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.

WBCS PYQS

Q1. A rocket leaves the earth, lands on the moon and returns to earth. It will require the maximum energy to:(WBCS Prelims 2004)

(A) Rise from the earth

(B) Land on the moon

(D) Land on the earth

Correct Answer: (A) Rise from the earth

Q2. When a body is raised upwards from the Surface of the earth, its weight shall:(WBCS Prelims 2007)

(A) Increase

(B) Remain the same

(D) Fluctuate

Correct Answer: (A) Increase / (C) Decrease (Note: As altitude increases, gravity g decreases, so weight w = mg decreases. However, in an accelerating lift upward, apparent weight increases.)

Q3. Given the Premises:
(i) Human cannot live without atmosphere
(ii) The moon is without atmosphere
Then which of the following conclusions is valid?(WBCS Prelims 2011)

(A) Some men cannot live on the moon

(B) There is no life possible on the moon

(D) All of these

Correct Answer: (C) The man cannot live on the moon / (D) All of these

Q4. A body is floating in a liquid partly immersed. The body and the liquid are taken to moon. The body:(WBCS Prelims 2018)

(A) Will continue to float as before

(B) Floats with greater part immersed

(D) Will sink

Correct Answer: (A) Will continue to float as before

Q5. A person on a swing stands up. The time period of the swing:(WBCS PRE 2021)

(A) Increases

(B) Remains the same

(D) Becomes zero

Correct Answer: (C) Decreases