Table of Contents
Gravitation, Gravity and Gravitational Fields
1. Introduction
Gravitation is the natural phenomenon by which objects possessing mass attract one another.
It governs the fall of objects towards Earth, the motion of planets around the Sun, the orbits of moons and artificial satellites, ocean tides and the large-scale structure of the universe.
Gravity is one of the fundamental interactions of nature and is particularly important in astronomy, geophysics, space science and engineering.
2. Gravitation and Gravity
Although the terms are often used interchangeably, they have slightly different meanings.
Gravitation refers to the universal attractive interaction between masses.
Gravity commonly refers to the gravitational effect experienced by an object near a planet, moon or other massive body.
For example, Earth’s gravitational field causes objects near its surface to accelerate towards its centre.
3. Newton’s Universal Law of Gravitation
Newton’s law of universal gravitation states that every two objects in the universe attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
[
F=G\frac{m_1m_2}{r^2}
]
Where:
- (F) = gravitational force
- (G) = universal gravitational constant
- (m_1,m_2) = masses of the two objects
- (r) = distance between their centres
The universal gravitational constant is approximately:
[
G=6.6743\times10^{-11}\ N,m^2/kg^2
]
Its SI unit can also be written as:
[
m^3kg^{-1}s^{-2}
]
3.1 Characteristics of gravitational force
- It is always attractive between ordinary positive masses.
- It acts along the line joining the centres of the two bodies.
- It is proportional to the product of the masses.
- It is inversely proportional to the square of the distance between their centres.
- It has an unlimited range, although its strength decreases with distance.
- It acts between all objects possessing mass.
3.2 Effect of changing mass and distance
If one mass doubles while all other factors remain unchanged, gravitational force doubles.
If both masses double, gravitational force becomes four times greater.
If the distance between the centres doubles, gravitational force becomes one-quarter.
If the distance triples, gravitational force becomes one-ninth.
4. Difference Between G and g
Two important symbols in gravitation are (G) and (g).
| Universal gravitational constant (G) | Acceleration due to gravity (g) |
|---|---|
| Universal constant | Varies with location |
| Approximately (6.6743\times10^{-11}) | Approximately 9.8 m/s² near Earth’s surface |
| Appears in Newton’s gravitation law | Describes gravitational acceleration |
| SI unit: N·m²/kg² | SI unit: m/s² |
| Same value throughout the universe | Depends on the gravitational field |
Examination point: (G) is universal, whereas (g) varies from place to place.
5. Acceleration Due to Gravity
Acceleration due to gravity is the acceleration acquired by an object when it falls under the influence of gravity alone.
Near Earth’s surface:
[
g\approx9.8\ m/s^2
]
The conventional standard acceleration of gravity is:
[
g_0=9.80665\ m/s^2
]
This standard value is a defined reference value, not the exact gravitational acceleration at every location on Earth.
5.1 Factors affecting local gravitational acceleration
Local gravitational acceleration varies with:
- Latitude.
- Altitude.
- Earth’s rotation.
- Earth’s non-spherical shape.
- Local geological density variations.
The variation is small for many everyday applications but important in precise geophysical measurements.
6. Variation of Gravity with Altitude
Gravitational acceleration decreases as altitude above Earth’s surface increases.
At a distance (r) from the centre of Earth:
[
g=\frac{GM}{r^2}
]
At Earth’s surface, (r=R), where (R) is Earth’s mean radius.
At height (h):
[
g_h=\frac{GM}{(R+h)^2}
]
Therefore:
[
g_h=g\left(\frac{R}{R+h}\right)^2
]
Where:
- (g_h) = gravitational acceleration at altitude (h)
- (g) = gravitational acceleration at Earth’s surface
- (R) = Earth’s radius
For altitudes much smaller than Earth’s radius, an approximate expression is:
[
g_h\approx g\left(1-\frac{2h}{R}\right)
]
This approximation is valid only when (h) is small compared with (R).
7. Variation of Gravity with Depth
For an idealised spherical Earth of uniform density, gravitational acceleration decreases linearly with depth below the surface.
[
g_d=g\left(1-\frac{d}{R}\right)
]
Where:
- (g_d) = gravitational acceleration at depth (d)
- (d) = depth below Earth’s surface
- (R) = Earth’s radius
At the centre of an idealised uniform-density Earth, gravitational acceleration is zero because the gravitational pulls from all directions balance.
Important qualification: Earth’s actual density is not uniform, so the simple linear relationship is an idealised model rather than an exact description of Earth’s interior.
8. Gravitational Field
A gravitational field is the region around a mass in which another mass experiences gravitational force.
The gravitational field strength at a point is defined as the gravitational force per unit mass placed at that point.
[
g=\frac{F}{m}
]
The SI unit is newton per kilogram (N/kg), which is equivalent to m/s².
8.1 Gravitational field lines
Gravitational field lines represent the direction of the gravitational force on a small test mass.
Their characteristics include:
- They point towards the attracting mass.
- They are directed radially inward around an ideal spherical body.
- Closer field lines indicate a stronger field.
- Field lines do not cross.
- The field becomes weaker with increasing distance from the source mass.
9. Gravitational Potential Energy
Gravitational potential energy is the energy associated with the position of an object in a gravitational field.
Near Earth’s surface:
[
PE=mgh
]
For two masses separated by a distance (r), taking gravitational potential energy to be zero at infinite separation:
[
U=-\frac{GMm}{r}
]
The negative sign indicates that the gravitational interaction is attractive and that energy must be supplied to separate the masses to infinity.
10. Gravitational Potential
Gravitational potential at a point is the gravitational potential energy per unit mass at that point.
[
V=\frac{U}{m}
]
For a point mass:
[
V=-\frac{GM}{r}
]
The SI unit is joule per kilogram (J/kg).
Gravitational field strength is related to the rate of change of gravitational potential with distance. In a radial field:
[
g_r=-\frac{dV}{dr}
]
The negative sign indicates that the field points in the direction of decreasing potential.
11. Weightlessness and Apparent Weight
Weightlessness refers to a condition in which an object experiences zero or very small apparent weight.
Astronauts orbiting Earth experience apparent weightlessness because they and their spacecraft are in continuous free fall around Earth.
Gravity is still acting on them. It provides the centripetal acceleration needed for orbital motion.
Important: Weightlessness in orbit does not mean that gravity has disappeared.
11.1 Apparent weight
Apparent weight is the normal contact force or supporting force experienced by an object.
In a freely falling lift, the floor and the person accelerate downward together. The normal force becomes zero, so the person experiences apparent weightlessness.
12. Escape Velocity
Escape velocity is the minimum initial speed required for an object to escape the gravitational field of a celestial body without further propulsion, assuming no air resistance and zero final speed at infinity.
[
v_e=\sqrt{\frac{2GM}{R}}
]
Where:
- (G) = universal gravitational constant
- (M) = mass of the celestial body
- (R) = distance from its centre at the starting point
For Earth, escape velocity from the surface is approximately:
[
v_e=11.2\ km/s
]
This value assumes an idealised calculation without atmospheric drag or Earth’s rotation.
Escape velocity does not depend on the mass of the escaping object.
13. Orbital Motion and Satellites
A satellite remains in orbit because gravity provides the centripetal force required for its curved motion.
For a circular orbit:
[
\frac{GMm}{r^2}=\frac{mv^2}{r}
]
Therefore:
[
v=\sqrt{\frac{GM}{r}}
]
Where (r) is the orbital radius measured from the centre of the attracting body.
13.1 Types of artificial satellites
Low Earth Orbit (LEO): Satellites orbit relatively close to Earth. They are used for Earth observation, scientific research and many communication systems.
Medium Earth Orbit (MEO): Used by several global navigation satellite systems.
Geostationary orbit: A circular equatorial orbit with a period equal to Earth’s rotation period relative to distant stars, approximately 23 hours 56 minutes. A satellite in this orbit appears stationary relative to Earth’s surface.
The commonly quoted altitude for a geostationary orbit is approximately 35,786 km above the equator.
13.2 Natural satellites
A natural satellite is a celestial body that orbits another body.
Examples include:
- Earth’s Moon.
- Phobos and Deimos, which orbit Mars.
- Europa and Ganymede, which orbit Jupiter.
14. Kepler’s Laws of Planetary Motion
Johannes Kepler formulated three laws describing planetary motion.
First law: Law of orbits
Planets move in elliptical orbits with the Sun at one focus.
Second law: Law of areas
A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
A planet moves faster when it is closer to the Sun and slower when it is farther away.
Third law: Law of periods
The square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit.
[
T^2\propto a^3
]
For objects orbiting the same central mass:
[
\frac{T^2}{a^3}=\text{constant}
]
15. Tides and Gravitation
Ocean tides are primarily caused by the differential gravitational effects of the Moon and, to a lesser extent, the Sun on Earth’s oceans.
The Moon’s gravitational influence creates tidal effects, while the Sun also contributes to tidal patterns.
Spring tides: Occur when the Sun, Moon and Earth are approximately aligned, producing a greater tidal range.
Neap tides: Occur when the Sun and Moon are approximately at right angles relative to Earth, producing a smaller tidal range.
Spring tides are not limited to the spring season. The term refers to the greater tidal range.
36 Key Takeaways for Exams
- Gravitation is the universal attractive interaction between masses.
- Gravity commonly refers to the gravitational effect near a celestial body.
- Newton formulated the universal law of gravitation.
- Gravitational force is proportional to the product of two masses.
- Gravitational force is inversely proportional to the square of the distance between their centres.
- The universal gravitational constant is represented by (G).
- The approximate value of (G) is (6.6743\times10^{-11}\ N,m^2/kg^2).
- (G) is universal, whereas (g) varies with location.
- Acceleration due to gravity near Earth’s surface is approximately 9.8 m/s².
- The standard acceleration of gravity is defined as 9.80665 m/s².
- Gravitational acceleration generally decreases with increasing altitude.
- Local gravitational acceleration varies with latitude and geological conditions.
- In a uniform-density spherical Earth model, gravity decreases linearly with depth.
- Gravitational acceleration is zero at the centre of an ideal spherical Earth.
- A gravitational field is a region in which a mass experiences gravitational force.
- Gravitational field strength is force per unit mass.
- The SI unit of gravitational field strength is N/kg.
- Gravitational field lines point towards an attracting mass.
- Gravitational potential is potential energy per unit mass.
- Gravitational potential outside a spherical mass is negative when zero is defined at infinity.
- Gravitational potential energy near Earth’s surface is (mgh).
- Apparent weightlessness does not mean the absence of gravity.
- Astronauts in orbit experience apparent weightlessness because they are in continuous free fall.
- Escape velocity is the minimum ideal speed needed to escape a celestial body’s gravitational field.
- Earth’s escape velocity is approximately 11.2 km/s.
- Escape velocity does not depend on the mass of the escaping object.
- Gravity provides the centripetal force that maintains satellite orbits.
- Orbital speed for a circular orbit depends on the central mass and orbital radius.
- A geostationary satellite orbits above Earth’s equator.
- Geostationary orbital altitude is approximately 35,786 km above Earth’s equator.
- Kepler’s first law states that planetary orbits are elliptical.
- Kepler’s second law states that equal areas are swept in equal times.
- Kepler’s third law relates the square of orbital period to the cube of semi-major axis.
- The Moon is the primary gravitational influence on Earth’s ocean tides.
- Spring tides occur when the Sun, Moon and Earth are approximately aligned.
- Neap tides occur when the Sun and Moon are approximately at right angles relative to Earth.