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PHYSICS CHAPTER 7 CHAPTER 7: Gravitation (2 Hours) 1 PHYSICS CHAPTER 7 Learning Outcome: 7.1 Gravitational Force and Field Strength(1 hour) At the end of this chapter, students should be able to: State and use the Newton’s law of gravitation, m1m2 Fg G 2 r Define gravitational field strength as gravitational force per unit mass ag F m Derive and use gravitational field strength, ag G M r2 Sketch a graph of ag against r and explain the change in ag with altitude and depth from the surface of the earth. 2 PHYSICS CHAPTER 7 7.1 Newton’s law of gravitation 7.1.1 Newton’s law of gravitation States that a magnitude of an …………………..between two point masses is directly ………………....... to the product of their masses and inversely proportional to the square of the …………………between them. OR mathematically, Fg m1m2 m1m2 Fg 2 r and 1 Fg 2 r where m1m2 Fg G 2 r Fg : Gravitational force m1 , m2 : masses of particle 1 and 2 r : distance between particle 1 and 2 G : Universal gravitatio nal Constant 6.67 x10 11 N m 2 kg 2 3 PHYSICS CHAPTER 7 The statement can also be shown by using the Figure 7.1. m1 m2 F12 F21 r Figure 7.1 m1m2 F21 F12 Fg G 2 r where F21 : Gravitational force by particle 2 on particle 1 F12 : Gravitatio nal force by particle 1 on particle 2 4 PHYSICS CHAPTER 7 Figures 7.2a and 7.2b show the gravitational force, Fg varies with the distance, r. Fg Fg gradient Gm1m2 0 Figure 7.2a r 0 Figure 7.2b 1 r2 Notes: Every spherical object with constant density can be reduced to a point mass at the centre of the sphere. The gravitational forces always attractive in nature and the forces always act along the line joining the two point masses. 5 PHYSICS CHAPTER 7 Example 7.1 : A spaceship of mass 9000 kg travels from the Earth to the Moon along a line that passes through the Earth’s centre and the Moon’s centre. The average distance separating Earth and the Moon is 384,000 km. Determine the distance of the spaceship from the Earth at which the gravitational force due to the Earth twice the magnitude of the gravitational force due to the Moon. (Given the mass of the Earth, mE=6.001024 kg, the mass of the Moon, mM=7.351022 kg and the universal gravitational constant, G=6.671011 N m2 kg2) 6 PHYSICS CHAPTER 7 Solution : mE 6.00 10 24 kg; mM 7.35 10 22 kg; ms 900 0 kg; rEM 3.84 108 m mE m FEs s FMs x rEM x mM rEM Given FEs 2FMs 7 PHYSICS CHAPTER 7 C Example 7.2 : 50 g 6 cm 3.2 kg A 8 cm 2.5 kg B Figure 7.3 Two spheres of masses 3.2 kg and 2.5 kg respectively are fixed at points A and B as shown in Figure 8.3. If a 50 g sphere is placed at point C, determine a. the resultant force acting on it. b. the magnitude of the sphere’s acceleration. (Given G = 6.671011 N m2 kg2) 8 PHYSICS Solution : a. CHAPTER 7 mA 3.2 kg; mB 2.5 kg; mC 50 10 3 kg rBC 6 10 2 m; rAC 10 10 2 m C θ sin θ 0.6 FA cos θ 0.8 10 10 2 m FB 2 6 10 m θ A 8 10 - 2 m B The magnitude of the forces on mC, 9 PHYSICS Solution : CHAPTER 7 mA 3.2 kg; mB 2.5 kg; mC 50 10 3 kg rBC 6 10 2 m; rAC 10 10 2 m GmB mC FB 2 rBC Force FA FB x-component (N) FA cos θ 0 y-component (N) FA sin θ FB 10 PHYSICS Solution : CHAPTER 7 F F x y The magnitude of the nett force is F F F 2 2 x and its direction is Fy θ tan F x θ 1 y 11 PHYSICS CHAPTER 7 Solution : b. By using the Newton’s second law of motion, thus F m a C and the direction of the acceleration in the …………………..of the nett force on the mC i.e……………………………………… 12 PHYSICS CHAPTER 7 Exercise 7.1 : Given G = 6.671011 N m2 kg2 1. Four identical masses of 800 kg each are placed at the corners of a square whose side length is 10.0 cm. Determine the nett gravitational force on one of the masses, due to the other three. ANS. : 8.2103 N; 45 2. Three 5.0 kg spheres are located in the xy plane as shown in Figure 7.4.Calculate the magnitude of the nett gravitational force on the sphere at the origin due to the other two spheres. ANS. : 2.1108 N Figure 7.4 13 PHYSICS CHAPTER 7 Exercise 7.1 : 3. Figure 7.5 In Figure 7.5, four spheres form the corners of a square whose side is 2.0 cm long. Calculate the magnitude and direction of the nett gravitational force on a central sphere with mass of m5 = 250 kg. ANS. : 1.68102 N; 45 14 PHYSICS CHAPTER 7 7.1.2 Gravitational Field is defined as a ……………………………surrounding a body that has the property of mass where the …………………….is experienced if a test mass placed in the region. Field lines are used to show gravitational field around an object with mass. For spherical objects (such as the Earth) the field is radial as shown in Figure 7.6. M Figure 7.6 15 PHYSICS CHAPTER 7 The gravitational field in small region near the Earth’s surface are ……………..and can be drawn ……………………..to each other as shown in Figure 7.7. Figure 7.7 The field lines indicate two things: The arrows – the ………………. of the field The spacing – the ………………. of the field Note: The gravitational field is a conservative field in which the work done in moving a body from one point to another is independent of the path taken. 16 PHYSICS CHAPTER 7 Gravitational field strength, ag is defined as the ……………………per ………………of a body (test mass) placed at a point. OR Fg ag m where Fg : Gravitatio nal force ag : Gravitatio nal field strength m : mass of a body (test mass) It is also known as gravitational acceleration (the free-fall acceleration). It is a ……………………….. The S.I. unit of the gravitational field strength is ……………..or …………... 17 PHYSICS CHAPTER 7 7.1.3 Gravitational Acceleration Its direction is in the ……………………..of the gravitational force. Another formula for the gravitational field strength at a point is given by ag ag Fg and m GMm Fg 2 r 1 GMm m r2 GM ag 2 r where M : mass of the point mass r : distance between te st mass and point mass 18 PHYSICS CHAPTER 7 Figure 7.8 shows the direction of the gravitational field strength on a point S at distance r from the centre of the planet. GM ag 2 r M Figure 7.8 r 19 PHYSICS CHAPTER 7 The gravitational field in the small region near the Earth’s surface( r R) are uniform where its strength is 9.81 m s2 and its direction can be shown by using the Figure 7.9. GM ag g 2 R Figure 7.9 where R : radius of the Earth g : gravitatio nal accelerati on 9.81 m s 2 20 PHYSICS CHAPTER 7 Example 7.3 : Determine the Earth’s gravitational field strength a. on the surface. b. at an altitude of 350 km. (Given G = 6.671011 N m2 kg2, mass of the Earth, M = 6.00 1024 kg and radius of the Earth, R = 6.40 106 m) Solution : a. g R r M r R 6.40 10 6 m; ag g The gravitational field strength is GM g 2 R 21 PHYSICS Solution : b. CHAPTER 7 ag h r Rh r 6.40 10 6 350 10 3 r 6.75 10 6 m The gravitational field strength is given by R M GM ag 2 r 22 PHYSICS CHAPTER 7 Example 7.4 : The gravitational field strength on the Earth’s surface is 9.81 N kg1. Calculate a. the gravitational field strength at a point C at distance 1.5R from the Earth’s surface where R is the radius of the Earth. b. the weight of a rock of mass 2.5 kg at point C. Solution : g 9.81 N kg 1 a. The gravitational field strength on the Earth’s surface is The distance of point C from the Earth’s centre is 23 PHYSICS CHAPTER 7 Solution : a. Thus the gravitational field strength at point C is given by ag GM rC 2 b. Given m 2.5 kg The weight of the rock is ag GM 2.5R 2 W ma g 24 PHYSICS CHAPTER 7 Example 7.5 : 5 km B A Figure 7.10 Figure 7.10 shows an object A at a distance of 5 km from the object B. The mass A is four times of the mass B. Determine the location of a point on the line joining both objects from B at which the nett gravitational field strength is zero. 25 PHYSICS Solution : CHAPTER 7 r 5 103 m; M A 4M B A a g1 C a g 2 rx At point C, a g nett r B x 0 a g1 a g 2 26 PHYSICS CHAPTER 7 7.1.4 Variation of gravitational field strength on the distance from the centre of the Earth Outside the Earth ( r > R) Figure 7.11 shows a test mass which is outside the Earth and at a distance r from the centre. M r R Figure 7.11 The gravitational field strength outside the Earth is GM ag 2 r 1 ag 2 r 27 PHYSICS CHAPTER 7 On the Earth ( r = R) Figure 7.12 shows a test mass on the Earth’s surface. M r R Figure 7.12 The gravitational field strength on the Earth’s surface is GM ag 2 g 9.81 m s 2 R 28 PHYSICS CHAPTER 7 Inside the Earth ( r < R) Figure 7.13 shows a test mass which is inside the Earth and at distance r from the centre. M M' where r R M ' : the mass of spherical portion of the Earth of radius, r Figure 7.13 The gravitational field strength inside the Earth is given by GM ' ag 2 r 29 PHYSICS CHAPTER 7 By assuming the Earth is a solid sphere and constant density, hence M ' V ' M V M ' 43 r 3 r3 4 3 3 M 3 R R 3 r M ' 3 M R Therefore the gravitational field strength inside the Earth is r3 G 3 M R ag r2 GM ag 3 r R ag r 30 PHYSICS CHAPTER 7 The variation of gravitational field strength, ag as a function of distance from the centre of the Earth, r is shown in Figure 7.14. R ag GM ag 2 g R ag r 0 1 ag 2 r r R Figure 7.14 31 PHYSICS Learning Outcome: CHAPTER 7 7.2 Gravitational potential (1 hour) At the end of this chapter, students should be able to: Define gravitational potential in a gravitational field. Derive and use the formulae, GM V r Sketch the variation of gravitational potential, V with distance, r from the centre of the earth. 32 PHYSICS 7.2.1 CHAPTER 7 Gravitational potential, V at a point is defined as the ………………...by an external force in bringing a test mass from infinity to a point per unit the test mass. OR mathematically, V is written as: W V m where m : mass of the test mass V : gravitatio nal potential at a point W : work done in bringing a test mass from infinity t o a point It is a ……………………... 33 PHYSICS CHAPTER 7 The S.I unit for gravitational potential is ……………or ………………. Another formula for the gravitational potential at a point is given by W V m and GMm 1 1 V m r1 r2 GMm 1 1 V m r GM V r where 1 1 W GMm r1 r2 where r1 and r2 r r : distance between th e point and the point mass, M 34 PHYSICS CHAPTER 7 The gravitational potential difference between point A and B (VAB) in the Earth’s gravitational field is defined as the ………………..in bringing a test mass from point B to point A per unit the test mass. OR mathematically, VAB is written as: WBA VAB VA -VB m where WBA : work done in bringing the test mass from point B to point A. VA : gravitatio nal potential at point A VB : gravitatio nal potential at point B 35 PHYSICS CHAPTER 7 Figure 7.15 shows two points A and B at a distance rA and rB from the centre of the Earth respectively in the Earth’s gravitational field. A rA B rB M Figure 7.15 The gravitational potential difference between the points A and B is given by VAB VA VB GM GM VAB rA rB 1 1 VAB GM rB rA 36 PHYSICS CHAPTER 7 The gravitational potential difference between point B and A in the Earth’s gravitational field is given by VBA WAB VB VA m The variation of gravitational potential, V when the test mass, m move away from the Earth’s surface is illustrated by the graph in Figure 7.16. V 0 GM R R r 1 V r Figure 7.16 Note: The Gravitational potential at infinity is zero. V 0 37 PHYSICS CHAPTER 7 Example 7.6 : When in orbit, a satellite attracts the Earth with a force of 19 kN and the satellite’s gravitational potential due to the Earth is 5.45107 J kg1. a. Calculate the satellite’s distance from the Earth’s surface. b. Determine the satellite’s mass. (Given G = 6.671011 N m2 kg2, mass of the Earth, M = 5.981024 kg and radius of the Earth , R = 6.38106 m) Solution : Fg 19 103 N; V 5.45 10 7 J kg 1 Fg h r R 38 PHYSICS Solution : CHAPTER 7 Fg 19 103 N; V 5.45 10 7 J kg 1 a. By using the formulae of gravitational potential, thus: GM V r Therefore the satellite’s distance from the Earth’s surface is: r hR 39 PHYSICS Solution : CHAPTER 7 Fg 19 103 N; V 5.45 10 7 J kg 1 b. From the Newton’s law of gravitation, hence: GMm Fg 2 r 40 PHYSICS CHAPTER 7 Example 7.7 : The gravitational potential at the surface of a planet of radius R is 12.8 MJ kg1. Determine the work done in overcoming the gravitational force when a space probe of mass 1000 kg is lifted to a height of 2R from the surface of the planet. Solution : m 1000 kg; r1 R R M r1 m h 2R r2 On the surface of the planet, the gravitational potential is GM V r1 41 PHYSICS CHAPTER 7 Solution : m 1000 kg; r1 R r2 R h 3R The final distance of the space probe from the centre of the Earth is The work done required is given by 1 1 W GMm r1 r2 42 PHYSICS CHAPTER 7 Example 7.8 : The Moon has a mass of 7.351022 kg and a radius of 1740 km. a. Determine the gravitational potential at its surface. b. A probe of mass 100 kg is dropped from a height 1 km onto the Moon’s surface. Calculate its change in gravitational potential energy. c. If all the gravitational potential energy lost is converted to kinetic energy, calculate the speed at which the probe hits the surface. (Given G = 6.671011 N m2 kg2) Solution : M 7.35 1022 kg; R 1.74 106 m a. The gravitational potential on the moon’s surface is GM V R 43 PHYSICS Solution : b. Given CHAPTER 7 M 7.35 1022 kg; R 1.74 106 m 6 m 100 kg; r2 R 1.74 10 m R M r2 h 1.00 10 m 3 m r1 r1 R h Hence the change in the gravitational potential energy is ΔU U f U i GMm GMm 1 1 GMm ΔU r2 r1 r1 r2 44 PHYSICS CHAPTER 7 Solution : c. Given Gravitational potential energy lost = kinetic energy The speed of the probe when hit the moon’s surface is given by ΔU K 1 2 ΔU mv 2 45 PHYSICS CHAPTER 7 Learning Outcome: 7.3 Satellite motion in a circular orbit (½ hour) At the end of this chapter, students should be able to: Explain satellite motion with: velocity, GM v r period, r3 T 2 GM 46 PHYSICS 7.3 CHAPTER 7 Satellite motion in a circular orbit 7.3.1 Tangential (linear/orbital) velocity, v Consider a satellite of mass, m travelling around the Earth of mass, M, radius, R, in a circular orbit of radius, r with …………………………….speed, v as shown in Figure 8.22. Figure 8.22 47 PHYSICS CHAPTER 7 The centripetal force, Fc is contributed by the gravitational force of attraction, Fg exerted on the satellite by the Earth. Fg Fc mac GMm mv 2 r r 2 Hence the tangential velocity, v is given by GM v r where r : distance of the satellite from the centre of the Earth M : mass of the Earth G : universal gravitatio nal constant 48 PHYSICS CHAPTER 7 For a satellite close to the Earth’s surface, and rR Therefore GM gR 2 v gR The relationship between tangential velocity and angular velocity is 2r v r T Hence , the period, T of the satellite orbits around the Earth is given by 2r GM T r r3 T 2 GM 49 PHYSICS 7.3.2 CHAPTER 7 Synchronous (Geostationary) Satellite Figure 7.17 shows a synchronous (geostationary) satellite which stays above the same point on the equator of the Earth. Figure 7.17 The satellite have the following characteristics: It revolves in the ……………………………as the Earth. It rotates with the …………………….of rotation as that of the Earth (24 hours). It moves directly above the ……………………... The centre of a synchronous satellite orbit is at the centre of the Earth. It is used as a ……………………………... 50 PHYSICS CHAPTER 7 Exercise 7.2 : Given G = 6.671011 N m2 kg2 1. A rocket is launched vertically from the surface of the Earth at speed 25 km s-1. Determine its speed when it escapes from the gravitational field of the Earth. (Given g on the Earth = 9.81 m s2, radius of the Earth , R = 6.38 106 m) ANS. : 2.24104 m s1 2. A satellite revolves round the Earth in a circular orbit whose radius is five times that of the radius of the Earth. The gravitational field strength at the surface of the Earth is 9.81 N kg1. Determine a. the tangential speed of the satellite in the orbit, b. the angular frequency of the satellite. (Given radius of the Earth , R = 6.38 106 m) ANS. : 3538 m s1 ; 1.11104 rad s1 51 PHYSICS CHAPTER 7 Exercise 7.2 : 3. A geostationary satellite of mass 2400 kg is placed 35.92 Mm from the Earth’s surface orbits the Earth along a circular path. Determine a. the angular velocity of the satellite, b. the tangential speed of the satellite, c. the acceleration of the satellite, d. the force of attraction between the Earth and the satellite, e. the mass of the Earth. (Given radius of the Earth , R = 6.38 106 m) ANS. : 7.27105 rad s1; 3.08103 m s1; 0.224 m s2; 537 N ; 6.001024 kg 52 PHYSICS CHAPTER 8 THE END… Next Chapter… CHAPTER 8 : Rotational of A Rigid Body 53

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