GRAVITATION MODULE - 1 OBJECTIVES. Notes

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1 Gavitation Motion, Foce and Enegy 5 GRAVIAION Have you eve thought why a ball thown upwad always comes back to the gound? O a coin tossed in ai falls back on the gound. Since times immemoial, human beings have wondeed about this phenomenon. he answe was povided in the 17th centuy by Si Isaac Newton. He poposed that the gavitational foce is esponsible fo bodies being attacted to the eath. He also said that it is the same foce which keeps the moon in its obit aound the eath and planets bound to the Sun. It is a univesal foce, that is, it is pesent eveywhee in the univese. In fact, it is this foce that keeps the whole univese togethe. In this lesson you will lean Newton s law of gavitation. We shall also study the acceleation caused in objects due to the pull of the eath. his acceleation, called acceleation due to gavity, is not constant on the eath. You will lean the factos due to which it vaies. You will also lean about gavitational potential and potential enegy. You will also study Keple s laws of planetay motion and obits of atificial satellites of vaious kinds in this lesson. Finally, we shall ecount some of the impotant pogammes and achievements of India in the field of space eseach. OBJECIVES Afte studying this lesson, you should be able to: state the law of gavitation; calculate the value of acceleation due to gavity of a heavenly body; analyse the vaiation in the value of the acceleation due to gavity with height, depth and latitude; distinguish between gavitaitonal potential and gavitational potential enegy; identify the foce esponsible fo planetay motion and state Keple s laws of planetay motion; 11

2 Gavitation calculate the obital velocity and the escape velocity; explain how an atificial satellite is launched; distinguish between pola and equatoial satellites; state conditions fo a satellite to be a geostationay satellite; calculate the height of a geostationay satellite and list thei applications; and state the achievements of India in the field of satellite technology. MODULE - 1 Motion, Foce and Enegy 5.1 LAW OF GRAVIAION It is said that Newton was sitting unde a tee when an apple fell on the gound. his set him thinking: since all apples and othe objects fall to the gound, thee must be some foce fom the eath acting on them. He asked himself: Could it be the same foce which keeps the moon in obit aound the eath? Newton agued that at evey point in its obit, the moon would have flown along a tangent, but is held back to the obit by some foce (Fig. 5.1). Could this continuous fall be due to the same foce which foces apples to fall to the gound? He had deduced fom Keple s laws that the foce between the Sun and planets vaies as 1/. Using this esult he was able to show that it is the same Moon Eath Fig. 5.1 : At each point on its obit, the moon would have flown off along a tangent but the attaction of the eath keeps it in its obit. foce that keeps the moon in its obit aound the eath. hen he genealised the idea to fomulate the univesal law of gavitation as. Evey paticle attacts evey othe paticle in the univese with a foce which vaies as the poduct of thei masses and invesely as the squae of the distance between them. hus, if m 1 and m ae the masses of the two paticles, and is the distance between them, the magnitude of the foce F is given by. F mm 1 o F G mm 1 (5.1) he constant of popotionality, G, is called the univesal constant of gavitation. Its value emains the same between any two objects eveywhee 11

3 Motion, Foce and Enegy Gavitation in the univese. his means that if the foce between two paticles is F on the eath, the foce between these paticles kept at the same distance anywhee in the univese would be the same. One of the extemely impotant chaacteistics of the gavitational foce is that it is always attactive. It is also one of the fundamental foces of natue. Remembe that the attaction is mutual, that is, paticle of mass m 1 attacts the paticle of mass m and m attacts m 1. Also, the foce is along the line joining the two paticles. Knowing that the foce is a vecto quantity, does Eqn. (5.1) need modification? he answe to this question is that the equation should eflect both magnitude and the diection of the foce. As stated, the gavitational foce acts along the line joining the two paticles. hat is, m attacts m 1 with a foce which is along the line joining the two paticles (Fig. 5.). If the foce of attaction exeted by m 1 on m is denoted by F 1 and the distance between them is denoted by 1, then the vecto fom of the law of gavitation is mm F 1 G 1 ˆ (5.) 1 1 F 1 F 1 m 1 1 m Fig. 5. : he masses m 1 and m ae placed at a distance 1 fom eact othe. he mass m 1 attacts m with a foce F 1. Hee ˆ 1 is a unit vecto fom m 1 to m In a simila way, we may wite the foce exeted by m on m 1 as mm F 1 G 1 1 ˆ 1 (5.) As ˆ1 ˆ, fom Eqns. (5.) and (5.) we find that 1 F1 F 1 (5.4) he foces F 1 and F 1 ae equal and opposite and fom a pai of foces of action and eaction in accodance with Newton s thid law of motion. Remembe that ˆ 1 and ˆ 1 have unit magnitude. Howeve, the diections of these vectos ae opposite to each othe. Unless specified, in this lesson we would use only the magnitude of the gavitational foce. he value of the constant G is so small that it could not be detemined by Newton o his contempoay expeimentalists. It was detemined by Cavendish fo the fist time about 100 yeas late. Its accepted value today is Nm kg. It is because of the smallness of G that the gavitational foce due to odinay objects is not felt by us. 114

4 Gavitation Example 5.1 : Keple s thid law states (we shall discuss this in geate details late) that if is the mean distance of a planet fom the Sun, and is its obital peiod, then / const. Show that the foce acting on a planet is invesely popotional to the squae of the distance. Solution : Assume fo simplicity that the obit of a planet is cicula. (In eality, the obits ae nealy cicula.) hen the centipetal foce acting on the planet is F mv MODULE - 1 Motion, Foce and Enegy π whee v is the obital velocity. Since v ω, whee is the peiod, we can ewite above expession as F m π o F 4π m But o K (Kelple s d law) whee K is a constant of popotionality. Hence 4π m F K 1 o F 4π m K (Q 4π m K 4π m K 1. is constant fo a planet) Befoe poceedins futhe, it is bette that you check you pogess. INEX QUESIONS he peiod of evolution of the moon aound the eath is 7. days. Remembe that this is the peiod with espect to the fixed stas (the peiod of evolution with espect to the moving eath is about 9.5 days; it is this peiod that is used to fix the duation of a month in some calendas). he adius of moon s obit is m (60 times the eath s adius). Calculate the centipetal acceleation of the moon and show that it is vey close to the value given by 9.8 ms divided by 600, to take account of the vaiation of the gavity as 1/.. Fom Eqn. (5.1), deduce dimensions of G. 115

5 Motion, Foce and Enegy Gavitation. Using Eqn. (5.1), show that G may be defined as the magnitude of foce between two masses of 1 kg each sepaated by a distance of 1 m. 4. he magnitude of foce between two masses placed at a cetain distance is F. What happens to F if (i) the distance is doubled without any change in masses, (ii) the distance emains the same but each mass is doubled, (iii) the distance is doubled and each mass is also doubled? 5. wo bodies having masses 50 kg and 60 kg ae sepeated by a distance of 1m. Calculate the gavitational foce between them. 5. ACCELERAION DUE O GRAVIY Fom Newton s second law of motion you know that a foce F exeted on an object poduces an acceleation a in the object accoding to the elation F ma (5.5) he foce of gavity, i.e., the foce exeted by the eath on a body lying on o nea its suface, also poduces an acceleation in the body. he acceleation poduced by the foce of gavity is called the acceleation due to gavity. It is denoted by the symbol g. Accoding to Eq. (5.1), the magnitude of the foce of gavity on a paticle of mass m on the eath s suface is given by mm F G R (5.6) whee M is the mass of the eath and R is its adius. Fom Eqns. (5.5) and (5.6), we get mm mg G R M o g G R (5.7) Remembe that the foce due to gavity on an object is diected towads the cente of the eath. It is this diection that we call vetical. Fig. 5. shows vetical diections at diffeent places on the eath. he diection pependicula to the vetical is called the hoizontal diection. Once we know the mass and the adius of the eath, o of any othe celestial body such as a planet, the value of g at its suface can be calculated using Eqn. (5.7). On the suface of the eath, the value of g is taken as 9.8 ms. 116

6 Gavitation Given the mass and the adius of a satellite o a planet, we can use Eqn. (5.7) to find the acceleation due to the gavitational attaction of that satellite o planet. Befoe poceeding futhe, let us look at Eqn. (5.7) again. he acceleation due to gavity poduced in a body is independent of its mass. his means that a heavy ball and a light ball will fall with the same velocity. If we dop these balls fom a cetain height at the same time, both would each the gound simultaneously. C B Vetical fo B Vetical fo C Vetical fo A Fig. 5. : he vetical diection at any place is the diection towads the cente of eath at that point A MODULE - 1 Motion, Foce and Enegy ACIVIY 5.1 ake a piece of pape and a small pebble. Dop them simultaneously fom a cetain height. Obseve the path followed by the two bodies and note the times at which they touch the gound. hen take two pebbles, one heavie than the othe. Release them simultaneously fom a height and obseve the time at which they touch the gound. Fall Unde Gavity he fact that a heavy pebble falls at the same ate as a light pebble, might appea a bit stange. ill sixteenth centuy it was a common belief that a heavy body falls faste than a light body. Howeve, the geat scientist of the time, Galileo, showed that the two bodies do indeed fall at the same ate. It is said that he went up to the top of the owe of Pisa and eleased simultaneously two ion balls of consideably diffeent masses. he balls touched the gound at the same time. But when feathe and a stone wee made to fall simultaneously, they eached the gound at diffeent times. Galileo agued that the feathe fell slowe because it expeienced geate foce of buoyancy due to ai. He said that if thee wee no ai, the two bodies would fall togethe. In ecent times, astonauts have pefomed the feathe and stone expeiment on the moon and veified that the two fall togethe. Remembe that the moon has no atmosphee and so no ai. Unde the influence of gavity, a body falls vetically downwads towads the eath. Fo small heights above the suface of the eath, the acceleation due to gavity does not change much. heefoe, the equations of motion fo initial and final velocities and the distance coveed in time t ae given by 117

7 Motion, Foce and Enegy v u + gt Gavitation s ut + ( 1 )gt and v u + gs. (5.8) It is impotant to emembe that g is always diected vetically downwads, no matte what the diection of motion of the body is. A body falling with an acceleation equal to g is said to be in fee-fall. Fom Eqn. (5.8) it is clea that if a body begins to fall fom est, it would fall a distance h (1/)gt in time t. So, a simple expeiment like dopping a heavy coin fom a height and measuing its time of fall with the help of an accuate stop watch could give us the value of g. If you measue the time taken by a five-upee coin to fall though a distance of 1 m, you will find that the aveage time of fall fo seveal tials is 0.45 s. Fom this data, the value of g can be calculated. Howeve, in the laboatoy you would detemine g by an indiect method, using a simple pendulum. You must be wondeing as to why we take adius of the eath as the distance between the eath and a paticle on its suface while calculating the foce of gavity on that paticle. When we conside two disceet paticles o mass points, the sepaation between them is just the distance between them. But when we calculate gavitational foce between extended bodies, what distance do we take into account? o esolve this poblem, the concept of cente of gavity of a body is intoduced. his is a point such that, as fa as the gavitational effect is concened, we may eplace the whole body by just this point and the effect would be the same. Fo geometically egula bodies of unifom density, such as sphees, cylindes, ectangles, the geometical cente is also the cente of gavity. hat is why we choose the cente of the eath to measue distances to othe bodies. Fo iegula bodies, thee is no easy way to locate thei centes of gavity. Whee is the cente of gavity of metallic ing located? It should lie at the cente the ing. But this point is outside the mass of the body. It means that the cente of gavity of a body may lie outside it. Whee is you own cente of gavity located? Assuming that we have a egula shape, it would be at the cente of ou body, somewhee beneath the navel. Late on in this couse, you would also lean about the cente of mass of a body. his is a point at which the whole mass of the body can be assumed to be concentated. In a unifom gavitational field, the kind we have nea the eath, the cente of gavity coincides with the cente of mass. he use of cente of gavity, o the cente of mass, makes ou calculations extemely simple. Just imagine the amount of calculations we would have to do if we have 118

8 Gavitation to calculate the foces between individual paticles a body is made of and then finding the esultant of all these foces. You should emembe that G and g epesent diffeent physical quantities. G is the univesal constant of gavitation which emains the same eveywhee, while g is acceleation due to gavity, which may change fom place to place, as we shall see in the next section. You may like to answe a few questions to check you pogess. MODULE - 1 Motion, Foce and Enegy INEX QUESIONS he mass of the eath is kg and its mean adius is m. Calculate the value of g at the suface of the eath.. Caeful measuements show that the adius of the eath at the equato is 678 km while at the poles it is 657 km. Compae values of g at the poles and at the equato.. A paticle is thown up. What is the diection of g when (i) the paticle is going up, (ii) when it is at the top of its jouney, (iii) when it is coming down, and (iv) when it has come back to the gound? 4. he mass of the moon is kg and its adius is m. Calculate the gavitational acceleation at its suface. 5. VARIAION IN HE VALUE OF G 5..1 Vaiation with Height he quantity R in the denominato on the ight hand side of Eqn. (5.7) suggests that the magnitude of g deceases as squae of the distance fom the cente of the eath inceases. So, at a distance R fom the suface, that is, at a distance R fom the cente of the eath, the value of g becomes (1/4) th of the value of g at the suface. Howeve, if the distance h above the suface of the eath, called altitude, is small compaed with the adius of the eath, the value of g, denoted by g h, is given by GM g h ( R+ h) GM h R 1+ R 119

9 Motion, Foce and Enegy g h 1+ R Gavitation (5.9) whee g GM/R is the value of acceleation due to gavity at the suface of the eath. heefoe, g g h h 1+ R 1 + h R + h R Since (h/r) is a small quantity, (h/r) will be a still smalle quantity. So it can be neglected in compaison to (h/r). hus g h g h 1+ R (5.10) Let us take an example to undestand how we apply this concept. Example 5. : Moden aicafts fly at heights upwad of 10 km. Let us calculate the value of g at an altitude of 10 km. ake the adius of the eath as 6400 km and the value of g on the suface of the eath as 9.8 ms. Solution : Fom Eqn. (5.8), we have g h g.(10) km km 9.8 ms ms. 5.. Vaiation of g with Depth Conside a point P at a depth d inside the eath (Fig. 5.4). Let us assume that the eath is a sphee of unifom density ρ. he distance of the point P fom the cente of the eath is (R d). Daw a sphee of adius ( d). A mass placed at P will expeience gavitational foce fom paticles in (i) the shell of thickness d, and (ii) the sphee of adius. It can be shown that the foces due to all the paticles in the shell cancel each othe. hat is, the net foce on the paticle at P due to the matte in the shell is zeo. heefoe, in calculating the acceleation due to gavity at P, we have to conside only the R d d Fig. 5.4 : A point at depth d is at a distance R d fom the cente of the eath R P 10

10 Gavitation mass of the sphee of adius ( d). he mass M of the sphee of adius ( d) is MODULE - 1 Motion, Foce and Enegy M 4 π ρ (R d) (5.10) he acceleation due to gavity expeienced by a paticle placed at P is, theefoe, M g d G ( R-d) 4 π G ρ (R d) (5.11) Note that as d inceases, (R d) deceases. his means that the value of g deceases as we go below the eath. At d R, that is, at the cente of the eath, the acceleation due to gavity will vanish. Also note that (R d) is the distance fom the cente of the eath. heefoe, acceleation due to gavity is linealy popotional to. he vaiation of g fom the cente of the eath to distances fa fom the eath s suface is shown in Fig ms g.45 ms O R R Fig. 5.5 : Vaiation of g with distance fom the cente of the eath We can expess g d in tems of the value at the suface by ealizing that at d 0, we get the suface value: g 4 πg ρr. It is now easy to see that g d g ( R d) R (5.1) g d 1 R, 0 d R On the basis of Eqns. (5.9) and (5.1), we can conclude that g deceases with both height as well as depth. 11

11 Motion, Foce and Enegy Intenal Stuctue of the Eath Gavitation 645 km 490 km Coe kg Mantle kg CRUS kg 5km Fig. 5.6 :Stuctue of the eath (not to scale). hee pominent layes of the eath ae shown along with thei estimated masses. Refe to Fig. 5.6 You will note that most of the mass of the eath is concentated in its coe. he top suface laye is vey light. Fo vey small depths, thee is hadly any decease in the mass to be taken into account fo calculating g, while thee is a decease in the adius. So, the value of g inceases up to a cetain depth and then stats deceasing. It means that assumption about eath being a unifom sphee is not coect. 5.. Vaiation of g with Latitude You know that the eath otates about its axis. Due to this, evey paticle on the eath s suface excecutes cicula motion. In the absence of gavity, all these paticles would be flying off the eath along the tangents to thei cicula obits. Gavity plays an impotant ole in keeping us tied to the eath s suface. You also know that to keep a paticle in cicula motion, it must be supplied centipetal foce. A small pat of the gavity foce is used in supplying this centipetal foce. As a esult, the foce of attaction of the eath on objects on its suface is slightly educed. he maximum effect of the otation of the eath is felt at the equato. At poles, the effect vanishes completely. We now quote the fomula fo vaiation in g with latitude without deivation. If g λ denotes the value of g at latitude λ and g is the value at the poles, then g λ g Rω cosλ, (5.1) whee ω is the angula velocity of the eath and R is its adius. You can easily see that at the poles, λ 90 degees, and hence g λ g. Example 5. : Let us calculate the value of g at the poles. Solution : he adius of the eath at the poles 657 km m 1

12 Gavitation he mass of the eath kg Using Eqn. (5.7), we get MODULE - 1 Motion, Foce and Enegy g at the poles [ / ( ) ] ms 9.85 ms Example 5.4 : Now let us calculate the value of g at λ 60º, whee adius of eath is 671 km. Solution : he peiod of otation of the eath, 4 hous ( ) s fequency of the eath s otation 1/ angula fequency of the eath ω π/ π/( ) Rω cos λ ( ) ms Since g 0 g Rω cos λ, we can wite g λ (at latitude 60 degees) ms INEX QUESIONS At what height must we go so that the value of g becomes half of what it is at the suface of the eath?. At what depth would the value of g be 80% of what it is on the suface of the eath?. he latitude of Delhi is appoximately 0 degees noth. Calculate the diffeence between the values of g at Delhi and at the poles. 4. A satellite obits the eath at an altitude of 1000 km. Calcultate the acceleation due to gavity acting on the satellite (i) using Eqn. (5.9) and (ii) using the elation g is popotional to 1/, whee is the distance fom the cente of the eath. Which method do you conside bette fo this case and why? 5.4 WEIGH AND MASS he foce with which a body is pulled towads the eath is called its weight. If m is the mass of the body, then its weight W is given by W mg (5.14) Since weight is a foce, its unit is newton. If you mass is 50 kg, you weight would be 50 kg 9.8 ms 490 N. 1

13 Motion, Foce and Enegy Gavitation Since g vaies fom place to place, weight of a body also changes fom place to place. he weight is maximum at the poles and minimum at the equato. his is because the adius of the eath is minimum at the poles and maximum at the equato. he weight deceases when we go to highe altitudes o inside the eath. he mass of a body, howeve, does not change. Mass is an intinsic popety of a body. heefoe, it stays constant wheeve the body may be situated. Note: In eveyday life we often use mass and weight intechangeably. Sping balances, though they measue weight, ae maked in kg (and not in N) Gavitational Potential and Potential enegy he Potential enegy of an object unde the influence of a consevative foce may be defined as the enegy stoed in the body and is measued by the wok done by an extenal agency in binging the body fom some standad position to the given position. If a foce F displaces a body by a small distance d aganist the consevative foce, without changing its speed, the small change in the potential enegy du is given by, du F.d In case of gavitational foce between two masses M and m sepaated by a distance, GMm F gavitational potential enegy GMm du d 1 GMm o U GMm d It shows that the gavitational potential enegy between two paticles of masses M and m sepaated by a distance is given by GMm U + a constant he gavitational potential enegy is zeo when appoaches infinity. So the constant is zeo and U GMm 14

14 Gavitation Gavitational Potential (V) of mass M is defined as the gavitational potential enegy of unit mass. Hence, MODULE - 1 Motion, Foce and Enegy U GM Gavitational potential, V m It is a scala quantity and its SI unit is J/kg. ACIVIY 5. Calculate the weight of an object of mass 50 kg at distances of R, R, 4R, 5R and 6R fom the cente of the eath. Plot a gaph showing the weight against distance. Show on the same gaph how the mass of the object vaies with distance. y the following questions to consolidate you ideas on mass and weight. INEX QUESIONS Suppose you land on the moon. In what way would you weight and mass be affected?. Compae you weight at Mas with that on the eath? What happens to you mass? ake the mass of Mas 6 10 kg and its adius as m.. You must have seen two types of balances fo weighing objects. In one case thee ae two pans. In one pan, we place the object to be weighed and in the othe we place weights. he othe type is a sping balance. Hee the object to be weighed is suspended fom the hook at the end of a sping and eading is taken on a scale. Suppose you weigh a bag of potatoes with both the balances and they give the same value. Now you take them to the moon. Would thee be any change in the measuements made by the two balances? 4. State the SI unit of Gavitational potential. 5.5 KEPLER S LAWS OF PLANEARY MOION In ancient times it was believed that all heavenly bodies move aound the eath. Geek astonomes lent geat suppot to this notion. So stong was the faith in the eath-cented univese that all evidences showing that planets evolved aound the Sun wee ignoed. Howeve, Polish Astonome Copenicus in the 15th centuy poposed that all the planets evolved aound the Sun. In the 16th centuy, Galileo, based on his astonomical obsevations, suppoted Copenicus. Anothe Euopean astonome, ycho Bahe, collected a lot of obsevations on the motion of planets. Based on these obsevations, his assistant Keple fomulated laws of planetay motion. 15

15 Motion, Foce and Enegy Johannes Keple Geman by bith, Johannes Keple, stated his caee in astonomy as an assistant to ycho Bahe. ycho eligiously collected the data of the positions of vaious planets on the daily basis fo moe than 0 yeas. On his death, the data was passed on to Keple who spent 16 yeas to analyse the data. On the basis of his analysis, Keple aived at the thee laws of planetay motion. Gavitation He is consideed as the founde of geometical optics as he was the fist peson to descibe the woking of a telescope though its ay diagam. Fo his assetion that the eath evolved aound the Sun, Galileo came into conflict with the chuch because the Chistian authoities believed that the eath was at the cente of the univese. Although he was silenced, Galileo kept ecoding his obsevations quietly, which wee made public afte his death. Inteestingly, Galileo was feed fom that blame ecently by the pesent Pope. Keple fomulated thee laws which goven the motion of planets. hese ae: 1. he obit of a planet is an ellipse with the Sun at one of the foci (Fig. 5.7). (An ellipse has two foci.) Planet C B Sun O A D Ellipse Fig. 5.7 : he path of a planet is an ellipse with the Sun at one of its foci. If the time taken by the planet to move fom point A to B is the same as fom point C to D, then accoding to the second law of Keple, the aeas AOB and COD ae equal.. he aea swept by the line joining the planet to the sun in unit time is constant though out the obit (Fig 5.7). he squae of the peiod of evolution of a planet aound the sun is popotional to the cube of its aveage distance fom the Sun. If we denote the peiod by and the aveage distance fom the Sun as, α. Let us look at the thid law a little moe caefully. You may ecall that Newton used this law to deduce that the foce acting between the Sun and the planets 16

16 Gavitation vaied as 1/ (Example 5.1). Moeove, if 1 and ae the obital peiods of two planets and 1 and ae thei mean distances fom the Sun, then the thid law implies that 1 1 (5.15) he constant of popotionality cancels out when we divide the elation fo one planet by the elation fo the second planet. his is a vey impotant elation. Fo example, it can be used to get, if we know 1, 1 and. Example 5.5 : Calculate the obital peiod of planet mecuy, if its distance fom the Sun is m. You ae given that the distance of the eath fom the Sun is m. Solution : We know that the obital peiod of the eath is 65.5 days. So, days and m. We ae told that m fo mecuy. heefoe, the obital peiod of mecuy is given by MODULE - 1 Motion, Foce and Enegy 1 1 On substituting the values of vaious quantities, we get 1 1 (65.5) ( ) m 11 ( ) m 9 days 87.6 days. In the same manne you can find the obital peiods of othe planets. he data is given below. You can also check you esults with numbes in able 5.1. able 5.1: Some data about the planets of sola system Name of Mean distance Radius Mass the planet fom the Sun (in tems (x10 km) (Eath Masses) of the distance of eath) Mecuy Venus Eath Mas Jupite Satun Uanus Neptune Pluto

17 Motion, Foce and Enegy Gavitation Keple s laws apply to any system whee the foce binding the system is gavitational in natue. Fo example, they apply to Jupite and its satellites. hey also apply to the eath and its satellites like the moon and atificial satellites. Example 5.6 : A satellite has an obital peiod equal to one day. (Such satellites ae called geosynchonous satellites.) Calculate its height fom the eath s suface, given that the distance of the moon fom the eath is 60 R E (R E is the adius of the eath), and its obital peiod is 7. days. [his obital peiod of the moon is with espect to the fixed stas. With espect to the eath, which itself is in obit ound the Sun, the obital peiod of the moon is about 9.5 day.] Solution : A geostationay satellite has a peiod equal to 1 day. Fo moon 1 7. days and 1 60 R E, 1 day. Using Eqn. (5.15), we have 1 1 1/ 1/ (60 R E ) (1 day ) 7. day 6.6 R E. Remembe that the distance of the satellite is taken fom the cente of the eath. o find its height fom the suface of the eath, we must subtact R E fom 6.6 R E. he equied distance fom the eath s suface is 5.6 R E. If you want to get this distance in km, multiply 5.6 by the adius of the eath in km Obital Velocity of Planets We have so fo talked of obital peiods of planets. If the obital peiod of a planet is and its distance fom the Sun is, then it coves a distance π in time. Its obital velocity is, theefoe, v ob π (5.16) hee is anothe way also to calculate the obital velocity. he centipetal foce expeienced by the planet is mυ ob /, whee m is its mass. his foce must be supplied by the foce of gavitation between the Sun and the planet. If M is the GmM mass of the Sun, then the gavitational foce on the planet is s. Equating the two foces, we get so that, mυ ob G M s, GM v ob s (5.17) 18

18 Gavitation Notice that the mass of the planet does not ente the above equation. he obital velocity depends only on the distance fom the Sun. Note also that if you substitute v fom Eqn. (5.16) in Eqn. (5.17), you get the thid law of Keple. MODULE - 1 Motion, Foce and Enegy INEX QUESIONS Many planetay systems have been discoveed in ou Galaxy. Would Keple s laws be applicable to them?. wo atificial satellites ae obiting the eath at distances of 1000 km and 000 km fom the suface of the eath. Which one of them has the longe peiod? If the time peiod of the fome is 90 min, find the time peiod of the latte.. A new small planet, named Sedna, has been discoveed ecently in the sola system. It is obiting the Sun at a distance of 86 AU. (An AU is the distance between the Sun and the eath. It is equal to m.) Calculate its obital peiod in yeas. 4. Obtain an expession fo the obital velocity of a satellite obiting the eath. 5. Using Eqns. (5.16) and (5.17), obtain Keple s thid law. 5.6 ESCAPE VELOCIY You now know that a ball thown upwads always comes back due to the foce of gavity. If you thow it with geate foce, it goes a little highe but again comes back. If you have a fiend with geat physical powe, ask him to thow the ball upwads. he ball may go highe than what you had managed, but it still comes back. You may then ask: Is it possible fo an object to escape the pull of the eath? he answe is yes. he object must acquie what is called the escape velocity. It is defined as the minimum velocity equied by an object to escape the gavitational pull of the eath. It is obvious that the escape velocity will depend on the mass of the body it is tying to escape fom, because the gavitational pull is popotional to mass. It will also depend on the adius of the body, because smalle the adius, stonge is the gavitational foce. he escape velocity fom the eath is given by v esc G M R (5.18) 19

19 Motion, Foce and Enegy Gavitation whee M is the mass of the eath and R is its adius. Fo calculating escape velocity fom any othe planet o heavenly body, mass and adius of that heavenly body will have to be substituted in the above expession. It is not that the foce of gavity ceases to act when an object is launched with escape velocity. he foce does act. Both the velocity of the object as well as the foce of gavity acting on it decease as the object goes up. It so happens that the foce becomes zeo befoe the velocity becomes zeo. Hence the object escapes the pull of gavity. y the following questions to gasp the concept. INEX QUESIONS he mass of the eath is kg and its adius is 671 km. Calculate the escape velocity fom the eath.. Suppose the eath shunk suddenly to one-fouth its adius without any change in its mass. What would be the escape velocity then?. An imaginay planet X has mass eight times that of the eath and adius twice that of the eath. What would be the escape velocity fom this planet in tems of the escape velocity fom the eath? 5.7 ARIFICIAL SAELLIES A cicket match is played in Sydney in Austalia but we can watch it live in India. A game of ennis played in Ameica is enjoyed in India. Have you eve wondeed what makes it possible? All this is made possible by atificial satellites obiting the eath. You may now ask : How is an atificial satellite put in an obit? Eath Fig. 5.8 : A pojectile to obit the eath 10

20 Gavitation You have aleady studied the motion of a pojectile. If you poject a body at an angle to the hoizontal, it follows a paabolic path. Now imagine launching bodies with inceasing foce. What happens is shown in Fig Pojectiles tavel lage and lage distances befoe falling back to the eath. Eventually, the pojectile goes into an obit aound the eath. It becomes an atificial satellite. Remembe that such satellites ae man-made and launched with a paticula pupose in mind. Satellites like the moon ae natual satellites. In ode to put a satellite in obit, it is fist lifted to a height of about 00 km to minimize loss of enegy due to fiction in the atmosphee of the eath. hen it is given a hoizontal push with a velocity of about 8 kms 1. he obit of an atificial satellite also obeys Keple s laws because the contolling foce is gavitational foce between the satellite and the eath. he obit is elliptic in natue and its plane always passes though the cente of the eath. Remembe that the obital velocity of an atificial satellite has to be less than the escape velocity; othewise it will beak fee of the gavitational field of the eath and will not obit aound the eath. Fom the expessions fo the obital velocity of a satellite close to the eath and the escape velocity fom the eath, we can wite vsec v ob (5.19) MODULE - 1 Motion, Foce and Enegy Pola plane GN equato equatoial plane GS Fig. 5.9: Equitoial and pola obits Atificial satellites have geneally two types of obits (Fig. 5.9) depending on the pupose fo which the satellite is launched. Satellites used fo tasks such as emote sensing have pola obits. he altitude of these obits is about 800 km. If the obit is at a height of less than about 00 km, the satellite loses enegy because of fiction caused by the paticles of the atmosphee. As a esult, it moves to a 11

21 Motion, Foce and Enegy Gavitation lowe height whee the density is high. hee it gets bunt. he time peiod of pola satellites is aound 100 minutes. It is possible to make a pola satellite sunsynchonous, so that it aives at the same latitude at the same time evey day. Duing epeated cossing, the satellite can scan the whole eath as it spins about its axis (Fig. 5.10). Such satellites ae used fo collecting data fo weathe pediction, monitoing floods, cops, bushfies, etc. Satellites used fo communications ae put in equatoial obits at high altitudes. Most of these satellites ae geo-synchonous, the ones which have the same obital peiod as the peiod of otation of the eath, equal to 4 hous. hei height, as you saw in Example 5.6 is fixed at aound 6000 km. Since thei obital peiod matches that of the eath, they appea to be hoveing above the same spot on the eath. A combination of such satellites coves the entie globe, and signals can be sent fom any place on the globe to any othe place. Since a geo-synchonous satellite obseves the same spot on the eath all the time, it can also be used fo monitoing any peculia happening that takes a long time to develop, such as sevee stoms and huicanes. Descending obit West looking Ascending obit East looking Fig. 5.10: A sun synchomous satellite scanning the eath Applications of Satellites Atificial satellites have been vey useful to mankind. Following ae some of thei applications: 1. Weathe Foecasting : he satellites collect all kinds of data which is useful in foecasting long tem and shot tem weathe. he weathe chat that you see evey day on the television o in newspapes is made fom the data sent by these satellites. Fo a county like India, whee so much depends on timely ains, the satellite data is used to watch the onset and pogess of monsoon. Apat fom weathe, satellites can watch unhealthy tends in cops ove lage aeas, can wan us of possible floods, onset and spead of foest fie, etc. 1

22 Gavitation. Navigation : A few satellites togethe can pinpoint the position of a place on the eath with geat accuacy. his is of geat help in locating ou own position if we have fogotten ou way and ae lost. Satellites have been used to pepae detailed maps of lage chunks of land, which would othewise take a lot of time and enegy.. elecommunication : We have aleady mentioned about the tansmission of television pogammes fom anywhee on the globe to eveywhee became possible with satillites. Apat fom television signals, telephone and adio signals ae also tansmitted. he communication evolution bought about by atificial satellites has made the wold a small place, which is sometimes called a global village. 4. Scientific Reseach : Satellites can be used to send scientific instuments in space to obseve the eath, the moon, comets, planets, the Sun, stas and galaxies. You must have head of Hubble Space elescope and Chanda X-Ray elescope. he advantage of having a telescope in space is that light fom distant objects does not have to go though the atmosphee. So thee is hadly any eduction in its intensity. Fo this eason, the pictues taken by Hubble Space elescope ae of much supeio quality than those taken by teestial telescopes. Recently, a goup of Euopeon scientists have obseved an eath like planet out-side ou sola system at a distance of 0 light yeas. 5. Monitoing Militay Activities : Atificial satellites ae used to keep an eye on the enemy toop movement. Almost all counties that can affod cost of these satellites have them. MODULE - 1 Motion, Foce and Enegy Vikam Ambalal Saabhai Bon in a family of industialists at Ahmedabad, Gujaat, India. Vikam Saabhai gew to inspie a whole geneation of scientists in India. His initial wok on time vaiation of cosmic ays bought him lauels in scientific fatenity. A founde of Physical Reseach Laboatoy, Ahmedabad and a pionee of space eseach in India, he was the fist to ealise the dividends that space eseach can bing in the fields of communication, education, metology, emote sensing and geodesy, etc Indian Space Reseach Oganization India is a vey lage and populous county. Much of the population lives in ual aeas and depends heavily on ains, paticulaly the monsoons. So, weathe foecast 1

23 Motion, Foce and Enegy Gavitation is an impotant task that the govenment has to pefom. It has also to meet the communication needs of a vast population. hen much of ou aea emains unexploed fo mineals, oil and gas. Satellite technology offes a cost-effective solution fo all these poblems. With this in view, the Govenment of India set up in 1969 the Indian Space Reseach Oganization (ISRO) unde the dynamic leadeship of D. Vikam Saabhai. D. Saabhai had a vision fo using satellitis fo educating the nation. ISRO has pusued a vey vigoous pogamme to develop space systems fo communication, television boadcasting, meteoological sevices, emote sensing and scientific eseach. It has also developed successfully launch vehicles fo pola satellites (PSLV) (Fig. 5.11) and geo-synchonous satellites (GSLV) (Fig. 5.1). In fact, it has launched satellites fo othe counties like Gemany, Belgium and Koea. and has joined the exclusive club of five counties. Its scientific pogamme includes studies of (i) climate, envionment and global change, (ii) uppe atmosphee, (iii) astonomy and astophysics, and (iv) Indian Ocean. Recently, ISRO launched an exclusive educational satillite EduSat, fist of its kind in the wold. It is being used to educate both young and adult students living in emote places. It is now making pepaation fo a mission to the moon. Fig. 5.11: PSLV Fig. 5.1: GSLV 14

24 Gavitation MODULE - 1 Motion, Foce and Enegy INEX QUESIONS Some science wites believe that some day human beings will establish colonies on the Mas. Suppose people living this desie to put in obit a Mas synchonous satellite. he otation peiod of Mas is 4.6 hous. he mass and adius of Mas ae kg and 400 km, espectively. What would be the height of the satellite fom the suface of Mas?. List the advantages of having a telescope in space. WHA YOU HAVE LEARN he foce of gavitation exists between any two paticles in the univese. It vaies as the poduct of thei masses and invesely as the squae of distance between them. he constant of gavitation, G, is a univesal constant. he foce of gavitation of the eath attacts all bodies towads it. he acceleation due to gavity nea the suface of the eath is 9.8 ms. It vaies on the suface of the eath because the shape of the eath is not pefectly spheical. he acceleation due to gavity vaies with height, depth and latitude. he weight of a body is the foce of gavity acting on it. the gavitational potential enegy between two paticles of masses M and m sepaated by a distance given by U GMm Keple s fist law states that the obit of a planet is elliptic with sun at one of its foci. Keple s second law states that the line joining the planet with the Sun sweeps equal aeas in equal intevals of time. Keple s thid law states that the squae of the obital peiod of a planet is popotional to the cube of its mean distance fom the Sun. A body can escape the gavitational field of the eath if it can acquie a velocity equal to o geate than the escape velocity. he obital velocity of a satellite depends on its distance fom the eath. 15

25 Motion, Foce and Enegy Gavitation ERMINAL EXERCISE 1. You have leant that the gavitational attaction is mutual. If that is so, does an apple also attact the eath? If yes, then why does the eath not move in esponse?. We set up an expeiment on eath to measue the foce of gavitation between two paticles placed at a cetain distance apat. Suppose the foce is of magnitude F. We take the same set up to the moon and pefom the expeiment again. What would be the magnitude of the foce between the two paticles thee?. Suppose the eath expands to twice its size without any change in its mass. What would be you weight if you pesent weight wee 500 N? 4. Suppose the eath loses its gavity suddenly. What would happen to life on this plant? 5. Refe to Fig. 5.6 which shows the stuctue of the eath. Calculate the values of g at the bottom of the cust (depth 5 km) and at the bottom of the mantle (depth 855 km). 6. Deive an expession fo the mass of the eath, given the obital peiod of the moon and the adius of its obit. 7. Suppose you weight is 500 N on the eath. Calculate you weight on the moon. What would be you mass on the moon? 8. A pola satellite is placed at a height of 800 km fom eath s suface. Calculate its obital peiod and obital velocity. ANSWERS O INEX QUESIONS Moon s time peiod 7.d s Radius of moon s obit R m Moon s obital speed v π R Centipetal accleation v /R 4π R. 1 R 4π R 16

26 Gavitation MODULE π m ( ) s Motion, Foce and Enegy 4π.84 (7..4.6) 10 ms.007 ms If we calculate centipetal acceleation on dividing g by 600, we get the same value : ms ms. F G m m 1 F is foce G Foce (mass) Nm kg mm 1. F G If m 1 1kg, m 1kg, 1m, then F G o G is equal to the foce between two masses of 1kg each placed at a distance of 1m fom each othe 4. (i) F α 1/, if is doubled, foce becomes one-fouth. (ii) F α m 1 m, if m 1 and m ae both doubled then F becomes 4 times. (iii) F α mm 1, if each mass is doubled, and distance is also doubled, then F emains unchanged. 50 kg 60 kg Nm 5. F G 1m ; G kg Nm 000 kg kg. 1m N 10 7 N 17

27 Motion, Foce and Enegy 5. GM 1. g R Gavitation 4 Nm kg kg ( ) m N kg 9.81 m s. g at poles GM R g pole pole 4 Nm kg kg ( ) m N kg 9.81 ms Similaly, g eguato N kg 9.79 ms. he value of g is always vetically downwads. Nm 4. g moon kg kg ( ) m N kg 1.61 m s Let g at distance fom the cente of the eath be called g 1. Outside the eath, then g g 1 R 18

28 Gavitation If g 1 g/ R R 1.41 R Height fom eath s suface R R R. Inside the eath g vaies as distance fom the cente of the eath. Suppose at depth d, g is called g d. hen g d g R d R MODULE - 1 Motion, Foce and Enegy If g d 80%, then R d R d 0. R. In example 5., we calculated ω ad s 1 Rω cos 0º ( ) s ms g at poles 9.85 m s (Calculated in example 5.) g at Delhi 9.85 ms 0.09 ms 4. Using fomula (5.9), g g h h 1+ R 9.84 ms 9.81m s 000 km km 9.81m s 871km 671km 7.47 m s Using vaiation with g GM ( R+ h) Nm kg kg ( ) m 7. ms his gives moe accuate esults because fomula (5.9) is fo the case h << R. In this case h is not << R. 19

29 Motion, Foce and Enegy 5.4 Gavitation 1. On the moon the value of g is only 1/6th that on the eath. So, you weight on moon will become 1/6th of you weight on the eath. he mass, howeve, emains constant.. Mass of Mas 6 10 kg Radius of Mas m M g Mas G R Nm kg kg ( ) m.16 Weight on Mas Weight on Eath m..16 m So, you weight will become oughly 1/4th that on the eath. Mass emains constant.. Balances with two pans actually compae masses because g acts on both the pans and gets cancelled. he othe type of balance, sping balance, measues weight. he balance with two pans gives the same eading on the moon as on the eath. Sping balance with give weight as 1/6th that on the eath fo a bag of potatoes. 4. SI unit of Gavational potential is J/kg Yes. Wheeve the foce between bodies is gavitational, Keple s laws will hold.. Accoding to Keple s thid law 1 1 o α α / So, the satellite which is fathe off has highe peiod. Let 1 90 min, km km 000 km km [Fom the cente of the eath] min (90 min) 871 km 771 km 140

30 Gavitation. Accoding to Keple s thid law MODULE - 1 Motion, Foce and Enegy eath sedna eath sedna [Distance fom the Sun] eath 1 y., eath 1 AU sedna (1y) (86 AU) (86) y (1AU) sedna y 4. If v is the obital velocity of the satellite of mass m at a distance fom the cente of the eath, then equating centipltal foce with the gavitational foce, we have m v G mm v GM whee M is the mass of the eath. 5. Fom Eqs. (5.16) and (5.17), 4π GM 4 π. G M. o α v esc GM R 4 11 Nm kg kg m ms ms kms 1. v esc α 1 R 141

31 Motion, Foce and Enegy If R becomes 1/4th, v esc becomes double. Gavitation. v esc α M R If M becomes eight times, and R twice, then v esc α 4 o v esc becomes double (R + h) 4 G π M ( R+ h) GM (R + h) 4 π ( ) 4 (.14) m R + h 000 km h 6900 km. (a) Images ae cleae (b) x-ay telescopy etc. also wok. Answes to eminal Poblems. 15 N 5. g, 5.5 ms 7. Weight 500 N, mass 50 kg on moon as well as on eath h, v 7.47 km s 1 14