JPJC 2026 Gravitational Field Tutorial

2026/JPJC/PHYSICS/9478 1 JURONG PIONEER JUNIOR COLLEGE 9478 H2 PHYSICS GRAVITATIONAL FIELD TUTORIAL Self-Check Questions S1 Define the gravitational field strength at a point. S2 Derive, from Newton's law of gravitation and the definition of gravitational field strength, the equation g = 2 GM r for the gravitational field strength of a point mass. S3 Define the gravitational potential at a point. S4 What is a geostationary orbit? How can a satellite be geostationary? Self-Practice Questions P1 Two point masses of 20 kg and 30 kg are placed 5.0 m apart as shown in Fig. 1. Fig. 1 (a) Calculate the magnitude of the gravitational force acting on the 20 kg mass by the 30 kg mass. (b) State the magnitude of the gravitational force acting on the 30 kg mass by the 20 kg mass. (c) On Fig. 1, draw vectors to represent the forces acting on the two masses respectively. (d) Explain the statement ‘gravitational forces are action-reaction pairs’. P2 [2011/1/15-modified] In this question, you can consider all the mass of the Earth to be concentrated at its centre. Work to five significant figures using the following data. mass of Earth = 5.9768×1024 kg distance of Singapore from centre of Earth = 6.3782×106 m gravitational constant G = 6.6730×10−11 N m2 kg−2 Determine the value of g (five significant figures), the gravitational field strength of the Earth, in Singapore. P3 A point mass of 1.0 kg is placed 5.0 m away from another point mass of 30.0 kg. (a) Calculate the gravitational potential energy of the 1.0 kg mass due to the 30.0 kg mass. (b) Comment on the statement that “The answer in part (a) is the gravitational potential.” (c) What is the work done in moving the 1.0 kg mass to infinity? (d) Thus, state the significance of the negative sign in gravitational potential values. 5.0 m 30 kg 20 kg

2026/JPJC/PHYSICS/9478 2 P4 [2007/1/14] A satellite is orbiting the Earth. The gravitational force on the satellite is FG. The centripetal force required to maintain the satellite in orbit is Fc. Which diagram shows the force, or the forces, acting on the orbiting satellite? P5 [2006/P1/Q10 modified] An astronomical gas cloud has mass M and radius R. The gravitational potential on its surface S is GM R− and at its centre O, it is 3 2 GM R− . A unit mass is moved slowly by means of an external force from the surface S to the centre O. Determine the work done, in terms of G, m and r, on the mass by the external force. A C B D

2026/JPJC/PHYSICS/9478 3 Discussion Questions Gravitational force 1 [2013/P2/Q3] A satellite travels in a circular orbit around the Earth at constant speed of 2500 m s−1. The mass of the Earth is 6.0 × 1024 kg and may be assumed to be a point mass at the centre of the Earth. Calculate the radius of the orbit. Explain your working. Gravitational field strength 2 Point X is 5.0 m and 2.0 from two 30 kg masses as shown in Fig. 2. Fig. 2 (a) Calculate the magnitude of the gravitational field strength at point X due to the two 30 kg mass. (b) Draw a vector at point X to represent the gravitational field strength in (a). (c) State the changes in magnitude and direction of the gravitational field strength at point X in comparison with the original gravitational field strength in (a), if (i) the 30 kg mass on the right is changed to 90 kg mass, (ii) the point X is moved 2.0 m to the left, (iii) a 30 kg mass is placed at 2.0 m perpendicularly above point X. Gravitational potential and gravitational potential energy 3 [2024/1/11] Two points P and Q are a fixed distance apart on a line joining them to an object considered as a point mass. The two points move away from the object, along the line, maintaining their fixed separation. What happens to the individual gravitational potentials and the gravitational potential difference between them? 5.0 m 30 kg point X 30 kg 2.0 m

2026/JPJC/PHYSICS/9478 4 Combined Questions 4 Fig. 4 shows a point X which is equidistant from points A and B. The distances AX and BX are both r and have an angle of 2 between them. Point Y is located along the midpoint of line AB. Fig. 4 Two identical masses M are placed at points A and B. (a) Express the gravitational potential in terms of G, M, r and  due to the two masses at (i) point X (ii) point Y (Hint: gravitational potential is a scalar.) (b) Express the gravitational field strength in terms of G, M, r and  due to the two masses. (i) point X (ii) point Y (Hint: gravitational field strength is a vector.) (c) Explain whether is it possible for a point to have zero field strength yet have a non- zero potential. 5 [2008/P2/Q3] A satellite of mass m orbits a planet of mass M and radius PR . The radius of the orbit is R. The satellite and the planet may be considered to be point masses with their masses concentrated at the centres. They may be assumed to be isolated in space. (a) Derive an expression, in terms of M, m and R, for the kinetic energy of the satellite. Explain your workings. (b) Show that, for the satellite in orbit, the ratio gravitational potential energy of satellite kinetic energy of satellite is equal to −2. X r r B A Y

2026/JPJC/PHYSICS/9478 5 (c) The variation with orbital radius R of the gravitational potential energy of the satellite shown in Fig. 5. Fig. 5 10.0 8.0 6.0 4.0 2.0 0 −2.0 −4.0 −6.0 −8.0 −10.0 0 PR 2 PR 3 PR 4 PR energy / 109 J R / m

2026/JPJC/PHYSICS/9478 6 (i) On Fig. 5, draw the variation with orbital radius of the kinetic energy of the satellite. Your line should extend from 1.5 PRR= to 4 PRR= (ii) The mass m of the satellite is 1600 kg. The radius of the orbit of the satellite is changed from 4 PRR= to 2 PRR= . Use Fig. 6 to determine the change in orbital speed of the satellite. 6 [2007/P3/Q5 part] (a) The Earth spins on its axis with a period of one day. (i) Show that the angular velocity of a point on the Earth’s surface is 7.27 × 10−5 rad s−1. (ii) Calculate the centripetal acceleration of a point on the Earth’s equator. The radius of the Earth’s equator is 6.38 × 106 m. (b) (i) Calculate how far a satellite needs to be from the centre of the Earth for its angular velocity to be equal to the angular velocity of the Earth. (ii) State two circumstances under a satellite at this distance will be geostationary satellite. 7 Certain meteorites (tektites) found on the Earth have a composition identical with that of lunar granite. It is thought that they may be debris from volcanic eruption on the Moon. Fig. 7 shows how the gravitational potential between the surface of the Moon and the surface of the Earth varies along the line joining their centres. At point P, the gravitational potential is a maximum.