Gravitational Fields

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수업 노트

Newton's Law of Gravitation

Newton's law of gravitation states that the gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation.
The equation is: F = Gm₁m₂ / r², where F is the gravitational force (N), G is Newton's gravitational constant, m₁ and m₂ are the masses (kg), and r is the distance between their centres (m).
The inverse square law means that if the separation is doubled, the force reduces by a factor of (½)² = ¼.
The law applies to point masses, but planets and stars can be treated as point masses because their separation is much larger than their radius.
When using the equation, ensure r is the distance between the centres of the masses, not the distance between their surfaces.

Gravitational Field Strength

A gravitational field is a region of space where a test mass experiences a force due to the gravitational attraction of another mass.
Gravitational field strength (g) at a point is defined as the force per unit mass experienced by a test mass at that point: g = F / m.
The units of g are N kg⁻¹.
The gravitational field strength depends only on the mass M of the body producing the field and the distance r from its centre: g = GM / r².
On a planet's surface, g depends on the planet's mass (or density) and radius.
An object's mass remains the same everywhere, but its weight (force due to gravity) changes with g.
g is gravitational field strength, while G is Newton's gravitational constant; they are not interchangeable.

Gravitational Field Strength

Gravitational Field Lines

Gravitational field lines show the direction of the gravitational force that would act on a mass placed at a point in the field.
Gravitational field lines are always directed towards the centre of mass of the body producing the field, because gravitational forces are always attractive.
The gravitational field around a point mass is radial, with field lines pointing inwards.
A uniform gravitational field has equally spaced parallel field lines, and the field strength is the same at all points; for example, near the Earth's surface.
Radial fields are non-uniform: the field strength changes with distance from the centre.
For a uniform sphere, the field outside it is the same as if all its mass were concentrated at its centre (point mass approximation).

Kepler's Laws of Planetary Motion

Kepler's first law: The orbit of a planet is an ellipse, with the Sun at one of the two foci.
Kepler's second law: A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals.
A consequence of Kepler's second law is that planets move faster when nearer the Sun and slower when further away.
Kepler's third law: For planets or satellites in circular orbits about the same central body, the square of the orbital period is proportional to the cube of the orbital radius: T² ∝ r³.
The full equation for Kepler's third law is: T² = (4π² r³) / (GM), where M is the mass of the central body.
A graph of log T against log r gives a straight line, because T² ∝ r³ implies 2 log T ∝ 3 log r.

Orbital Motion and Period

For a satellite in a circular orbit, the gravitational force provides the centripetal force: GMm / r² = mv² / r.
The orbital speed is related to the period by v = 2πr / T.
Combining these gives T² = (4π² r³) / (GM), which shows that the period depends only on the orbital radius and the mass of the central body.
The mass of the orbiting object does not affect its orbital period.

A hand swinging a ball on a rope in a circle, illustrating centripetal force and orbital motion.

Gravitational Field Strength and Density

The gravitational field strength at the surface of a planet can be expressed in terms of its density ρ and radius r: g = (4/3)πGρr.
This shows that g ∝ ρr for a planet of uniform density.
If two planets have the same density, the one with the larger radius has a greater surface gravitational field strength.
For a planet of mass M and radius r, the density is ρ = M / (4/3 π r³).

Gravitational Fields of Celestial Bodies

The gravitational field strength at a distance r from the centre of a planet is g = GM / r².
At the surface of a planet, r is the planet's radius R, so g = GM / R².
The gravitational field strength decreases with distance from the planet's centre according to the inverse square law.
The mass of a planet can be determined from its surface gravitational field strength and radius: M = gR² / G.

The direction of gravity

Key Definitions and Equations

Newton's law of gravitation: F = Gm₁m₂ / r²
Gravitational field strength: g = F / m
Gravitational field strength due to a point mass or uniform sphere: g = GM / r²
Kepler's third law: T² = (4π² r³) / (GM)
Density: ρ = M / V, and for a sphere V = (4/3)πr³

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연습 문제

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1.What is the definition of gravitational field strength at a point?
Easy
- AThe force per unit mass experienced by a test mass at that point
- BThe force experienced by a test mass at that point
- CThe mass per unit force experienced by a test mass at that point
- DThe gravitational potential energy per unit mass at that point
2.Gravitational field lines are always directed towards the centre of mass of the body producing the field.
Easy
True or false?
3.Which of the following is the correct equation for Newton's law of gravitation?
Medium
- AF = G m1 m2 / r²
- BF = G m1 m2 / r
- CF = G m1 m2 r²
- DF = G (m1 + m2) / r²
4.Which of the following statements about gravitational fields are correct? (select all that apply)
Medium
- AGravitational field strength is a vector quantity.
- BGravitational field lines can be repulsive.
- CThe gravitational field strength at a point depends on the mass of the test mass placed there.
- DThe gravitational field strength is inversely proportional to the square of the distance from the centre of the mass producing the field.
- EGravitational forces have an infinite range.
5.Match each term with its correct definition.
Medium
- Gravitational field
- Gravitational field strength
- Newton's law of gravitation
- Kepler's first law
- A region of space where a test mass experiences a force due to the gravitational attraction of another mass
- The force per unit mass experienced by a test mass at a point
- The gravitational force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation
- The orbit of a planet is an ellipse, with the Sun at one of the two foci
6.A satellite orbits a planet at a distance r from the planet's centre. If the distance is doubled, by what factor does the gravitational force change?
Medium
- AIt is reduced by a factor of 4
- BIt is reduced by a factor of 2
- CIt is increased by a factor of 2
- DIt is increased by a factor of 4
7.Which of the following best describes the gravitational field lines around a point mass?
Medium
- ARadial and pointing inwards towards the mass
- BRadial and pointing outwards away from the mass
- CParallel and equally spaced
- DCircular around the mass
8.The gravitational field strength at the surface of a planet depends on both the mass and the radius of the planet.
Easy
True or false?

더 많은 문제

9.Match each Kepler's law with its correct description.
Medium
- Kepler's first law
- Kepler's second law
- Kepler's third law
- The orbit of a planet is an ellipse, with the Sun at one of the two foci
- A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals
- For planets or satellites in a circular orbit about the same central body, the square of the time period is proportional to the cube of the radius of the orbit
10.Arrange the steps to derive the relationship between orbital period and orbital radius for a satellite in circular orbit, starting from the given equations.
Medium
- Equate gravitational force to centripetal force: GMm/r² = mv²/r
- Cancel m and r to get v² = GM/r
- Substitute v = 2πr/T into v² = GM/r
- Square and rearrange to obtain T² = 4π² r³ / (GM)
11.Which of the following statements about gravitational field lines are correct? (select all that apply)
Medium
- AThey point radially inwards towards the centre of mass of the body producing the field
- BThey can be repulsive if the mass is negative
- CThey are equally spaced parallel lines in a uniform field
- DThey show the direction of acceleration of a mass placed in the field
- EThey become further apart as you move away from a point mass
12.The gravitational field strength at the surface of a planet is directly proportional to the planet's radius if its density is constant.
Easy
True or false?
13.A satellite is in a circular orbit at radius r from the centre of a planet. What is the relationship between the orbital period T and the orbital radius r?
Medium
- AT² ∝ r³
- BT ∝ r²
- CT² ∝ r
- DT ∝ r³
14.Which equation correctly gives the orbital period T of the moon?
Medium
- AT = 2π√(r³/(GM))
- BT = 2π√(GM/r³)
- CT = 2πr/(GM)
- DT = 2π√(r/(GM))
15.Which of the following are assumptions made when applying Newton's law of gravitation to planets orbiting the Sun? (select all that apply)
Hard
- APlanets can be treated as point masses
- BThe separation between planet and Sun is much larger than the planet's radius
- CThe gravitational force is repulsive
- DThe mass of the planet is negligible compared to the Sun
- EThe orbit is perfectly circular
16.Planet X has mass M and radius R. Planet Y has mass 4M and radius 2R. What is the ratio of the gravitational field strength at the surface of Y to that at the surface of X?
Hard
- A1
- B2
- C1/2
- D4
17.A satellite of mass 6500 kg orbits at 2000 km above the Earth's surface. The gravitational force between the Earth and the satellite is 37 kN. The radius of the Earth is 6400 km. Calculate the mass of the Earth.
Medium
- A5.9 × 10²⁴ kg
- B1.1 × 10²⁵ kg
- C3.4 × 10²⁴ kg
- D7.3 × 10²² kg
18.The moon Europa orbits Jupiter at a distance of 670 900 km. The mass of Jupiter is 1.898 × 10²⁷ kg and the mass of Europa is 4.8 × 10²² kg. Calculate the gravitational force between Jupiter and Europa.
Medium
- A1.35 × 10²² N
- B1.35 × 10¹⁹ N
- C9.0 × 10²¹ N
- D2.7 × 10²² N
19.Venus has an orbital period of 0.61 years and is at a mean distance of 0.72 AU from the Sun. Using Kepler's third law T² = (4π²r³)/(GM), which statement correctly shows that Kepler's third law is valid for Venus?
Hard
- AT²/r³ gives the same constant as for any planet orbiting the Sun
- BT²/r³ is different for Venus because Venus is closer to the Sun
- CT²/r³ increases as the orbital radius increases
- DT²/r³ is only valid for circular orbits, not elliptical ones
20.The diagram shows the gravitational field produced by planet P. The field lines are radial and their spacing increases with distance from the planet. How does this diagram show that the gravitational field strength of planet P decreases with distance from the surface?
Medium
- AThe field lines are further apart, so the field strength is weaker
- BThe field lines point away from the planet, so the field strength is weaker
- CThe field lines are curved, so the field strength is weaker
- DThe field lines are closer together, so the field strength is weaker
21.A small horizontal region on the surface of Venus is shown. P is a point underneath the surface which contains material of higher density than the material surrounding it. Which diagram correctly shows the gravitational field lines on the surface of Venus?
Hard
- AField lines point radially inwards towards the centre of the planet, unaffected by the dense region
- BField lines point radially outwards from the centre of the planet
- CField lines converge towards the dense region P
- DField lines are parallel and equally spaced across the surface
22.The ISS orbits the Earth at an average distance of 408 km from the surface. Data: average distance between the centre of the Earth and the centre of the Moon = 3.80 × 10⁸ m; mass of the Earth = 5.97 × 10²⁴ kg; radius of the Earth = 6.37 × 10⁶ m. Calculate the maximum average gravitational field strength experienced by the ISS. You may assume that both the Moon and the ISS can be positioned at any point on their orbital path.
Hard
- A8.66 N kg^-1
- B8.68 N kg^-1
- C9.81 N kg^-1
- D8.63 N kg^-1
23.Show that the gravitational field strength g is proportional to the radius r of a planet and its density ρ. Which equation correctly represents this relationship?
Hard
- Ag ∝ ρr
- Bg ∝ ρ/r
- Cg ∝ ρr²
- Dg ∝ ρ/r²
24.Two planets P and Q are in concentric circular orbits about a star S. The radius of P's orbit is R and the radius of Q's orbit is 2R. The gravitational force between P and Q is F when angle SPQ is 90°. Derive an equation for the gravitational force between P and Q, in terms of F, when they are nearest to each other.
Hard
- AF/5
- BF/2
- CF/4
- DF
25.Planet P is twice the mass of planet Q. Which statement correctly describes the gravitational field lines between the two planets and the neutral point?
Hard
- AField lines point from P to Q; the neutral point is closer to Q
- BField lines point from Q to P; the neutral point is closer to P
- CField lines point from P to Q; the neutral point is closer to P
- DField lines point from Q to P; the neutral point is closer to Q
26.A kilogram mass rests on the surface of the Earth. A spherical region of the Earth's interior has a higher density than its surroundings. Which statement about the gravitational field at the surface directly above this region is correct?
Hard
- AThe field strength is slightly greater than at other points on the surface
- BThe field strength is slightly less than at other points on the surface
- CThe field strength is unchanged because only the total mass matters
- DThe field strength is zero at that point
27.The gravitational force between two masses is inversely proportional to the square of the distance between their centres.
Medium
True or false?
28.Gravitational field lines can be used to find the direction of the gravitational force on a mass placed in the field.
Medium
True or false?
29.Which of the following statements about gravitational field strength g are correct? (select all that apply)
Medium
- Ag is the force per unit mass experienced by a test mass
- Bg depends on the mass of the test mass placed in the field
- Cg is inversely proportional to the square of the radial distance from the centre of the body producing the field
- Dg is a vector quantity directed towards the centre of the body producing the field
- Eg is measured in newtons (N)
30.A satellite orbits a planet in a circular orbit. Which of the following statements are correct? (select all that apply)
Hard
- AThe gravitational force provides the centripetal force
- BThe orbital period squared is proportional to the orbital radius cubed
- CThe satellite's speed is independent of its orbital radius
- DThe gravitational field strength at the satellite is GM/r²
- EThe satellite's mass affects its orbital period
31.Match each quantity or relationship with its correct expression or description.
Medium
- Gravitational force between two point masses
- Gravitational field strength at distance r from a point mass M
- Kepler's third law for circular orbits
- Gravitational field strength in terms of density and radius
- F = Gm1m2/r²
- g = GM/r²
- T² ∝ r³
- g ∝ ρr
32.Arrange the steps to derive the relationship between orbital period and orbital radius for a satellite in circular orbit.
Medium
- Equate gravitational force to centripetal force: GMm/r² = mv²/r
- Cancel m and rearrange to get v² = GM/r
- Substitute v = 2πr/T into v² = GM/r
- Square and rearrange to obtain T² = 4π²r³/(GM)
33.A student uses the equation g = GM/r² to calculate the gravitational field strength at the surface of a planet. Which experimental measurement is most important to determine r accurately?
Medium
- AThe radius of the planet
- BThe mass of the planet
- CThe gravitational constant G
- DThe mass of the test mass
34.In an experiment to determine the gravitational field strength on a planet, a 2.0 kg mass is found to have a weight of 16 N. What is the gravitational field strength?
Medium
- A8.0 N kg⁻¹
- B32 N kg⁻¹
- C0.125 N kg⁻¹
- D16 N kg⁻¹
35.Which of the following are valid assumptions when using Newton's law of gravitation to model the orbit of a satellite around a planet? (select all that apply)
Medium
- AThe planet and satellite can be treated as point masses.
- BThe gravitational force is the only force acting on the satellite.
- CThe satellite moves in a circular orbit.
- DThe mass of the satellite is negligible compared to the planet.
- EThe gravitational field is uniform.
36.The gravitational field strength g at a point depends on the mass of the test mass placed at that point.
Easy
True or false?
37.Match each term with its correct definition or description.
Medium
- Gravitational field strength
- Newton's law of gravitation
- Kepler's first law
- Kepler's third law
- The force per unit mass at a point
- The gravitational force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation
- The orbit of a planet is an ellipse with the Sun at one focus
- The square of the orbital period is proportional to the cube of the orbital radius
38.Arrange the steps to determine the mass of the Earth using a satellite in circular orbit, starting from the given equations.
Medium
- Equate gravitational force to centripetal force: GMm/r² = mv²/r
- Cancel m and rearrange to v² = GM/r
- Substitute v = 2πr/T into v² = GM/r
- Rearrange to T² = 4π²r³/(GM)
- Solve for M: M = 4π²r³/(GT²)
39.What does the gradient of the graph represent?
Medium
- AGravitational field strength
- BMass of the planet
- CGravitational constant
- DWeight of the masses
40.A satellite orbits a planet in a circular orbit. If the orbital radius is doubled, what happens to the orbital period?
Hard
- AIt increases by a factor of 2√2
- BIt increases by a factor of 2
- CIt increases by a factor of 4
- DIt decreases by a factor of 2
41.Complete the statement of Kepler's third law.
Easy
For planets or satellites in a circular orbit about the same central body, the square of the time period is proportional to the ____ of the radius of the orbit.
42.An astronaut of mass 80 kg stands on the Moon where the gravitational field strength is 1.6 N kg^-1. Calculate the weight of the astronaut on the Moon.
Medium
43.Calculate the gravitational force between the Earth (mass 6.0 × 10²⁴ kg) and a satellite of mass 5000 kg orbiting at a distance of 8500 km from the centre of the Earth. Give your answer in newtons to 2 significant figures.
Medium
44.Calculate the orbital period of the moon if r = 4.0 × 10⁸ m and M = 6.0 × 10²⁴ kg. Give your answer in seconds to 2 significant figures.
Hard
45.State the name of the law that states: The gravitational force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation.
Easy
46.The moon Ganymede orbits Jupiter at a distance approximately twice that of Europa and is about 3 times as massive. By what factor is the gravitational force between Jupiter and Ganymede smaller than the force between Jupiter and Europa? Give your answer as a decimal to 2 significant figures.
Medium
47.By considering the gravitational force acting on a mass m at the surface of Venus, determine an equation for the mass M of Venus in terms of the radius R, the gravitational field strength g and the gravitational constant G. Then calculate the mass of Venus given: radius of Venus = 6.05 × 10⁶ m; gravitational field strength at Venus's surface = 8.80 N kg^-1. Give your answer in kg to 3 significant figures.
Hard
48.NASA's spacecraft Messenger orbited Mercury at an altitude of 270 km. The radius of Mercury is 2.40 × 10⁶ m and the gravitational field strength at Mercury's surface is 3.70 N kg^-1. Calculate the gravitational field strength of Mercury at an altitude of 270 km. Give your answer in N kg^-1 to 3 significant figures.
Hard
49.The gravitational field strength on the Moon's surface is 1.63 N kg^-1 and the Moon has a diameter of 3480 km. Calculate the mass of the Moon. Give your answer in kg to 3 significant figures.
Hard
50.Planet X has a density ρ and the gravitational field strength on its surface is g. Planet Y has a density three times that of planet X and the gravitational field strength on its surface is 9 times that of planet X. Using g ∝ ρr, show that the mass of planet Y is roughly 80 times larger than the mass of planet X. Give the ratio M_Y/M_X to the nearest integer.
Hard
51.The gravitational field strength on the surface of a particular moon is 2.5 N kg^-1. The moon orbits a planet of similar density, but the diameter of the planet is 50 times greater than the moon. Calculate the gravitational field strength at the surface of the planet. Give your answer in N kg^-1.
Medium
52.The distance between the Sun and Mercury varies from 4.60 × 10¹⁰ m to 6.98 × 10¹⁰ m. The gravitational attraction between them is F when they are closest together. Show that the minimum gravitational force between the Sun and Mercury is about 43% of F. Give the percentage to the nearest whole number.
Hard
53.State Newton's law of gravitation.
Medium
54.Outline why Newton's law of gravitation can be applied to planets orbiting the Sun, even though planets are not point masses.
Medium
55.Explain how the gravitational field lines around a planet can be used to determine the direction of the gravitational force on a mass placed in the field.
Hard
56.Explain why gravitational field lines are always directed towards the centre of mass of the body producing the field.
Hard
57.Explain why the gravitational field strength at the surface of a planet is independent of the mass of an object placed on its surface.
Hard
58.An astronaut of mass 80 kg stands on the Moon where the gravitational field strength is 1.6 N kg^-1. Calculate the weight of the astronaut on the Moon.
Medium
- A128 N
- B115 N
- C141 N
- D154 N
59.Calculate the gravitational force between the Earth (mass 6.0 × 10²⁴ kg) and a satellite of mass 5000 kg orbiting at a distance of 8500 km from the centre of the Earth. Give your answer in newtons to 2 significant figures.
Medium
- A33600 N
- B30800 N
- C25200 N
- D28000 N
60.Calculate the orbital period of the moon if r = 4.0 × 10⁸ m and M = 6.0 × 10²⁴ kg. Give your answer in seconds to 2 significant figures.
Hard
- A1890000 s
- B2310000 s
- C2100000 s
- D2520000 s
61.The moon Ganymede orbits Jupiter at a distance approximately twice that of Europa and is about 3 times as massive. By what factor is the gravitational force between Jupiter and Ganymede smaller than the force between Jupiter and Europa? Give your answer as a decimal to 2 significant figures.
Medium
- A0.75
- B2.75
- C-0.25
- D1.75
62.By considering the gravitational force acting on a mass m at the surface of Venus, determine an equation for the mass M of Venus in terms of the radius R, the gravitational field strength g and the gravitational constant G. Then calculate the mass of Venus given: radius of Venus = 6.05 × 10⁶ m; gravitational field strength at Venus's surface = 8.80 N kg^-1. Give your answer in kg to 3 significant figures.
Hard
- A5.313e+24 kg
- B4.83e+24 kg
- C5.796e+24 kg
- D4.347e+24 kg
63.NASA's spacecraft Messenger orbited Mercury at an altitude of 270 km. The radius of Mercury is 2.40 × 10⁶ m and the gravitational field strength at Mercury's surface is 3.70 N kg^-1. Calculate the gravitational field strength of Mercury at an altitude of 270 km. Give your answer in N kg^-1 to 3 significant figures.
Hard
- A2 N kg^-1
- B3 N kg^-1
- C4 N kg^-1
- D5 N kg^-1
64.The gravitational field strength on the Moon's surface is 1.63 N kg^-1 and the Moon has a diameter of 3480 km. Calculate the mass of the Moon. Give your answer in kg to 3 significant figures.
Hard
- A7.45e+22 kg
- B8.94e+22 kg
- C6.705e+22 kg
- D8.195e+22 kg
65.Planet X has a density ρ and the gravitational field strength on its surface is g. Planet Y has a density three times that of planet X and the gravitational field strength on its surface is 9 times that of planet X. Using g ∝ ρr, show that the mass of planet Y is roughly 80 times larger than the mass of planet X. Give the ratio M_Y/M_X to the nearest integer.
Hard
- A97
- B81
- C73
- D89
66.The gravitational field strength on the surface of a particular moon is 2.5 N kg^-1. The moon orbits a planet of similar density, but the diameter of the planet is 50 times greater than the moon. Calculate the gravitational field strength at the surface of the planet. Give your answer in N kg^-1.
Medium
- A125 N kg^-1
- B151 N kg^-1
- C112 N kg^-1
- D138 N kg^-1
67.The distance between the Sun and Mercury varies from 4.60 × 10¹⁰ m to 6.98 × 10¹⁰ m. The gravitational attraction between them is F when they are closest together. Show that the minimum gravitational force between the Sun and Mercury is about 43% of F. Give the percentage to the nearest whole number.
Hard
- A47 %
- B39 %
- C51 %
- D43 %

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