Gravitation — Study Notes

Overview

Gravitation is one of the fundamental forces of nature and a high-yield topic in RRB Group D General Science. This chapter explains why objects fall to the ground, how planets orbit the Sun, and how satellites stay in orbit. Questions typically test Newton's law of universal gravitation, the distinction between mass and weight, concepts of free fall and acceleration due to gravity, and satellite motion. Expect 2–3 direct questions in the exam covering formula-based calculations, conceptual understanding of g-values at different locations, and practical applications like satellite launches. Mastery requires understanding the inverse-square law, memorizing key formulas, and applying them to numerical problems involving weight, gravitational force, and orbital motion.

Key Concepts

• **Universal Law of Gravitation**: Every object in the universe attracts every other object with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This is the foundation of planetary motion and satellite dynamics.

• **Mass vs Weight**: Mass is the quantity of matter in an object (measured in kg, constant everywhere). Weight is the gravitational force acting on that mass (measured in N, varies with location). Weight = mass × acceleration due to gravity.

• **Acceleration due to Gravity (g)**: On Earth's surface, g ≈ 9.8 m/s² (often approximated as 10 m/s² for quick calculations). This value decreases with altitude and varies slightly with latitude due to Earth's rotation and non-spherical shape.

• **Free Fall**: When an object falls under gravity alone with no air resistance, it is in free fall. All objects in free fall near Earth's surface accelerate at the same rate (g), regardless of their mass—a feather and a hammer fall at the same rate in vacuum.

• **Variation of g**: The value of g decreases as we go above the Earth's surface (on mountains, in aeroplanes) or below the surface (in mines). At the centre of Earth, g = 0.

• **Gravitational Potential Energy**: The energy possessed by an object due to its position in a gravitational field. Near Earth's surface: PE = mgh, where h is height above reference level.

• **Satellites**: Objects that revolve around a planet in a fixed orbit. Artificial satellites stay in orbit because their tangential velocity balances the gravitational pull, creating a continuous free fall around Earth.

• **Escape Velocity**: The minimum velocity needed for an object to break free from a planet's gravitational field without further propulsion. For Earth, escape velocity ≈ 11.2 km/s.

Formulas / Key Facts

• **Newton's Law of Universal Gravitation**: F = G(m₁m₂)/r², where F is gravitational force, G is universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are masses, r is distance between centres.

• **Weight Formula**: W = mg, where W is weight in newtons, m is mass in kg, g is acceleration due to gravity (9.8 m/s²).

• **Acceleration due to Gravity on Earth's Surface**: g = GM/R², where G is gravitational constant, M is Earth's mass, R is Earth's radius. Standard value: g = 9.8 m/s² or approximately 10 m/s².

• **Free Fall Equations**: v = u + gt; s = ut + ½gt²; v² = u² + 2gs (where u = initial velocity, v = final velocity, t = time, s = distance, g = acceleration).

• **Weight on Moon**: Moon's gravity is 1/6th of Earth's, so weight on Moon = (1/6) × weight on Earth.

• **Orbital Velocity of Satellite**: v = √(GM/r), where r is orbital radius from Earth's centre. For low Earth orbit, v ≈ 8 km/s.

• **Escape Velocity**: vₑ = √(2GM/R). For Earth, vₑ ≈ 11.2 km/s.

• **Gravitational Potential Energy**: PE = mgh (near surface); PE = -GMm/r (at distance r from centre).

Worked Examples

**Example 1: Calculate weight** *A person has a mass of 60 kg. What is their weight on Earth? (Take g = 10 m/s²)*

**Solution**: Weight W = mg W = 60 kg × 10 m/s² W = 600 N

The person weighs 600 newtons on Earth.

**Example 2: Weight on Moon** *If the same person (60 kg) goes to the Moon, what will be their weight? (Moon's g = 1.6 m/s²)*

**Solution**: Weight on Moon = mg_moon = 60 kg × 1.6 m/s² = 96 N

Alternatively: Moon's gravity = 1/6 Earth's Weight on Moon = 600 N / 6 = 100 N (using approximation)

**Example 3: Free Fall Calculation** *A stone is dropped from a height. What is its velocity after 3 seconds? (Take g = 10 m/s²)*

**Solution**: Initial velocity u = 0 (dropped, not thrown) Using v = u + gt v = 0 + 10 × 3 v = 30 m/s

After 3 seconds, the stone's velocity is 30 m/s downward.

Common Mistakes

• **Confusing mass and weight**: Students often use these interchangeably. Remember: mass is constant (70 kg on Earth = 70 kg on Moon), but weight changes with gravity (700 N on Earth, ~117 N on Moon). Always check the unit—kg means mass, N means weight.

• **Forgetting the square in inverse-square law**: The gravitational force depends on 1/r², not 1/r. If distance doubles, force becomes 1/4th (not 1/2). If distance triples, force becomes 1/9th.

• **Assuming g is always exactly 10 m/s²**: While 10 is convenient for mental math, the standard value is 9.8 m/s². Use 9.8 when precision matters in calculations, especially in formula-based questions.

• **Thinking heavier objects fall faster**: In the absence of air resistance, all objects fall at the same rate (g). A 1 kg ball and a 10 kg ball dropped simultaneously will hit the ground together. Mass does not affect free fall acceleration.

• **Misapplying escape velocity**: Escape velocity is the speed needed to leave Earth's gravitational influence completely, not just to reach space or orbit. Satellites don't need escape velocity—they need orbital velocity (about 8 km/s for low orbit vs 11.2 km/s to escape).

Quick Reference

• **Universal gravitation law**: F = G(m₁m₂)/r²; force decreases as square of distance.
• **Weight formula**: W = mg; weight is force, measured in newtons.
• **Earth's g = 9.8 m/s²**; Moon's g = 1.6 m/s² (≈ 1/6 of Earth's).
• **Free fall**: All objects fall at rate g regardless of mass (in vacuum).
• **g decreases** with altitude above Earth and depth below surface; zero at Earth's centre.
• **Escape velocity from Earth ≈ 11.2 km/s**; orbital velocity ≈ 8 km/s.