path. Once the satellite is in obit, it does not need any ocket moto to keep it in obit. Thee ae many examples of natually occuing obital motion in space. Fo instance, the Moon obits aound the Eath; all the nine planets including the Eath obit aound the Sun. 6.1 Kinematics of cicula obits Satellites and planets may move in obits that ae cicula o elliptical. In this syllabus, we shall only deal with poblems that involve cicula obits. In cicula obits, the satellites and planets move with constant speed. Ms Lim YH (009) 13

14 Tampines Junio College Gavitational Field Conside a satellite of mass m moving aound the Eath of mass M in a cicula obit of adius as shown: Impotant F Thee is only one foce acting on the satellite, the gavitational foce exeted by the Eath. The gavitational foce on the satellite is used as the centipetal foce that keeps the satellite in a cicula obit. As a esult, the satellite does not fall towads the Eath. gavitational foce = centipetal foce GM m mv = = m How is T elated to? gavitational foce = centipetal foce GM m = 3 = m = GM T 4 m T 6. Enegies Associated with Satellites Imagine a satellite of mass m moves in a cicula obit about the Eath of mass M. The adius of the obit is. Deive an expession fo (a) the kinetic enegy T of the satellite, Gavitational foce = centipetal foce GM m kinetic enegy, T = = 1 v mv m = GM m Ms Lim YH (009) 14

15 Tampines Junio College Gavitational Field (b) the gavitational potential enegy V of the satellite, GM m Gavitational potential enegy, V = (c) the total enegy E of the satellite. Impotant Total enegy, E = T + V GM m G M m = + = GM m Note: A negative total enegy indicates that the satellite does not have sufficient enegy to escape fom Eath s gavitational field to infinity. 6.3 Geostationay (o Geosynchonous) obit If the peiod of the satellite s obit is 4 hous, and if the satellite obits ove the equato and in the same diection as the Eath s otation, then the satellite will always appea above the same point on the Eath. Such a satellite is called a geosynchonous o geostationay satellite, and its obit a geosynchonous o geostationay obit. To an obseve on the gound, the geostationay satellite seems to be stationay elative to the obseve. Note that a geostationay obit can (i) only occu above the equato and (ii) tavels fom west to east. Since the peiod of obit of a satellite depends on the adius of the obit (T 3 ), a geostationay satellite is placed at a paticula obital adius. The following calculations show how this obital adius is found. To find the geostationay obit: gavitational foce = centipetal foce GM m = = m = G M T m T Fo geostationay obit, we substitute T = 4 hous = s, M = kg, and obtain = ( ) ( ) (86400) 1 3 = m The adius of a geostationay obit is m. Ms Lim YH (009) 15

16 Tampines Junio College Gavitational Field Summay: (i) Geneal Equations: Note: The fomulas inside the boxes ae applicable only to point masses and unifom sphees while the othe fomulas (inside ellipses) ae valid in all situations. (ii) Obital Motion gavitational foce = centipetal foce GM m = mv = m Ms Lim YH (009) 16

17 Tampines Junio College Gavitational Field Appendices Appendix 1 - SIR ISSAC NEWTON AND GRAVITATION Thee is a popula stoy that Newton was sitting unde an apple tee, an apple fell on his head, and he suddenly thought of the Univesal Law of Gavitation. As in all such legends, this is almost cetainly not tue in its details, but the stoy contains elements of what actually happened. What Really Happened with the Apple? Pobably the moe coect vesion of the stoy is that Newton, upon obseving an apple fall fom a tee, began to think along the following lines: The apple acceleated since it stated with zeo velocity. Thus thee must be a foce that acts on the apple to cause this acceleation. Let's call this foce "gavity", and the associated acceleation the "acceleation due to gavity". Then imagine the apple tee is twice as high. Again, we expect the apple to be acceleated towad the gound, so this suggests that gavity eaches to the top of the tallest apple tee. Fig. 1 Si Isaac's Most Excellent Idea Now came Newton's tuly billiant insight: if the foce of gavity eaches to the top of the highest tee, might it not each all the way to the obit of the Moon! Then, the obit of the Moon about the Eath could be a consequence of the gavitational foce, because the acceleation due to gavity could change the velocity of the Moon in just such a way that it followed an obit aound the eath. This can be illustated with the thought expeiment shown in Fig.. Suppose we fie a cannon hoizontally fom a high mountain; the pojectile will eventually fall to eath, as indicated by the shotest tajectoy in the figue, because of the gavitational foce diected towad the cente of the Eath and the associated acceleation. (Remembe that an acceleation is a change in velocity and that velocity is a vecto, so it has both a magnitude and a diection. Thus, an acceleation occus if eithe o both the magnitude and the diection of the velocity change.) But as we incease the muzzle velocity fo ou imaginay cannon, the pojectile will tavel futhe and futhe befoe etuning to eath. Finally, Newton easoned that if the cannon pojected the cannon ball with exactly the ight velocity, the pojectile would tavel completely aound the Eath, always falling in the Fig. gavitational field but neve eaching the Eath, which is cuving away at the same ate that the pojectile falls. That is, the cannon ball would have been put into obit aound the Eath. Newton concluded that the obit of the Moon was of exactly the same natue: the Moon continuously "fell" in its path aound the Eath because of the acceleation due to gavity, thus poducing its obit. Ms Lim YH (009) 17

18 Tampines Junio College Gavitational Field By such easoning, Newton came to the conclusion that any two objects in the Univese exet gavitational attaction on each othe, with the foce having a univesal fom: Law of Univesal Gavitation Evey object in the univese attacts evey othe object with a foce diected along the line of centes fo the two objects that is popotional to the poduct of thei masses and invesely popotional to the squae of the sepaation between the two objects. F G m m G 1 The constant of popotionality G is known as the univesal gavitational constant. It is temed a "univesal constant" because it is thought to be the same at all places and all times, and thus univesally chaacteizes the intinsic stength of the gavitational foce. Appendix - Expeiment to measue g using falling body The diagam shows the expeimental setup used to measue the acceleation of fee fall using a falling body. ball P s to electonic time light beam light gate A steel ball is eleased fom P and an electonic time is tiggeed to stat by the elease. Afte falling a distance s, it passes a light gate, which causes the time to stop. By measuing the distance s and the time t of fall as ecoded by the time, the fee-fall acceleation g can be detemined fom the equation g = s = ½ g t s t Appendix 3 - Tue weight and appaent weight The tue weight of a body is equal to the gavitational foce on the body and is detemined only by the position of the body. The appaent weight is the foce that the body exets on its suppot. It will not be equal to the tue weight if the body is undegoing acceleation. The appaent weight is zeo i.e. the body is weightless fo the following cases: Ms Lim YH (009) 18

19 Tampines Junio College Gavitational Field (i) (ii) a body falling feely unde gavity a space vehicle obiting the Eath In the second case, the gavitational foce on the man is used entiely to povide the centipetal foce equied to keep him in cicula motion in his obit. The astonaut does not exet any foce on the vehicle. He expeiences weightlessness. Note: A body's tue weight is zeo only at the point whee thee is no gavitational field. This can happen in oute space (o deep space) whee the gavity effects of the Eath and othe planets ae zeo. Anothe example of such a point is to be found between the Eath and the Moon whee the two gavitational fields cancel. Appendix 4 - Factos affecting g on Eath The factos affecting g ae: (a) altitude and latitude (b) density (c) otation of the eath It is assumed in calculations that the Eath is spheical, has unifom density (homogeneous) and that it does not otate. Thus in finding g, we only conside the distance fom the cente of the Eath. Howeve, the eath is actually ellipsoidal i.e. it is flattened at the poles and bulges at the Equato (points at the poles ae close than points on the equato). It has non-unifom density due to unequal deposits of mineal in diffeent pats of the Eath (vaiations in density enable oil pospecting). The Eath otates about a pola axis with an angula speed ω= /T = /(4 x 60 x 60) = 7.7 x 10-5 ad s -1. All objects on the suface of the Eath ae undegoing cicula motion with same value of ω except at the poles whee adius of cicle is zeo. Pat of the gavitational foce (and thus g o) is used to povide the centipetal foce fo the objects. The amount of foce depends on the latitude since the adius of path is diffeent. Appendix 5 - Effect of the otation of the Eath on g The Eath otates fom the West to the East about the Noth South axis with a peiod of 4 hous. Any object on the suface at the equato is otating with the Eath with an angula velocity ω. It thus expeiences a centipetal acceleation. F = gavitational foce of attaction acting on object. N = nomal foce acting on object. By Newton's second law, net foce = F - N = mrω whee R is adius of Eath N F But GmM F = mg (tue weight) R whee g = acceleation of fee fall at Eath's suface = 9.81 m s -. Thus, N = mg - mrω = m(g - Rω ) Ms Lim YH (009) 19

20 Tampines Junio College Gavitational Field By Newton's thid law, N = foce exeted by object on Eath's suface. appaent weight, mg' = N = m(g - Rω ) i.e. appaent acceleation g' = g - Rω Appendix 6 Vaiation of g The Eath may be taken to be unifom sphee of adius and density ρ. Fo less than the adius of the eath, the Mass is NOT CONSTANT but depends on the adius. 3 Hence GM g = 4 G 3 = 4G 3 a linea elationship with Appendix 7 Deivation of elations By definition, U F d Hence, du F d Since = U/m, we have U F d d m m g Thus, d g d Ms Lim YH (009) 0