# FXA Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

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1 Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing a cicle. 1 Deive fom fist pinciples, the equation : Select and apply the equation : fo planets and satellites (natual and atificial). T 2 = 4π 2 3 GM T 2 = 4π 2 3 GM Use gavitational field lines to epesent a gavitational field. Select and apply Keple s thid law to solve poblems. T 2 α 3 State Newton s law of gavitation. Select and use the equation fo the foce between two point o spheical objects : F = - GMm Define geostationay obit of a satellite and state the uses of such satellites. GRAVITATIONAL FIELDS Select and apply the equation fo the gavitational field stength (g) of a point mass : g = - GM The mass of an object ceates a GRAVTATIONAL FIELD aound it and this foce field exets an attactive foce on any othe mass which is placed in the field egion. All masses, fom the smallest paticles of matte to the lagest stas, have a gavitational field aound them. Select and use the equation g = - GM to detemine the mass of the Eath o anothe simila object. When an object is dopped, the Eath and the object exet equal and oppositely diected foces on each othe, but because The object s mass is minute in compaison to that of the Eath, it is the object which is pulled towads the Eath. Explain that close to the Eath s suface the gavitational field stength is unifom and appoximately equal to the acceleation of fee fall. Analyse cicula obits in an invese squae law field by Relating the gavitational foce to the centipetal acceleation it causes. A GRAVITATIONAL FIELD is a egion in space in which any mass will expeience a foce of attaction. All masses have a gavitational field aound them.

2 2 GRAVITATIONAL FIELD STRENGTH (g) The FIELD STRENGTH (g) at a point in a gavitational field is the FORCE (F) pe UNIT MASS (m) expeienced by a small* test mass placed at the point. The weight of an object is the foce of gavity acting on it. If an object of mass (m) is in a gavitational field of stength (g), the gavitational foce (F) on the object is : F = mg * The test mass must be small enough so as not to cause a significant change in the gavitational field being measued. If the object is allowed to fall feely unde the action of this foce, it Acceleates with an acceleation : a = F = mg = g m m FIELD STRENGTH (g) is expessed mathematically as : (N) g = F m (N kg -1 ) (kg) Field stength at any point = The acceleation of fee fall in a gavitational field (N kg -1 ) expeienced by an object at that point (m s -2 ) Show that N kg -1 is the same as m s -2. POINTS TO NOTE Fo a planet, g is the foce exeted by the planet s gavity on a 1 kg mass placed on its suface. The value of g vaies slightly fom place to place on the Eath s suface due to : Non-unifomities in the Eath s shape and composition. The effect of the Eath s spin, which educes g by an amount vaying fom zeo at the poles to a maximum at the equato. The aveage value of the Eath s gavitational field stength is 9.81 N kg -1. GRAVITATIONAL FIELD STRENGTH is a vecto quantity.

3 3 The stength of the field is indicated by the sepaation of the field lines. GRAVITATIONAL FIELD LINES The concept of field stength gives us a measue of the foce involved in any paticula gavitational inteaction, and field lines enables us to pictue the shape of the field as well as the diection of the foces aound the body. In a RADIAL field, the sepaation of the field lines inceases with distance fom the cente, indicating that the field stength is deceasing as the distance inceases. The diagam below uses field lines to show the Eath s gavitational field. Gavitational field lines Close to the suface and ove an aea small in compaison with the oveall aea of the planet, the field can be assumed to be UNIFORM (i.e. constant stength and diection). This is indicated By PARALLEL field lines. PRACTICE QUESTIONS (1) On a planetay scale the field lines divege with distance fom the Eath s suface. The field is RADIAL. Close to the Eath s suface the field lines can be assumed to be paallel. 1 (a) What is a gavitational field? (b) Define gavitational field stength. POINTS TO NOTE The diection of the field lines indicates the diection of the gavitational foce acting on a mass situated in the field. This is the diection in which a feely-falling mass will acceleate and defines the vetical diection. The field lines ae diected towads the cente of the planet Which tells us that the gavitational field is ATTRACTIVE. (c) What does a field line indicate in a gavitational field? (d) With the aid of a diagam in each case, explain what is meant by : (i) A RADIAL field. (ii) A UNIFORM field.

4 4 2 (a) What is the gavitational foce acting on an object of mass 48 kg on the luna suface whee the field stength is 1.67 N kg -1? (b) Calculate the field stength at a point in a gavitational field whee an object of mass 5.0 kg expeiences a foce of 75 N. 3 An object of mass (m) is situated at a point in a gavitational field whee the field stength is (g). Show that the acceleation of fee fall of the object at this point is also (g). Inseting a constant of popotionality tuns this into a mathematical equation which expesses NEWTON S LAW OF GRAVITATION : (N) (N m 2 kg -2 ) (kg) F = - G m 1 m 2 (m) G = univesal gavitational constant = 6.67 x N m 2 kg -2 NEWTON S LAW OF GRAVITATION 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 sepaation. m 1 m 2 F Conside two point masses (m 1 and m 2 ) whose centes ae distance () apat. Then, using Newton s law of gavitation, the gavitational attaction foce (F) which each mass exets on the othe is given by : F POINTS TO NOTE The minus sign is thee because it is conventional in field theoy to egad foces exeted by attactive fields as negative, and gavity is attactive eveywhee in the univese. Anothe eason is that is measued outwads fom the attacting body and F acts in the opposite diection. Gavitational foces ae extemely weak, unless at least one of the objects is of planetay mass o lage. Gavitational foces act at a distance, without the need fo an intevening medium. Newton s law is expessed in tems of point masses. Fo eal bodies, the law can be applied by assuming all the m 1 F F m 2 mass of a body to be concentated at its cente of mass. The sepaation () is then the distance between the centes of mass. F α m 1 m 2

5 2 5 Newton s law of gavitation is an example of an invese squae law. Complete the table below which will aid you undestanding of the invese squae natue of the law. distance apat Gavitational foce F The diagam above shows a spacecaft of mass 3000 kg at vaious distances fom Eath, coesponding to R, 2R, 4R and 8R, whee R is the adius of the Eath (6.4 x 10 6 m). Calculate the gavitational foce on the spacecaft on the Eath s suface, assuming the mass of the Eath to be 6.0 x kg, and G = 6.67 x N m 2 kg -2. PRACTICE QUESTIONS (2) 1 Calculate the gavitational foce between the following pais of objects. Take G = 6.67 x N m 2 kg -2. (a) A man of mass 95 kg on the Eath s suface, given that the mass of the Eath is 6.0 x kg and its adius is 6400 km. (b) Two spacecaft of masses 2500 kg and 3200 kg, when thei centes of mass ae 12 m apat. (c) Two potons, each of mass 1.67 x kg, whose centes ae 1.0 x m apat. Calculate the foce on the spacecaft at each position shown, and expess these foces as factions of the foce at the Eath s suface. Do you answes suppot the invese squae law of gavitation. 3 A spacecaft of total mass 2500 kg is at the halfway point between the Eath and the Moon. Calculate : (a) The gavitational attaction foce on the spacecaft : (i) Due to the Eath, (ii) Due to the Moon. (b) The magnitude and diection of the esultant gavity foce. Eath mass = 6.0 x kg Moon mass = 7.4 x10 22 kg Distance between centes of Eath and Moon = 3.8 x 10 8 m. Univesal gavitational constant, G = 6.67 x N m 2 kg -2.

6 6 GRAVITATIONAL FIELD STRENGTH OF A POINT MASS VARIATION OF g WITH DISTANCE FROM EARTH S CENTRE Conside a mass (m) at a distance () fom the cente of a planet o sta of mass (M), whee the gavitational field stength is (g). EARTH Fom the definition of field stength, the foce (F) acting on (m) is : planet of mass = M F m F = mg...(1) And applying Newton s law of gavitation, the foce (F) is : g-value at Eath suface F = -G Mm..(2) Combining equations (1) and (2) gives : Fom which : g = - GM mg = -G Mm (N kg-1) ( N m 2 kg -2 ) (m) (kg) The above gaph shows the elationship between gavitational field stength (g) and distance fom the cente of the Eath (). It shows that : Below the suface : g is diectly popotional to. At the cente : g = 0. Fo > R (Eath adius) : g is invesely popotional to. NOTE : All the above applies to any planet o sta.

7 PRACTICE QUESTIONS (3) Assume G = 6.67 x N m 2 kg Calculate the mass of the Moon, given that its adius is 1.74 x 10 6 m and the gavitational field stength at is suface is 1.70 N kg 1. 5 Instuments in a spacecaft ae used to find values fo the 7 gavitational field stength (g) due to the Moon. Conside the gaph shown below g/n kg -1 2 The Sun has a mass of 2.0 x kg and a mean adius of 1.4 x 10 9 m. Calculate : 0.40 (a) The gavitational field stength at : (i) Its suface, (ii) The Eath s obit, which is at a distance of 1.5 x m fom the Sun. (b) The Eath has a mass of 6.0 x kg. Show that at a distance of km fom the Eath s cente, its gavitational field stength is equal and opposite to that of the Sun. 3 The gavitational field stength on the Eath s suface is 9.81 N kg -1. If the Eath has a mass (M) and a mean adius (R), Calculate the field stength : (a) At a point which is at a distance of 4R fom the cente of the Eath. (b) At the suface of a planet having a mass = 2M and a adius = 3R. 4 X is a point on a spheical planet of adius 2000 km. Y is a point 1000 km above the suface of the planet. Calculate the atio g x /g y of the acceleations of fee fall measued at X and Y x x x x Fo points outside the Moon, the field is consideed to be that of a point mass, equal to the mass of the Moon, at the Moon s cente. (a) Calculate the numeical value of the gadient of the gaph. (b) Show that the gadient is equivalent to GM, whee G is the univesal gavitational constant, and M is the mass of the Moon. (c) Hence detemine the mass (M) of the Moon. 1 m 2

8 8 6 Use the intenet to find the suface gavitational field stength and the diamete of the planets in the sola system. Use the data obtained to calculate the mass of each planet. Then use the intenet to check you calculated values. centipetal foce povided by attactive gavitational foce between Eath and Sun. v SATELLITE ORBITS SUN M EARTH m Any body obiting a planet is a satellite of that planet. Ou Moon, fo example, is a natual satellite of planet Eath, while Eath itself is a natual satellite of the Sun. cicula obit Some planets have many natual satellites, and even the paticles which constitute the ings of planets like Satun can be thought of as satellites of thei planet. The diagam above shows the Eath of mass (m) obiting the Sun of mass (M) with a speed (v) at an obital adius (). The centipetal foce needed fo the cicula motion is povided by the gavitational foce acting between the Sun and Eath. Theefoe : gavitational foce = centipetal foce G Mm = mv 2 Atificial satellites ae becoming inceasingly numeous and they maintain thei obits due to the gavitational attaction between themselves and the Eath, at sufficient heights to escape atmospheic fiction that would dissipate thei enegy and send cashing back to Eath. Fom which : v 2 = GM. (1) But speed, v = distance tavelled in one complete obit = 2π Time taken fo one complete obit (PERIOD) T Then, substituting fo v in equation (1) gives : (2π) 2 = GM T 2 Expanding and eaanging gives : T 2 = 4π 2 3 GM

9 9 The equation opposite shows that fo a given planet o sta, the atio (T 2 / 3 ) is a constant fo all of its satellites, egadless of thei mass. To pove the above, NEWTON had assumed that : The Sun and the planets wee point masses. T 2 = 4π 2 * The gavitational foce between the Sun and the planets was diectly popotional to thei masses and invesely popotional to the squae of thei distance apat. Foty yeas o so ealie, the astonome JOHANNES KEPLER, had made vey caeful obsevations of the TIME PERIOD (T) and the AVERAGE ORBITAL RADIUS () fo each of the planets in the sola system. Based on these measuements, KEPLER had poposed his THIRD LAW of planetay motion : 3 GM The atio (T 2 / 3 ) is the same (i.e. constant) fo all the planets. PRACTICE QUESTIONS (4) 1 A satellite is moving in a cicula obit at a speed of 3.5 x 10 3 m s -1 aound a planet of mass (M). The time peiod of the satellite is 100 minutes. Calculate : (a) The obit adius. T = 2π 3 GM This equation allows us to calculate the PERIOD (T) of any satellite fom its ORBIT RADIUS () and the MASS (M) of the planet o sta it obits. (b) The centipetal acceleation of the satellite. (c) The mass (M) of the planet. (Assume G = 6.67 x N m 2 kg -2 ) So NEWTON was able to use his THEORY OF GRAVITATION to pove KEPLER S THIRD LAW. Equation * above can be eaanged to give the following foms : M = 4π 2 3 GT 2 This equation allows us to calculate the MASS (M) of the cental planet o sta fom the PERIOD (T) and ORBIT RADIUS () of one of its satellites. 2 A satellite is in a cicula obit aound the Eath at a height of 110 km above the suface. Given that the adius of the Eath is 6400 km and that G = 6.67 x N m 2 kg -2, calculate : (a) The Eath s gavitational field stength at the obit height. (b) The satellite speed. (c) The time peiod of the satellite.

10 10 3 Calculate the mass of the Sun fom the data given below : Mean adius of Eath s obit aound the Sun = 1.5 x m. Eath s peiodic time = days. Gavitational constant, G = 6.67 x N m 2 kg -2. satellite Eath otation km GEOSTATIONARY SATELLITE - GEOSYNCHRONOUS ORBIT geosynchonous obit In the ealy days of satellite communication, the satellites wee in faily close Eath obits and so they wee visible ove the hoizon fo only shot peiods of time. This was of limited value because boadcasts wee only possible when the satellite was in ange of both the tansmitte and the eceive. In 1945, the sci-fi wite Athu C. Clake pedicted the value of satellites which would obit the Eath with the same angula speed and diection as the Eath. These would appea to be stationay ove a point on the Eath s suface and theefoe always be available fo eceiving o tansmitting adio waves anywhee on the side of the planet facing the satellite. Thee ae now well ove 130 of these GEOSTATIONARY satellites in GEOSYNCHRONOUS obit of the Eath, most of which ae used fo telecommunication, paticulaly television boadcasting. The pictue below gives some idea of the incedible numbe of atificial satellites which now cicle ou planet. A GEOSTATIONARY SATELLITE is in a GEOSYNCHRONOUS ORBIT. This means that it : Has an obit cented on the Eath s cente. Tavels above the equato in the same diection as that of the Eath (west to east). Has an obital peiod the same as that of the Eath s otation about its own axis (24 hous). Always appeas to be above the same point on the Eath s suface. Geostationay satellites foming a cicle above Eath s equato

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