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 the masses and inversely proportional to the square of their separation.
Equation: 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 radii.
The distance r is always measured from the centre of mass of each body, so for an object on or above a planet's surface, r = planet's radius + height above surface.

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 unit of g is N kg⁻¹.
From equating F = mg with Newton's law, g = GM / r², where M is the mass producing the field and r is the distance from its centre.
The field strength g depends only on the mass M producing the field, so all objects in that field experience the same g regardless of their own mass.
g is inversely proportional to the square of the radial distance: g ∝ 1/r².
Factors affecting g at a planet's surface are its radius and its mass (or density).

Gravitational Field Strength

Gravitational Field Lines

Gravitational field lines show the direction of the gravitational force that would act on a mass placed at that point.
Field lines are always directed towards the centre of mass of the body because gravitational forces are always attractive.
The gravitational field around a point mass is radial, with field lines pointing radially inwards.
A uniform field (e.g. near the Earth's surface) is represented by equally spaced parallel field lines; g is the same throughout.
Radial fields are non-uniform: g depends on the distance from the centre of mass.
For a uniform sphere, the field outside it is identical to that of a point mass at its centre.

Gravitational Potential

Gravitational potential (Vg) at a point is the work done per unit mass in bringing a test mass from infinity to that point.
Unit: J kg⁻¹.
Equation: Vg = − GM / r, where M is the mass producing the field and r is the distance from its centre.
Gravitational potential is always negative because it is defined as zero at infinity, and work must be done to move a mass away from the body towards infinity.
As r decreases (closer to the planet), Vg becomes more negative (smaller).
As r increases (further away), Vg becomes less negative (larger), approaching zero at infinity.
Gravitational potential is a scalar quantity, unlike gravitational field strength which is a vector.

Gravitational Potential Energy

Gravitational potential energy (Ep) of two point masses is the work done to assemble the system from infinite separation: Ep = − Gm₁m₂ / r.
Near the Earth's surface, where the field is approximately uniform, Ep = mgΔh.
The change in GPE when a mass moves between two distances r₁ and r₂ is: ΔEp = Gm₁m₂ (1/r₁ − 1/r₂).
The change in gravitational potential between two points is: ΔVg = Gm₁ (1/r₁ − 1/r₂).
The work done in moving a mass m through a potential difference ΔVg is: ΔW = mΔVg.
The area under a force–distance graph for a gravitational field represents the work done (change in GPE).
Work is done when a mass moves against the gravitational field lines (away from the planet).

Gravitational Potential Gradient

The gravitational field strength at a point is equal to the negative gradient of a potential–distance graph at that point: g = − ΔVg / Δr.
The potential gradient is the rate of change of gravitational potential with respect to displacement in the direction of the field.
For a planet, the graph of Vg against r follows a −1/r relation and all values of Vg are negative.
The gradient of the Vg–r graph at any point gives the value of g at that point.
To find g from the graph, draw a tangent at the point and calculate its gradient.

Gravitational Equipotential Surfaces

Equipotential lines (2D) or surfaces (3D) join points that have the same gravitational potential.
They are always perpendicular to the gravitational field lines.
In a radial field, equipotential lines are concentric circles around the planet and become further apart as distance increases.
In a uniform field, equipotential lines are horizontal, parallel, and equally spaced.
No work is done when moving along an equipotential surface; work is only done when moving between different equipotential surfaces (ΔV = 0 along an equipotential).

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; consequently, planets move faster nearer the Sun and slower further away.
Kepler's Third Law: For planets or satellites in circular orbit about the same central body, the square of the time period is proportional to the cube of the orbital radius: T² ∝ r³.
The relationship between T and r 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, since 2 log T ∝ 3 log r.

Escape Speed

Escape speed is the minimum speed that will allow an object to escape a gravitational field with no further energy input.
It is the same for all masses in the same gravitational field (e.g. a rocket and a tennis ball on Earth).
Escape speed is derived by equating kinetic energy to gravitational potential energy: ½ m v_esc² = GMm / r.
The mass m cancels, giving: v_esc = √(2GM / r).
Escape speed depends on the mass and radius of the object creating the field (planet, moon, or black hole).
Rockets launched from Earth do not need to reach escape velocity to enter orbit; they are continuously given energy by fuel and thrust.

Orbital Motion, Speed & Energy

For a satellite in circular orbit, the gravitational force provides the centripetal force: GMm / r² = mv² / r.
The mass of the satellite cancels, giving orbital speed: v = √(GM / r).
All satellites, regardless of their mass, travel at the same speed in a particular orbit radius.
An orbiting satellite has both kinetic energy (Ek) and gravitational potential energy (Ep); its total energy is constant.
For a smaller orbital radius: gravitational force is larger, speed is higher, Ek is greater, Ep is lower, and orbital period T is shorter.
For a larger orbital radius: gravitational force is smaller, speed is lower, Ek is lower, Ep is greater, and orbital period T is longer.
Total energy = Ek + Ep; as orbital radius decreases, Ek increases and Ep decreases.

Effects of Drag on Orbital Motion

Satellites in low orbits (<600 km) may experience viscous drag (air resistance) due to the very low but non-zero density of the upper atmosphere.
Drag causes a small dissipation of kinetic energy into thermal energy due to friction between air particles and the satellite's surface.
As a satellite loses energy, its orbital radius decreases and it spirals towards the Earth.
As the orbit becomes lower, some potential energy is transferred to kinetic energy, so the satellite's speed increases.
The increased speed in a lower orbit leads to greater air resistance and greater dissipation of kinetic energy.
The satellite's total energy decreases if the overall decrease in potential energy is larger than the overall increase in kinetic energy: ΔE_total < 0 if ΔEp > ΔEk.

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연습 문제

무료 미리 보기 — 58개 중 8개 문제. 가입하면 전부 볼 수 있어요.

1.Which of the following best defines a gravitational field?
Easy
- AA region of space where a test mass experiences a force due to the gravitational attraction of another mass
- BA region of space where a test mass experiences a repulsive force
- CThe force per unit mass experienced by a test mass
- DThe work done per unit mass in bringing a test mass from infinity
2.Which equation correctly represents Newton's law of gravitation?
Easy
- AF = G m1 m2 / r²
- BF = G m1 m2 / r
- CF = G m1 m2 r²
- DF = m g
3.Which of the following statements about gravitational field lines are correct? (select all that apply)
Medium
- AThey point towards the centre of mass of a body
- BThey are always directed radially outwards
- CThey can be used to represent the direction of the gravitational force on a mass placed in the field
- DThey are equally spaced in a uniform field
- EThey are always straight lines
4.Gravitational forces are always attractive.
Easy
True or false?
5.Gravitational potential is a vector quantity.
Easy
True or false?
6.Match each term with its correct definition.
Medium
- Gravitational field strength
- Gravitational potential
- Escape speed
- The force per unit mass experienced by a test mass at a point
- The work done per unit mass in bringing a test mass from infinity to a point
- The minimum speed that will allow an object to escape a gravitational field with no further energy input
7.Place the following steps in the correct order to derive the equation for escape speed from a planet of mass M and radius r.
Medium
- Set kinetic energy equal to gravitational potential energy: 1/2 m v_esc² = GMm/r
- Cancel the mass m from both sides
- Rearrange to make v_esc the subject: v_esc = √(2GM/r)
8.Match each gravitational quantity with its correct SI unit.
Medium
- Gravitational force
- Gravitational field strength
- Gravitational potential
- Gravitational potential energy
- N
- N kg⁻¹
- J kg⁻¹
- J

더 많은 문제

9.Place the following steps in the correct order to derive the equation for the orbital speed of a satellite in a circular orbit.
Medium
- Equate the gravitational force to the centripetal force: GMm/r² = mv²/r
- Cancel the satellite mass m from both sides
- Rearrange to make v² the subject: v² = GM/r
- Take the square root to obtain v = √(GM/r)
10.The gravitational force between two point masses is F when they are separated by a distance r. What is the force when the separation is doubled?
Easy
- AF/4
- BF/2
- C2F
- D4F
11.The escape speed from a planet is greater for a more massive object.
Medium
True or false?
12.Which of the following statements about equipotential surfaces in a gravitational field are correct? (select all that apply)
Medium
- AThey are always perpendicular to the gravitational field lines
- BThey are represented by solid lines with arrows
- CNo work is done when moving along an equipotential surface
- DIn a radial field they are concentric circles around the planet
- EThey become closer together further away from the planet in a radial field
13.A satellite is in a circular orbit around a planet. If the satellite's mass is doubled while its orbital radius stays the same, what happens to its orbital speed?
Medium
- AIt stays the same
- BIt doubles
- CIt halves
- DIt increases by a factor of √2
14.Gravitational field strength is a vector quantity.
Easy
True or false?
15.A satellite in a low orbit experiences a small viscous drag. Which of the following best describes the effect on the satellite's orbit over time?
Hard
- AIts orbital radius decreases and its speed increases
- BIts orbital radius decreases and its speed decreases
- CIts orbital radius increases and its speed decreases
- DIts orbital radius increases and its speed increases
16.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 focus
- A line from the Sun to a planet sweeps out equal areas in equal times
- The square of the orbital period is proportional to the cube of the orbital radius
17.Place the following stages of a satellite's orbit in order of increasing orbital radius, starting from the smallest.
Medium
- Geostationary orbit (radius 42 000 km)
- Low Earth orbit (radius 6 700 km)
- Medium Earth orbit (radius 20 000 km)
- High Earth orbit (radius 100 000 km)
18.Which of the following are correct for a satellite in a circular orbit? (select all that apply)
Hard
- AThe gravitational force provides the centripetal force
- BThe satellite's kinetic energy is constant
- CThe satellite's gravitational potential energy is constant
- DThe satellite's total energy is constant
- EThe satellite's speed varies along the orbit
19.A satellite orbits a planet at a distance r from the planet's centre. The mass of the planet is M. Which expression gives the orbital speed of the satellite?
Easy
- A√(GM/r)
- B√(2GM/r)
- CGM/r²
- D√(GM/r²)
20.A satellite is moved from a circular orbit of radius r to a circular orbit of radius 4r around the same planet. What is the ratio of its new orbital speed to its original orbital speed?
Medium
- A1/2
- B1/4
- C2
- D1/16
21.A satellite in a low orbit experiences a small viscous drag. Which of the following correctly describes the effect on the satellite's orbit over time?
Medium
- AIt spirals towards the Earth, its orbital radius decreases and its speed increases
- BIt spirals away from the Earth, its orbital radius increases and its speed decreases
- CIt maintains the same orbital radius but its speed decreases
- DIt spirals towards the Earth, its orbital radius decreases and its speed decreases
22.A satellite of mass m orbits a planet of mass M at radius r. Which of the following expressions gives the total energy of the satellite?
Medium
- A−GMm/(2r)
- BGMm/r
- C−GMm/r
- DGMm/(2r)
23.The gravitational potential at a point is −4.0 × 10⁷ J kg⁻¹. What is the gravitational potential energy of a 2.0 kg mass at that point?
Medium
- A−8.0 × 10⁷ J
- B−2.0 × 10⁷ J
- C8.0 × 10⁷ J
- D−4.0 × 10⁷ J
24.A planet has mass M and radius R. What is the escape speed from its surface?
Medium
- A√(2GM/R)
- B√(GM/R)
- C√(GM/(2R))
- D2GM/R
25.A satellite of mass 500 kg is in a circular orbit at a radius of 7.0 × 10⁶ m around a planet of mass 6.0 × 10²⁴ kg. What is the gravitational force on the satellite? (G = 6.67 × 10⁻¹¹ N m² kg⁻²)
Hard
- A4.1 × 10³ N
- B2.0 × 10³ N
- C8.2 × 10³ N
- D1.0 × 10³ N
26.Which of the following statements about equipotential surfaces in a gravitational field is correct?
Medium
- ANo work is done when moving along an equipotential surface
- BEquipotential surfaces are always parallel to the field lines
- CThe potential difference between two points on the same equipotential surface is large
- DEquipotential surfaces are always equally spaced in a radial field
27.A satellite orbits a planet in a circular orbit. If the satellite's mass is doubled while its orbital radius stays the same, what happens to its orbital speed?
Medium
- AIt stays the same
- BIt doubles
- CIt halves
- DIt increases by a factor of √2
28.A satellite is in a circular orbit around the Earth. It moves from an orbit of radius r to an orbit of radius 2r. Which of the following statements is correct?
Hard
- AIts kinetic energy decreases and its gravitational potential energy increases
- BIts kinetic energy increases and its gravitational potential energy decreases
- CBoth its kinetic energy and gravitational potential energy increase
- DBoth its kinetic energy and gravitational potential energy decrease
29.The gravitational field strength at the surface of a planet is 5.0 N kg⁻¹. What is the weight of a 60 kg astronaut on this planet?
Medium
- A300 N
- B12 N
- C60 N
- D0.083 N
30.A planet has a radius R and a gravitational field strength g at its surface. What is the gravitational field strength at a distance 2R from the centre of the planet?
Medium
- Ag/4
- Bg/2
- C2g
- D4g
31.Which of the following statements about Kepler's second law is correct?
Medium
- AA line segment joining the Sun to a planet sweeps out equal areas in equal time intervals
- BThe orbit of a planet is an ellipse with the Sun at one focus
- CThe square of the orbital period is proportional to the cube of the orbital radius
- DPlanets move faster when they are furthest from the Sun
32.A satellite orbits a planet with a period T and orbital radius r. What is the period of a second satellite orbiting the same planet with orbital radius 4r?
Hard
- A8T
- B2T
- C4T
- D16T
33.A satellite is in a circular orbit around the Earth. Which of the following provides the centripetal force required to keep the satellite in orbit?
Medium
- AThe gravitational force between the Earth and the satellite
- BThe normal force from the Earth
- CThe tension in a cable attached to the Earth
- DThe satellite's inertia
34.Which of the following statements about gravitational potential is correct?
Medium
- AIt is a scalar quantity and is always negative
- BIt is a vector quantity and is always negative
- CIt is a scalar quantity and is always positive
- DIt is a vector quantity and can be positive or negative
35.The gravitational potential at a point is −5.0 × 10⁶ J kg⁻¹. What is the gravitational potential difference when moving a 3.0 kg mass from this point to a point where the gravitational potential is −2.0 × 10⁶ J kg⁻¹?
Medium
- A3.0 × 10⁶ J kg⁻¹
- B−3.0 × 10⁶ J kg⁻¹
- C7.0 × 10⁶ J kg⁻¹
- D−7.0 × 10⁶ J kg⁻¹
36.A satellite of mass 1000 kg moves from an orbit where the gravitational potential is −8.0 × 10⁷ J kg⁻¹ to an orbit where it is −6.0 × 10⁷ J kg⁻¹. What is the work done in moving the satellite?
Medium
- A2.0 × 10¹⁰ J
- B−2.0 × 10¹⁰ J
- C1.4 × 10¹¹ J
- D−1.4 × 10¹¹ J
37.A planet has a mass M and a radius R. A rocket is launched from its surface with a speed equal to half the escape speed. What is the maximum height reached above the surface? (Neglect air resistance and rotation of the planet.)
Hard
- AR/3
- BR/2
- CR
- D3R
38.The gravitational field strength at the surface of a planet is 8.0 N kg⁻¹. If the planet's mass is doubled and its radius is also doubled, what is the new gravitational field strength at its surface?
Medium
- A4.0 N kg⁻¹
- B8.0 N kg⁻¹
- C16 N kg⁻¹
- D2.0 N kg⁻¹
39.Which of the following statements about gravitational field lines is correct?
Medium
- AThey are always directed towards the centre of mass of the body producing the field
- BThey are always directed away from the centre of mass of the body producing the field
- CThey are parallel and equally spaced around a point mass
- DThey can be either attractive or repulsive depending on the mass
40.A planet has a radius R and a gravitational field strength g at its surface. What is the gravitational field strength at a distance R/2 from the centre of the planet? (Assume the planet is a uniform sphere.)
Medium
- A4g
- B2g
- Cg/2
- Dg/4
41.State the name of the force that provides the centripetal force required to keep a satellite in a circular orbit around a planet.
Easy
42.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.
Medium
43.Complete the definition of gravitational potential.
Easy
The gravitational potential at a point is the work done per unit ____ in bringing a test mass from ____ to that point.
44.Calculate the force required to keep the satellite in this orbit. Give your answer in newtons.
Medium
45.Calculate the gravitational field strength at the surface of the Moon. Mass of the Moon = 7.35 × 10²² kg, radius of the Moon = 1.74 × 10⁶ m, G = 6.67 × 10⁻¹¹ N m² kg⁻². Give your answer to 3 significant figures.
Medium
46.What is the name of the constant G in the equation F = Gm₁m₂/r²?
Easy
47.Calculate the gravitational potential difference between orbit A and orbit B. Give your answer in J kg⁻¹.
Medium
48.Calculate the work done in moving the satellite from orbit A to orbit B. Give your answer in J.
Medium
49.Calculate the escape speed from the surface of Mars. Mass of Mars = 6.42 × 10²³ kg, radius of Mars = 3.39 × 10⁶ m, G = 6.67 × 10⁻¹¹ N m² kg⁻². Give your answer in km/s to 3 significant figures.
Hard
50.Complete the statement about gravitational field strength.
Medium
Gravitational field strength at a point is defined as the ____ per ____ experienced by a test mass at that point.
51.Explain why gravitational potential is always negative.
Hard
52.Explain why all satellites in geostationary orbit travel at the same speed.
Hard
53.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.
Medium
- A33240 N
- B27700 N
- C30470 N
- D24930 N
54.Calculate the force required to keep the satellite in this orbit. Give your answer in newtons.
Medium
- A1908 N
- B1431 N
- C1590 N
- D1749 N
55.Calculate the gravitational field strength at the surface of the Moon. Mass of the Moon = 7.35 × 10²² kg, radius of the Moon = 1.74 × 10⁶ m, G = 6.67 × 10⁻¹¹ N m² kg⁻². Give your answer to 3 significant figures.
Medium
- A2.62 N/kg
- B1.62 N/kg
- C0.62 N/kg
- D3.62 N/kg
56.Calculate the gravitational potential difference between orbit A and orbit B. Give your answer in J kg⁻¹.
Medium
- A33000000 J/kg
- B27000000 J/kg
- C30000000 J/kg
- D36000000 J/kg
57.Calculate the work done in moving the satellite from orbit A to orbit B. Give your answer in J.
Medium
- A45000000000 J
- B40500000000 J
- C49500000000 J
- D54000000000 J
58.Calculate the escape speed from the surface of Mars. Mass of Mars = 6.42 × 10²³ kg, radius of Mars = 3.39 × 10⁶ m, G = 6.67 × 10⁻¹¹ N m² kg⁻². Give your answer in km/s to 3 significant figures.
Hard
- A7.02 km/s
- B6.02 km/s
- C5.02 km/s
- D4.02 km/s

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