13.3. Gravitational field of a point mass
A subsection of Physics, 9702, through 13. Gravitational fields
Listing 10 of 19 questions
State what is meant by a line of force in a gravitational field. By reference to the pattern of the lines of gravitational force near to the surface of the Earth, explain why the acceleration of free fall near to the Earth’s surface is approximately constant. The Moon may be considered to be a uniform sphere that is isolated in space. It has radius 1.74 × 103 km and mass 7.35 × 1022 kg. Calculate the gravitational field strength at the Moon’s surface. gravitational field strength = N kg–1 A satellite is in a circular orbit about the Moon at a height of 320 km above its surface. Calculate the time for the satellite to complete one orbit of the Moon. time = s
9702_m18_qp_42
THEORY
2018
Paper 4, Variant 2
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The point P in represents a point mass. On , draw lines to represent the gravitational field around P. P A moon is in circular orbit around a planet. Explain why the path of the moon is circular. Many moons are in circular orbit about a planet. The angular velocity of a moon is ω when the orbit of the moon has a radius r about the planet. shows the variation of r 3 with 1 / ω 2 for these moons. / 107 rad–2 s2 ω r3 / 1023 m3 Show that the mass M of the planet is given by the expression M = gradient G where G is the gravitational constant. Use and the expression in to show that the mass M of the planet is 1.0 × 1026 kg. Determine the speed of a moon in orbit around the planet with an orbital radius of 1.2 × 108 m. speed = m s–1
9702_m22_qp_42
THEORY
2022
Paper 4, Variant 2
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Define gravitational field strength. The mass of a spherical comet of radius 3.6 km is approximately 1.0 × 1013 kg. Assuming that the comet has constant density, calculate the gravitational field strength on the surface of the comet. field strength = N kg–1 A probe having a weight of 960 N on Earth lands on the comet. Using your answer in , determine the weight of the probe on the surface of the comet. weight = N A second comet has a length of approximately 4.5 km and a width of approximately 2.6 km. Its outline is illustrated in . Suggest one similarity and one difference between the gravitational fields at the surface of this comet and at the surface of the comet in . similarity: difference:
9702_s17_qp_42
THEORY
2017
Paper 4, Variant 2
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Define gravitational field strength. An isolated planet is a uniform sphere of radius 3.39 × 106 m. Its mass of 6.42 × 1023 kg may be considered to be a point mass concentrated at its centre. The planet rotates about its axis with a period of 24.6 hours. For an object resting on the surface of the planet at the equator, calculate, to three significant figures: the gravitational field strength field strength = N kg−1 the centripetal acceleration acceleration = m s−2 the force per unit mass exerted on the object by the surface of the planet. force per unit mass = N kg−1
9702_s21_qp_42
THEORY
2021
Paper 4, Variant 2
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State Newton’s law of gravitation. Use Newton’s law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by g = GM r 2 . The Earth has a mass of 5.98 × 1024 kg and a radius of 6.37 × 106 m. The Moon has a mass of 7.35 × 1022 kg and a radius of 1.74 × 106 m. The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of 3.84 × 108 m apart. Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is 1.62 N kg–1. Explain why there is a point X on the line between the centres of the Earth and the Moon where the resultant gravitational field strength due to the Earth and the Moon is zero. Calculate the distance x of point X from the centre of the Moon. x = m
9702_s22_qp_41
THEORY
2022
Paper 4, Variant 1
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State Newton’s law of gravitation. Use Newton’s law of gravitation to show that the gravitational field strength g at a distance r away from a point mass M is given by g = GM r 2 . The Earth has a mass of 5.98 × 1024 kg and a radius of 6.37 × 106 m. The Moon has a mass of 7.35 × 1022 kg and a radius of 1.74 × 106 m. The Earth and the Moon can both be considered as point masses at their centres. Their centres are a distance of 3.84 × 108 m apart. Show that the gravitational field strength at the surface of the Moon due to the mass of the Moon is 1.62 N kg–1. Explain why there is a point X on the line between the centres of the Earth and the Moon where the resultant gravitational field strength due to the Earth and the Moon is zero. Calculate the distance x of point X from the centre of the Moon. x = m
9702_s22_qp_43
THEORY
2022
Paper 4, Variant 3
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State Newton’s law of gravitation. A star and a planet are isolated in space. The planet orbits the star in a circular orbit of radius R, as illustrated in . star mass M R planet t The angular speed of the planet about the star is ω. By considering the circular motion of the planet about the star of mass M, show that ω and R are related by the expression R 3ω2 = GM where G is the gravitational constant. Explain your working. The Earth orbits the Sun in a circular orbit of radius 1.5 × 108 km. The mass of the Sun is 2.0 × 1030 kg. A distant star is found to have a planet that has a circular orbit about the star. The radius of the orbit is 6.0 × 108 km and the period of the orbit is 2.0 years. Use the expression in to calculate the mass of the star. mass = kg
9702_w13_qp_43
THEORY
2013
Paper 4, Variant 3
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State Newton’s law of gravitation. The planet Jupiter and one of its moons, Io, may be considered to be uniform spheres that are isolated in space. Jupiter has radius R and mean density ρ. Io has mass m and is in a circular orbit about Jupiter with radius nR, as illustrated in . nR Io Jupiter radius R density ρ The time for Io to complete one orbit of Jupiter is T. Show that the time T is related to the mean density ρ of Jupiter by the expression ρT 2 = 3πn3 G where G is the gravitational constant. The radius R of Jupiter is 7.15 × 104 km and the distance between the centres of Jupiter and Io is 4.32 × 105 km. The period T of the orbit of Io is 42.5 hours. Calculate the mean density ρ of Jupiter. ρ = kg m–3 The Earth has a mean density of 5.5 × 103 kg m–3. It is said to be a planet made of rock. By reference to your answer in , comment on the possible composition of Jupiter.
9702_w17_qp_42
THEORY
2017
Paper 4, Variant 2
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Questions Discovered
19