13.2. Gravitational force between point masses
A subsection of Physics, 9702, through 13. Gravitational fields
Listing 10 of 55 questions
Two point masses are isolated in space and are separated by a distance x. State an expression relating the gravitational force F between the two masses to the magnitudes M and m of the masses. State the name of any other symbol used. A spacecraft is to be put into a circular orbit about a spherical planet. The planet may be considered to be isolated in space. The mass of the planet, assumed to be concentrated at its centre, is 7.5 × 1023 kg. The radius of the planet is 3.4 × 106 m. The spacecraft is to orbit the planet at a height of 2.4 × 105 m above the surface of the planet. At this altitude, there is no atmosphere. Show that the speed of the spacecraft in its orbit is 3.7 × 103 m s –1. One possible path of the spacecraft as it approaches the planet is shown in . 5.00 × 107 m planet mass 7.5 × 1023 kg 3.64 × 106 m A B (not to scale) The spacecraft enters the orbit at point A with speed 3.7 × 103 m s–1. At point B, a distance of 5.00 × 107 m from the centre of the planet, the spacecraft has a speed of 4.1 × 103 m s–1. The mass of the spacecraft is 650 kg. For the spacecraft moving from point B to point A, show that the change in gravitational potential energy of the spacecraft is 8.3 × 109 J. By considering changes in gravitational potential energy and in kinetic energy of the spacecraft, determine whether the total energy of the spacecraft increases or decreases in moving from point B to point A. A numerical answer is not required.
9702_s19_qp_41
THEORY
2019
Paper 4, Variant 1
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Two point masses are separated by a distance x in a vacuum. State an expression for the force F between the two masses M and m. State the name of any other symbol used. A small sphere S is attached to one end of a rod, as shown in . 8.0 cm thread rod small sphere S view from side (not to scale) The rod hangs from a vertical thread and is horizontal. The distance from the centre of sphere S to the thread is 8.0 cm. A large sphere L is placed near to sphere S, as shown in . 8.0 cm 6.0 cm 1.2 mm thread initial position of rod final position of rod small sphere S large sphere L θ view from above (not to scale) There is a force of attraction between spheres S and L, causing sphere S to move through a distance of 1.2 mm. The line joining the centres of S and L is normal to the rod. Show that the angle θ through which the rod rotates is 1.5 × 10–2 rad. The rotation of the rod causes the thread to twist. The torque T (in N m) required to twist the thread through an angle β (in rad) is given by T = 9.3 × 10–10 × β. Calculate the torque in the thread when sphere L is positioned as shown in . torque = N m The distance between the centres of spheres S and L is 6.0 cm. The mass of sphere S is 7.5 g and the mass of sphere L is 1.3 kg. By equating the torque in to the moment about the thread produced by gravitational attraction between the spheres, calculate a value for the gravitational constant. gravitational constant = N m2 kg–2 Suggest why the total force between the spheres may not be equal to the force calculated using Newton’s law of gravitation.
9702_s19_qp_42
THEORY
2019
Paper 4, Variant 2
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Two point masses are isolated in space and are separated by a distance x. State an expression relating the gravitational force F between the two masses to the magnitudes M and m of the masses. State the name of any other symbol used. A spacecraft is to be put into a circular orbit about a spherical planet. The planet may be considered to be isolated in space. The mass of the planet, assumed to be concentrated at its centre, is 7.5 × 1023 kg. The radius of the planet is 3.4 × 106 m. The spacecraft is to orbit the planet at a height of 2.4 × 105 m above the surface of the planet. At this altitude, there is no atmosphere. Show that the speed of the spacecraft in its orbit is 3.7 × 103 m s –1. One possible path of the spacecraft as it approaches the planet is shown in . 5.00 × 107 m planet mass 7.5 × 1023 kg 3.64 × 106 m A B (not to scale) The spacecraft enters the orbit at point A with speed 3.7 × 103 m s–1. At point B, a distance of 5.00 × 107 m from the centre of the planet, the spacecraft has a speed of 4.1 × 103 m s–1. The mass of the spacecraft is 650 kg. For the spacecraft moving from point B to point A, show that the change in gravitational potential energy of the spacecraft is 8.3 × 109 J. By considering changes in gravitational potential energy and in kinetic energy of the spacecraft, determine whether the total energy of the spacecraft increases or decreases in moving from point B to point A. A numerical answer is not required.
9702_s19_qp_43
THEORY
2019
Paper 4, Variant 3
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State two conditions for an object to be in equilibrium. 1. 2. A sphere of weight 2.4 N is suspended by a wire from a fixed point P. A horizontal string is used to hold the sphere in equilibrium with the wire at an angle of 53° to the horizontal, as shown in . T F 53° horizontal P string wire weight 2.4 N sphere (not to scale) Calculate: 1. the tension T in the wire T = N 2. the force F exerted by the string on the sphere. F = N The wire has a circular cross-section of diameter 0.50 mm. Determine the stress σ in the wire. σ = Pa The string is disconnected from the sphere in . The sphere then swings from its initial rest position A, as illustrated in . h 53° 75 cm P A B (not to scale) The sphere reaches maximum speed when it is at the bottom of the swing at position B. The distance between P and the centre of the sphere is 75 cm. Air resistance is negligible and energy losses at P are negligible. Show that the vertical distance h between A and B is 15 cm. Calculate the change in gravitational potential energy of the sphere as it moves from A to B. change in gravitational potential energy = J Use your answer in to determine the speed of the sphere at B. Show your working. speed = m s–1
9702_s20_qp_21
THEORY
2020
Paper 2, Variant 1
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State what is meant by a gravitational force. A binary star system consists of two stars S1 and S2, each in a circular orbit. The orbit of each star in the system has a period of rotation T. Observations of the binary star from Earth are represented in . t = T S1 S2 t = 0 S1 S2 S1 t = — T S2 S1 t = — T S2 S1 t = — 3T S2 (not to scale) Observed from Earth, the angular separation of the centres of S1 and S2 is 1.2 × 10–5 rad. The distance of the binary star system from Earth is 1.5 × 1017 m. Show that the separation d of the centres of S1 and S2 is 1.8 × 1012 m. The stars S1 and S2 rotate with the same angular velocity ω about a point P, as illustrated in . S1 S2 P d x (not to scale) Point P is at a distance x from the centre of star S1. The period of rotation of the stars is 44.2 years. Calculate the angular velocity ω. ω = rad s–1 By considering the forces acting on the two stars, show that the ratio of the masses of the stars is given by mass of S1 mass of S2 = d – x x . The mass M1 of star S1 is given by the expression GM1 = d 2 (d – ω 2 where G is the gravitational constant. The ratio in is found to be 1.5. Use data from and your answer in to determine the mass M1. M1 = kg
9702_s20_qp_41
THEORY
2020
Paper 4, Variant 1
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State what is meant by a gravitational force. A binary star system consists of two stars S1 and S2, each in a circular orbit. The orbit of each star in the system has a period of rotation T. Observations of the binary star from Earth are represented in . t = T S1 S2 t = 0 S1 S2 S1 t = — T S2 S1 t = — T S2 S1 t = — 3T S2 (not to scale) Observed from Earth, the angular separation of the centres of S1 and S2 is 1.2 × 10–5 rad. The distance of the binary star system from Earth is 1.5 × 1017 m. Show that the separation d of the centres of S1 and S2 is 1.8 × 1012 m. The stars S1 and S2 rotate with the same angular velocity ω about a point P, as illustrated in . S1 S2 P d x (not to scale) Point P is at a distance x from the centre of star S1. The period of rotation of the stars is 44.2 years. Calculate the angular velocity ω. ω = rad s–1 By considering the forces acting on the two stars, show that the ratio of the masses of the stars is given by mass of S1 mass of S2 = d – x x . The mass M1 of star S1 is given by the expression GM1 = d 2 (d – ω 2 where G is the gravitational constant. The ratio in is found to be 1.5. Use data from and your answer in to determine the mass M1. M1 = kg
9702_s20_qp_43
THEORY
2020
Paper 4, Variant 3
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A pendulum consists of a solid sphere suspended by a string from a fixed point P, as shown in . h θ X Y P string momentum 0.72 N s 0.93 m sphere (not to scale) The sphere swings from side to side. At one instant the sphere is at its lowest position X, where it has kinetic energy 0.86 J and momentum 0.72 N s in a horizontal direction. A short time later the sphere is at position Y, where it is momentarily stationary at a maximum vertical height h above position X. The string has a fixed length and negligible weight. Air resistance is also negligible. On , draw a solid line to represent the displacement of the centre of the sphere at position Y from position X. Show that the mass of the sphere is 0.30 kg. Calculate height h. h = m The distance between point P and the centre of the sphere is 0.93 m. When the sphere is at position Y, the string is at an angle θ to the vertical. Show that θ is 47°. For the sphere at position Y, calculate the moment of its weight about point P. moment = N m State and explain whether the sphere is in equilibrium when it is stationary at position Y.
9702_s21_qp_21
THEORY
2021
Paper 2, Variant 1
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State Newton’s law of gravitation. A satellite is in a circular orbit around a planet. The radius of the orbit is R and the period of the orbit is T. The planet is a uniform sphere. Use Newton’s law of gravitation to show that R and T are related by 4π2R 3 = GMT 2 where M is the mass of the planet and G is the gravitational constant. The Earth may be considered to be a uniform sphere of mass 5.98 × 1024 kg and radius 6.37 × 106 m. A geostationary satellite is in orbit around the Earth. Use the expression in to determine the height of the satellite above the Earth’s surface. height = m Another satellite is in a circular orbit around the Earth with the same orbital radius and period as the satellite in . Calculate the angular speed of the satellite in this orbit. Give a unit with your answer. angular speed = unit Despite having the same orbital period, the orbit of this satellite is not geostationary. Suggest two ways in which the orbit of this satellite could be different from the orbit of the satellite in .
9702_s23_qp_42
THEORY
2023
Paper 4, Variant 2
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A spherical planet has mass M and radius R. The planet may be assumed to be isolated in space and to have its mass concentrated at its centre. The planet spins on its axis with angular speed ω, as illustrated in . R mass m pole of planet equator of planet A small object of mass m rests on the equator of the planet. The surface of the planet exerts a normal reaction force on the mass. State formulae, in terms of M, m, R and ω, for the gravitational force between the planet and the object, the centripetal force required for circular motion of the small mass, the normal reaction exerted by the planet on the mass. Explain why the normal reaction on the mass will have different values at the equator and at the poles. The radius of the planet is 6.4 × 106 m. It completes one revolution in 8.6 × 104 s. Calculate the magnitude of the centripetal acceleration at 1. the equator, acceleration = m s–2 2. one of the poles. acceleration = m s–2 Suggest two factors that could, in the case of a real planet, cause variations in the acceleration of free fall at its surface. 1. 2.
9702_w08_qp_4
THEORY
2008
Paper 4, Variant 0
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