GRAVITATION
Gravitation is the fundamental attractive force that pulls any two masses together in the universe. It holds planets in orbit around the Sun, keeps our atmosphere bound to Earth, and causes dropped objects to fall to the ground.
1. Newton’s Universal Law of Gravitation
Statement: Every particle in the universe attracts every other particle with a mutual force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between their centers.
| Core Formula: F = G × (m₁ × m₂) / r² • F = Gravitational force of attraction (in Newtons, N) • m₁, m₂ = Masses of the two interacting objects (in kg) • r = Distance between the centers of both objects (in meters) • G = Universal Gravitational Constant |
Key Properties of Universal Gravitational Constant (G):
• Numerical Value: 6.674 × 10⁻¹¹ N·m²/kg² (SI) | 6.674 × 10⁻⁸ dyne·cm²/g² (CGS)
• Universal Nature: G is truly constant across the entire cosmos. It never changes with temperature, pressure, chemical nature, or the medium separating the masses.
2. Acceleration Due to Gravity (g)
Definition: The steady acceleration experienced by any freely falling body due solely to Earth’s gravitational attraction (neglecting air friction).
• Standard Values: 9.8 m/s² (SI unit) | 980 cm/s² (CGS unit)
| Formula for Earth’s Surface: g = (G × M) / R² • M = Mass of Earth (5.972 × 10²⁴ kg) • R = Radius of Earth (~6,371 km) • Note: Notice that the mass of the falling body (m) does NOT appear in this formula! |
| Crucial Exam Concept: Because ‘g’ is completely independent of the mass of the falling object, in a vacuum (where there is no air resistance), a light feather and a heavy iron cannonball dropped from the same height will accelerate at the exact same rate and hit the ground simultaneously. |
How the Value of ‘g’ Changes Around the Earth
| Condition / Location | Value of ‘g’ | Underlying Explanation |
| Poles vs. Equator | Maximum at Poles (~9.83 m/s²) Minimum at Equator (~9.78 m/s²) | Earth is an oblate spheroid (flattened at poles, bulging at equator). Radius at poles is smaller (R_poles < R_equator), so gravity is stronger. |
| Going Up (Altitude) | Decreases with height | As you rise, the distance (r = R + h) from Earth’s center increases: g’ = g / (1 + h/R)². |
| Going Down (Depth) | Decreases with depth | As you descend into a mine, mass of the Earth layer above you pulls upward, reducing net downward gravity. |
| Center of the Earth | g = 0 m/s² (Zero) | Mass of Earth surrounds you uniformly and pulls equally in every direction, completely cancelling out gravitational pull. |
3. Difference Between Mass and Weight
A common point of confusion is mistaking mass for weight. Here is a clear side-by-side comparison:
| Parameter | Mass (m) | Weight (W) |
| Definition | Actual quantity of matter contained within a body. | Gravitational pull exerted by a celestial body (e.g., Earth) on that mass. |
| Formula | Inherent property of matter (constant). | W = m × g |
| Nature / Type | Scalar quantity (has magnitude only). | Vector quantity (always directed toward center of Earth). |
| SI Unit | Kilogram (kg) | Newton (N) or Kilogram-force (kgf) |
| CGS Unit | Gram (g) | Dyne |
| Measuring Device | Beam balance / Pan balance (compares unknown to known mass). | Spring balance (measures tension produced by gravity). |
| Constancy | Constant everywhere in the universe; never zero. | Varies by location; becomes exactly zero where g = 0 (e.g., at Earth’s center or deep space). |
4. Key Applications and Scenarios
A. Gravity on the Moon
The Moon has significantly less mass and radius than Earth, resulting in surface gravity that is roughly 1/6th of Earth’s:
• g_moon ≈ g_earth / 6 ≈ 1.63 m/s²
• Example: An astronaut weighing 60 kgf on Earth weighs only 10 kgf on the Moon. However, their mass remains exactly 60 kg.
B. Apparent Weight in an Elevator / Lift
When standing on a weighing scale in a lift, the scale measures normal reaction force (R), also called apparent weight:
• Stationary or moving at constant speed: Acceleration a = 0 → Apparent Weight = True Weight (W = mg)
• Accelerating Upwards (rate a): Floor pushes harder → Apparent Weight increases [W = m(g + a)]
• Accelerating Downwards (rate a): Floor recedes slightly → Apparent Weight decreases [W = m(g – a)]
• Free Fall (cable snaps, a = g): Floor falls at the same rate as you → Apparent Weight = 0 (Total Weightlessness)
C. Escape Velocity (v_e)
Definition: The absolute minimum speed an object must be projected with from the surface of a body to break away from its gravitational pull forever without needing ongoing thrust.
• Formula: v_e = √(2gR)
• On Earth: 11.2 km/s (~40,320 km/h)
• On the Moon: 2.4 km/s
• Why the Moon has no atmosphere: Because the Moon’s escape velocity is extremely low (2.4 km/s), thermal speeds of gas molecules exceed escape velocity and escape into space.
D. Kepler’s Laws of Planetary Motion
1. Law of Orbits: All planets move in elliptical orbits with the Sun located at one focus.
2. Law of Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time (orbital speed is fastest when closest to Sun).
3. Law of Periods: The square of a planet’s orbital time period is directly proportional to the cube of the semi-major axis of its orbit (T² ∝ r³).
5. High-Yield Exam Facts Checklist
- At Earth’s center, g = 0, so weight is 0 N, but mass is completely unaffected.
- Weight of any object is maximum at the Poles and minimum at the Equator.
- A beam balance measures mass; a spring balance measures weight.
- Earth’s escape velocity is 11.2 km/s (independent of the mass of the projectile).
- Big ‘G’ is constant throughout the universe; small ‘g’ depends on planetary mass, radius, and local altitude/depth.