DHS 07 Gravitational Field (Tutorial)

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -1 Tutorial: Gravitational Field Summary 1 Gravitational field strength g at a point is the gravitational force per unit mass exerted on a ______________ mass placed at that point. i.e. g = ______ Force acting on an object of mass m placed there, F = mg 2 Newton’s law of gravitation states that the gravitational force between two point masses is directly proportional to the product of the masses and ______________ proportional to the ____________ of their separation. i.e. F = ____________ 3 Gravitational field strength g due to a point mass M at a distance r from it is given by g = −___________ where G = 6.67 × 10−11 N m2 kg−2 (negative sign indicates that the field strength is directed towards M) Force acting on an object of mass m placed there, F = −____________ 4 Gravitational potential φ at a point is the work done per unit mass in bringing a __________________ mass from _____________ to that point. i.e. φ = 5 Gravitational potential φ due to a point mass M at a distance r from it is given by φ = −____________ 6 Gravitational potential energy U of an object of mass m placed at a point is given by U = mφ 7 Gravitational potential energy U of an object of mass m placed at a point distance r from M is given by U = −____________ 8 Satellite of mass m round the Earth (mass ME) with orbit radius r : W m

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -2 Gravitational force acting on the satellite by the Earth, F = This gravitational force provides the centripetal force: = mrω2 = mr  T2 = r3 (Kepler’s 3rd law) 9 Energies of satellite of mass m in orbit of radius r round the Earth (mass ME): From =  KE, K = = PE, U =  TE, E = K + U = + = i.e. E = 10 Geostationary orbit, for satellite to appear stationary when viewed from a fixed point on the rotating Earth, (a) has period T = Earth’s period of rotation = ____________ (b) is in the plane of ______________ (c) moves from ________ to ________, same sense as self-rotation of the Earth 11 Relation between gravitational force F and potential energy U F = − Divide both sides by m, g = − ______ 2 EGM m r 2 EGM m r 2 2 T π  24 EGM π   2mv r 2 EGM m r 2 2 mv 2 EGM m r EGM m r− 2 EGM m r EGM m r − 2 EGM m r− 2 U dU dr

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -3 Self-Attempt Questions S1 The force of attraction between two masses 3 kg and 2 kg is F when the distance between them is r. Express the new force of attraction, in terms of F, if the distance between the two masses is doubled. (F/4) [2] S2 The radius of the Earth may be taken as 5760 km. A sate llite is in circular orbit 144 km above the Earth’s surface. What is the approximate value of the gravitational force on the satellite compared with that when it is at the Earth’s surface? A greater by 10% B greater by 5% C less by 5% D less by 10% S3 If a body of mass m was released in a vacuum just above the surface of a planet of mass M and radius R, what would be its gravitational acceleration? A B C D E S4 The diagram (not to scale) represents the relative positions of the Earth and the Moon. The line XY joins the surface of the Earth and the surface of the Moon. Which graph represents the variation of gravitational field strength g along the line XY? S5 A body is moved from a point P on the Earth’s surface to another point Q further from the Earth’s centre. Which one of the following statements about the gravitational potential energy of the body at the two points is correct? [Take the gravitational potential energy of the body as zero when it is at an infinite distance from the Earth.] GmM R Gm R 2 GM R 2 GmM R GM R g A B C D g g g

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -4 A It is positive at both points and greater at Q than at P. B It is positive at both points and lower at Q than at P. C It is zero at P but positive at Q. D It is negative at both points and greater at Q than at P. E It is negative at both points and lower at Q than at P. S6 A satellite of mass 50 kg is initially placed at a point where the gravitational potential due to the Earth is −20 MJ kg−1. It is then moved to another point where the gravitational potential is −60 MJ kg−1. In which direction did it move and what is its change in potential energy? A closer to the Earth and a loss of 2000 MJ of potential energy B closer to the Earth and a loss of 40 MJ of potential energy C further from the Earth and a gain of 2000 MJ of potential energy D further from the Earth and a gain of 40 MJ of potential energy S7 Two stationary particles of masses M1 and M2 are at distance d apart as shown below. (a) (i) Draw on the figure above, the direction of the gravitational field which M1 causes at Y. [1] (ii) What is the gravitational field strength which M1 causes at Y? [1] (iii) What is the force which M1 exerts on M2? [1] (b) A third particle, M 3 lying on the line joining the two particles, experiences no resultant gravitational force. The distance between M 1 and M3 is x. Express x in terms of d, M1 and M2. [2] S8 Earth has a mass of 6.0 × 1024 kg and Moon has a mass of 7.4 × 1022 kg. The distance between their centres is 3.8 × 105 km. Find the resultant gravitational field strength due to Earth and Moon at point X which is 2.0 × 10 5 km away from Earth as shown below. (9.85 × 10−3 N kg−1 towards Earth) [2] M1 M2 Y X d 3.8 × 105 km 2.0 × 105 km X

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -5 S9 The diagram below shows the variation of the gravitational potential energy Ep of a body with distance r from the centre of the planet. What is represented by the gradient at any point on the curve? A The gravitational potential at that value of r B The gravitational field strength at that value of r C The force pulling the body towards the planet D The acceleration of the body towards the planet S10 The radius of the Earth’s orbit around the Sun is 1.5 × 1011 m. The mass of the Earth is 6.0 × 1024 kg and period of revolution of the Earth around the Sun is 1 year. (a) Calculate (i) the speed of the Earth in its orbit around the Sun, (2.99 × 104 m s−1) [2] (ii) the acceleration of the Earth, (5.96 × 10−3 m s−2) [2] (iii) the force the Sun exerts on the Earth, (3.58 × 1022 N) [2] (iv) the gravitational field strength of the Sun at the Earth, (5.96 × 10−3 N kg−1) [1] (b) The radius of the orbit of Mars around the Sun is 2.3 × 1011 m. Determine the period of revolution of Mars about the Sun in terms of years. (1.90 years) [3] Discussion Questions D1 The Earth may be assumed to be spherical with radius r and density ρ. Which equation correctly relates the gravitational field strength g at its surface to these quantities and the gravitational constant G? A B C D D2 Which graph shows how the gravitational field strength g due to a point mass varies with the distance r from the point mass? (Arbitrary values are used on the graph axes. On all the graphs, g has the value 64 units when r = 1 unit.) 2 Gg r= ρ 3 4 Gg r= πρ 4 3 rGg = πρ 24 3 rGg = πρ Ep r

Dunman High School (Senior High Physics) 9749 Physics (2024) Topic 7: Gravitational Field 7 – Tutorial Questions -6 D3 P is a planet with centre O, as shown in the figure below. X and M are two points of equal gravita