13.4. Gravitational potential
A subsection of Physics, 9702, through 13. Gravitational fields
Listing 10 of 30 questions
Define gravitational potential at a point. A rocket is launched from the surface of a planet and moves along a radial path, as shown in . R R 4R A B rocket path planet mass M The planet may be considered to be an isolated sphere of radius R with all of its mass M concentrated at its centre. Point A is a distance R from the surface of the planet. Point B is a distance 4R from the surface. Show that the difference in gravitational potential Δφ between points A and B is given by the expression Δφ GM R = where G is the gravitational constant. The rocket motor is switched off at point A. During the journey from A to B, the rocket has a constant mass of 4.7 × 104 kg and its kinetic energy changes from 1.70 TJ to 0.88 TJ. For the planet, the product GM is 4.0 × 1014 N m2 kg–1. It may be assumed that resistive forces to the motion of the rocket are negligible. Use the expression in to determine the distance from A to B. distance = m
9702_m17_qp_42
THEORY
2017
Paper 4, Variant 2
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Define gravitational potential at a point. The Earth may be considered to be an isolated sphere of radius R with its mass concentrated at its centre. The variation of the gravitational potential φ with distance x from the centre of the Earth is shown in . –2.0 / 107 J kg–1 –4.0 –6.0 –8.0 R 2R distance x 3R 4R 5R The radius R of the Earth is 6.4 × 106 m. By considering the gravitational potential at the Earth’s surface, determine a value for the mass of the Earth. mass = kg A meteorite is at rest at infinity. The meteorite travels from infinity towards the Earth. Calculate the speed of the meteorite when it is at a distance of 2R above the Earth’s surface. Explain your working. speed = m s–1 In practice, the Earth is not an isolated sphere because it is orbited by the Moon, as illustrated in . Earth Moon initial path of meteorite (not to scale) The initial path of the meteorite is also shown. Suggest two changes to the motion of the meteorite caused by the Moon. 1. 2.
9702_s10_qp_42
THEORY
2010
Paper 4, Variant 2
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Define gravitational potential at a point. The Earth may be considered to be an isolated sphere of radius R with its mass concentrated at its centre. The variation of the gravitational potential φ with distance x from the centre of the Earth is shown in . –2.0 / 107 J kg–1 –4.0 –6.0 –8.0 R 2R distance x 3R 4R 5R The radius R of the Earth is 6.4 × 106 m. By considering the gravitational potential at the Earth’s surface, determine a value for the mass of the Earth. mass = kg A meteorite is at rest at infinity. The meteorite travels from infinity towards the Earth. Calculate the speed of the meteorite when it is at a distance of 2R above the Earth’s surface. Explain your working. speed = m s–1 In practice, the Earth is not an isolated sphere because it is orbited by the Moon, as illustrated in . Earth Moon initial path of meteorite (not to scale) The initial path of the meteorite is also shown. Suggest two changes to the motion of the meteorite caused by the Moon. 1. 2.
9702_s10_qp_43
THEORY
2010
Paper 4, Variant 3
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Define gravitational potential at a point. The gravitational potential φ at distance r from point mass M is given by the expression φ = – GM r where G is the gravitational constant. Explain the significance of the negative sign in this expression. A spherical planet may be assumed to be an isolated point mass with its mass concentrated at its centre. A small mass m is moving near to, and normal to, the surface of the planet. The mass moves away from the planet through a short distance h. State and explain why the change in gravitational potential energy ΔEP of the mass is given by the expression ΔEP = mgh where g is the acceleration of free fall. The planet in has mass M and diameter 6.8 × 103 km. The product GM for this planet is 4.3 × 1013 N m2 kg–1. A rock, initially at rest a long distance from the planet, accelerates towards the planet. Assuming that the planet has negligible atmosphere, calculate the speed of the rock as it hits the surface of the planet. speed = m s–1
9702_s12_qp_41
THEORY
2012
Paper 4, Variant 1
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Define gravitational potential at a point. The gravitational potential φ at distance r from point mass M is given by the expression φ = – GM r where G is the gravitational constant. Explain the significance of the negative sign in this expression. A spherical planet may be assumed to be an isolated point mass with its mass concentrated at its centre. A small mass m is moving near to, and normal to, the surface of the planet. The mass moves away from the planet through a short distance h. State and explain why the change in gravitational potential energy ΔEP of the mass is given by the expression ΔEP = mgh where g is the acceleration of free fall. The planet in has mass M and diameter 6.8 × 103 km. The product GM for this planet is 4.3 × 1013 N m2 kg–1. A rock, initially at rest a long distance from the planet, accelerates towards the planet. Assuming that the planet has negligible atmosphere, calculate the speed of the rock as it hits the surface of the planet. speed = m s–1
9702_s12_qp_43
THEORY
2012
Paper 4, Variant 3
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Define gravitational potential at a point. A stone of mass m has gravitational potential energy EP at a point X in a gravitational field. The magnitude of the gravitational potential at X is φ. State the relation between m, EP and φ. An isolated spherical planet of radius R may be assumed to have all its mass concentrated at its centre. The gravitational potential at the surface of the planet is − 6.30 × 107 J kg−1. A stone of mass 1.30 kg is travelling towards the planet such that its distance from the centre of the planet changes from 6R to 5R. Calculate the change in gravitational potential energy of the stone. change in energy = J
9702_s14_qp_41
THEORY
2014
Paper 4, Variant 1
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Define gravitational potential at a point. A stone of mass m has gravitational potential energy EP at a point X in a gravitational field. The magnitude of the gravitational potential at X is φ. State the relation between m, EP and φ. An isolated spherical planet of radius R may be assumed to have all its mass concentrated at its centre. The gravitational potential at the surface of the planet is − 6.30 × 107 J kg−1. A stone of mass 1.30 kg is travelling towards the planet such that its distance from the centre of the planet changes from 6R to 5R. Calculate the change in gravitational potential energy of the stone. change in energy = J
9702_s14_qp_43
THEORY
2014
Paper 4, Variant 3
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By reference to the definition of gravitational potential, explain why gravitational potential is a negative quantity. Two stars A and B have their surfaces separated by a distance of 1.4 × 1012 m, as illustrated in . star A star B 1.4 = 1012 m P x Point P lies on the line joining the centres of the two stars. The distance x of point P from the surface of star A may be varied. The variation with distance x of the gravitational potential φ at point P is shown in . –2 –4 –6 –8 –10 –12 –14 –16 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x / 1012 m / 108 J kg–1 q A rock of mass 180 kg moves along the line joining the centres of the two stars, from star A towards star B. Use data from to calculate the change in kinetic energy of the rock when it moves from the point where x = 0.1 × 1012 m to the point where x = 1.2 × 1012 m. State whether this change is an increase or a decrease. change = J At a point where x = 0.1 × 1012 m, the speed of the rock is v. Determine the minimum speed v such that the rock reaches the point where x = 1.2 × 1012 m. minimum speed = m s−1
9702_s16_qp_41
THEORY
2016
Paper 4, Variant 1
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By reference to the definition of gravitational potential, explain why gravitational potential is a negative quantity. Two stars A and B have their surfaces separated by a distance of 1.4 × 1012 m, as illustrated in . star A star B 1.4 = 1012 m P x Point P lies on the line joining the centres of the two stars. The distance x of point P from the surface of star A may be varied. The variation with distance x of the gravitational potential φ at point P is shown in . –2 –4 –6 –8 –10 –12 –14 –16 0.2 0.4 0.6 0.8 1.0 1.2 1.4 x / 1012 m / 108 J kg–1 q A rock of mass 180 kg moves along the line joining the centres of the two stars, from star A towards star B. Use data from to calculate the change in kinetic energy of the rock when it moves from the point where x = 0.1 × 1012 m to the point where x = 1.2 × 1012 m. State whether this change is an increase or a decrease. change = J At a point where x = 0.1 × 1012 m, the speed of the rock is v. Determine the minimum speed v such that the rock reaches the point where x = 1.2 × 1012 m. minimum speed = m s−1
9702_s16_qp_43
THEORY
2016
Paper 4, Variant 3
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Define gravitational potential at a point. Starting from the equation for the gravitational potential due to a point mass, show that the gravitational potential energy EP of a point mass m at a distance r from another point mass M is given by EP = – GMm r where G is the gravitational constant. shows the path of a comet of mass 2.20 × 1014 kg as it passes around a star of mass 1.99 × 1030 kg. X Y 34.1 km s–1 comet mass 2.20 × 1014 kg star mass 1.99 × 1030 kg path of comet (not to scale) At point X, the comet is 8.44 × 1011 m from the centre of the star and is moving at a speed of 34.1 km s–1. At point Y, the comet passes its point of closest approach to the star. At this point, the comet is a distance of 6.38 × 1010 m from the centre of the star. Both the comet and the star can be considered as point masses at their centres. Calculate the magnitude of the change in the gravitational potential energy ΔEP of the comet as it moves from position X to position Y. ΔEP = J State, with a reason, whether the change in gravitational potential energy in is an increase or a decrease. Use your answer in to determine the speed, in km s–1, of the comet at point Y. speed = km s–1 A second comet passes point X with the same speed as the comet in and travelling in the same direction. This comet is gradually losing mass. The mass of this comet when it passes point X is the same as the mass of the comet in . Suggest, with a reason, how the path of the second comet compares with the path shown in .
9702_s22_qp_42
THEORY
2022
Paper 4, Variant 2
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Questions Discovered
30