Gravitation: From Newton to Satellites
Gravitation is the invisible cosmic glue that governs the motion of planets, stars, and galaxies. It is a non-contact force of attraction that acts between any two bodies in the universe, regardless of their size or distance. On Earth, this force manifests as gravity—the pull that draws everything toward the planet’s center. Understanding gravitation is not only fundamental to physics but also crucial for competitive exams and higher education in science and engineering. This comprehensive guide covers everything from Newton’s law to satellite mechanics, complete with detailed explanations, formulas, and practice questions.
What is Gravitational Force?
Gravitational force is one of the four fundamental forces of nature, and it has several distinct characteristics:
- Action at a Distance: It does not require physical contact between objects.
- Action-Reaction Pair: The force between two bodies is equal in magnitude and opposite in direction.
- The Weakest Force: It is approximately 1036 times weaker than electrostatic force and 1038 times weaker than nuclear force.
- Conservative and Constant: Gravitational force is conservative, meaning the work done is path-independent, and it follows the inverse square law.
Mathematically, the gravitational force between two point masses is given by:
F = G × (m1 × m2) / r2
Where:
- F = Gravitational force (in Newtons, N)
- G = Universal gravitational constant = 6.67 × 10-11 N·m2/kg2
- m1, m2 = Masses of the two bodies (in kilograms, kg)
- r = Distance between the centers of the two bodies (in meters, m)
Newton’s Law of Universal Gravitation
In 1687, Sir Isaac Newton formulated the Universal Law of Gravitation, which states:
“Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.”
This law applies universally—to apples falling from trees, the Moon orbiting Earth, and galaxies moving through space.
Key Points:
- G is a Universal Constant: Its value does not change with location, altitude, or planetary system.
- Dimensional Formula of G: [M-1 L3 T-2]
- Range: Gravitational force has an infinite range but diminishes rapidly with distance.
Acceleration Due to Gravity (g)
When an object falls freely under Earth’s gravity, it experiences an acceleration known as acceleration due to gravity (g). On Earth’s surface, g ≈ 9.8 m/s2, directed toward the center of the planet.
Derivation of g from Newton’s Law:
For an object of mass m on Earth’s surface:
Gravitational force: F = G × (Me × m) / Re2
From Newton’s second law: F = m × g
Equating the two: m × g = G × (Me × m) / Re2
Simplifying: g = G × Me / Re2
Where:
- Me = Mass of Earth ≈ 5.98 × 1024 kg
- Re = Radius of Earth ≈ 6.37 × 106 m
Note: The value of g is independent of the object’s mass. In a vacuum, a feather and a hammer fall at the same rate.
Variations in Acceleration Due to Gravity
The value of g is not constant everywhere. It varies due to several factors:
| Factor | Effect on g | Formula (if applicable) |
|---|---|---|
| Altitude (h) Above Earth’s surface |
Decreases | gh = g × [Re2 / (Re + h)2] For h ≪ Re: gh ≈ g × (1 – 2h/Re) |
| Depth (d) Below Earth’s surface |
Decreases | gd = g × (1 – d/Re) At Earth’s center: g = 0 |
| Latitude (λ) Equator vs. Poles |
Maximum at poles, minimum at equator | gλ = g – Re × ω2 × cos2λ ω = Angular velocity of Earth ≈ 7.27 × 10-5 rad/s |
| Earth’s Shape Oblate spheroid |
Polar radius < equatorial radius → gpole > gequator | gpole ≈ 9.832 m/s2 gequator ≈ 9.780 m/s2 |
Real-World Examples:
- Pendulum Clocks: A pendulum clock runs slower on a mountain due to decreased g, increasing its time period (T = 2π√(L/g)).
- Sports: A tennis ball bounces higher on hills where g is lower.
- Amusement Rides: On a merry-go-round, apparent weight changes due to centrifugal force variations.
Mass vs. Weight: A Critical Distinction
Many confuse mass and weight, but they are fundamentally different:
| Property | Mass | Weight |
|---|---|---|
| Definition | Quantity of matter in a body; measure of inertia | Force with which a body is attracted toward Earth |
| Nature | Scalar (magnitude only) | Vector (magnitude and direction) |
| SI Unit | Kilogram (kg) | Newton (N) |
| Variation | Constant everywhere | Changes with location (depends on g) |
| Formula | m = F/a (Newton’s second law) | W = m × g |
| Example | Mass on Earth = 60 kg; on Moon = 60 kg | Weight on Earth ≈ 588 N; on Moon ≈ 98 N |
Weightlessness occurs when the effective weight of a body becomes zero, such as in free fall, inside an orbiting spacecraft, or at Earth’s center.
Gravitation on the Moon and Other Planets
The gravitational acceleration on the Moon is about 1/6th of Earth’s (gmoon ≈ 1.63 m/s2). This is because:
gmoon = G × Mmoon / Rmoon2
Given:
- Mmoon ≈ 7.35 × 1022 kg (≈ 1/81 of Me)
- Rmoon ≈ 1.74 × 106 m (≈ 1/4 of Re)
Thus, weight on the Moon = (1/6) × weight on Earth. This lower gravity allows astronauts to jump higher and move more easily.
Comparative Table: Gravity in the Solar System
| Celestial Body | g (m/s2) | Relative to Earth (gearth = 1) |
|---|---|---|
| Sun | 274 | ≈ 27.9 |
| Earth | 9.8 | 1 |
| Moon | 1.63 | ≈ 1/6 |
| Mars | 3.7 | ≈ 0.38 |
| Jupiter | 24.8 | ≈ 2.53 |
Satellites: Orbiting the Earth
Satellites are objects that orbit planets. Earth’s natural satellite is the Moon; artificial satellites are human-made and serve communication, weather, and research purposes.
Types of Satellites:
- Geostationary Satellites (GEO)
- Orbit: Equatorial plane, circular
- Altitude: ≈ 35,786 km (orbital radius ≈ 42,164 km)
- Period: 24 hours (synchronous with Earth’s rotation)
- Use: Communication, broadcasting (e.g., INSAT series)
- Appear stationary from Earth’s surface
- Polar Satellites (LEO)
- Orbit: Polar (pass over poles)
- Altitude: ≈ 500–2,000 km (commonly ≈ 880 km)
- Period: ≈ 84–100 minutes
- Use: Weather forecasting, remote sensing, environmental monitoring (e.g., PSLV series)
- Cover entire Earth’s surface over time
Orbital Velocity (vo)
The velocity required for a satellite to maintain a stable circular orbit:
vo = √(G × Me / r)
For a near-Earth orbit (h ≪ Re, r ≈ Re):
vo ≈ √(g × Re) ≈ 7.9 km/s (≈ 28,440 km/h)
If a satellite’s speed is less than vo, it falls back to Earth; if equal, it maintains orbit; if greater, it may enter an elliptical path or escape.
Energy of a Satellite
Total mechanical energy (binding energy) in orbit:
E = – (G × Me × m) / (2r)
Where:
- Kinetic energy (KE) = (G × Me × m) / (2r)
- Potential energy (PE) = – (G × Me × m) / r
Note: Total energy is negative, indicating a bound system.
Kepler’s Laws of Planetary Motion
Before Newton, Johannes Kepler (1571–1630) derived three empirical laws describing planetary motion:
- Law of Orbits: Planets move in elliptical orbits with the Sun at one focus.
- Law of Areas: A line joining a planet to the Sun sweeps equal areas in equal times.
- Law of Periods: The square of the orbital period (T) is proportional to the cube of the semi-major axis (a):
T2 ∝ a3 or T2 = (4π2 / G×M) × a3
These laws apply to any system where a smaller body orbits a much larger one (e.g., satellites around Earth).
Escape Velocity (ve)
Escape velocity is the minimum speed needed for an object to break free from a planet’s gravitational pull without further propulsion:
ve = √(2gR) = √(2 × G × M / R)
For Earth:
- ve ≈ 11.2 km/s (≈ 40,320 km/h)
- Relation with orbital velocity: ve = √2 × vo
Why the Moon has no atmosphere: The Moon’s escape velocity is only ≈ 2.4 km/s, allowing gases to escape into space easily.
Practice Questions with Detailed Solutions
Here are some important questions for competitive exams, along with explanations:
- Who gave the law of gravitation?
- Answer: (d) Isaac Newton
- Newton published the law in his work Philosophiæ Naturalis Principia Mathematica in 1687.
- In F = G × M×m / d2, what is G?
- Answer: (d) Universal gravitational constant
- G is a fundamental constant with value 6.67 × 10-11 N·m2/kg2.
- Which statement is true for F = G×m1×m2 / r2?
- Answer: (d) G is a universal constant
- G does not depend on local g, Earth’s surface, or any other variable.
- Which statement about gravitational force is NOT correct?
- Answer: (d) It is the same for all pairs of bodies in the universe
- Gravitational force depends on masses and distance, so it varies between different pairs.
- How does gravitational force differ from electric and magnetic forces?
- Answer: (b) Gravitational force is attractive only; electric/magnetic forces can be attractive or repulsive
- Gravity is always attractive because mass is always positive.
Additional Solved Problem:
Suppose the gravitational force between two equal masses is F. If each mass is doubled and distance unchanged, the new force becomes:
F' = G × (2m × 2m) / r2 = 4 × [G×m2/r2] = 4F (Answer: c)
Quick Reference Table: Important Values
| Parameter | Symbol | Value (SI Units) |
|---|---|---|
| Universal Gravitational Constant | G | 6.67 × 10-11 N·m2/kg2 |
| Mass of Earth | Me | 5.98 × 1024 kg |
| Radius of Earth | Re | 6.37 × 106 m |
| Acceleration due to Gravity (surface) | g | 9.8 m/s2 |
| Orbital Velocity (near Earth) | vo | 7.9 km/s |
| Escape Velocity (Earth) | ve | 11.2 km/s |
| Geostationary Orbit Altitude | hgeo | 35,786 km |
| Moon’s Gravity (relative) | gmoon/gearth | ≈ 1/6 |
Gravitation is a cornerstone of classical physics, explaining phenomena from falling apples to galactic motions. Mastering its principles—Newton’s law, variations in g, satellite dynamics, and Kepler’s laws—is essential for competitive exams like JEE, NEET, SSC, and UPSC. Remember the key distinctions: mass vs. weight, orbital vs. escape velocity, and geostationary vs. polar satellites. With consistent practice and conceptual clarity, you can tackle any gravitation problem with confidence.
Pro Tip: Always check units in calculations, use correct dimensional formulas for verification, and relate formulas to real-world examples for better retention.
Tarik Aziz
Politically neutral Blogger exploring historical controversies. Political Science and International Relations aspirant, anti-extremism advocate, Educator (BA, MA Political Science, B.Ed), with skills in Web Development and MS Office.