Gravitation Class 11 Physics Quick Recall Sheet (Short Notes 2026-27)

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• Newton's Law: F = G m1 m2 / r². G = 6.67 × 10⁻¹¹ N m²/kg².
• Gravity (g): g = GM/R². Variation: Deep g' = g(1 - h/R); Height g' = g(1 - 2h/R).
• Kepler's 3rd Law: T² ∝ r³. (T²/r³ = Constant).
• Escape Velocity: v_e = √(2GM/R) = √(2gR). For Earth, v_e ≈ 11.2 km/s.
• Orbital Velocity: v_o = √(GM/r). v_e = √2 v_o. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

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Introduction

Gravitation is the universal force of attraction that binds the universe together. From the simple falling of an apple observed by Newton to the complex orbital dance of the planets described by Kepler, gravitation is the invisible thread of cosmic order. In this "Comprehensive" guide, we provide a mathematically rigorous expansion of Chapter 7, featuring formal derivations for the variation of 'g', gravitational potential energy, and the physics of satellites—providing the depth required for high-stakes exams like JEE and NEET.

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1. Newton's Law of Universal Gravitation

Every particle in the universe attracts every other particle with a force that is:

1. Directly proportional to the product of their masses (m1 m2).
2. Inversely proportional to the square of the distance between them (r²).

Formula: F = G (m1 m2) / r²

• G (Gravitational Constant): 6.67 × 10⁻¹¹ N m²/kg².
• Dimensions: [M⁻¹ L³ T⁻²].

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2. Derivation Master-Sheet: Variations in 'g'

The acceleration due to gravity (g = GM/R²) is not a constant; it varies with altitude and depth.

I. Variation with Altitude (h)

Let g be the acceleration at surface and gh at height h.

1. gh = GM / (R + h)²
2. gh = (GM/R²) · [R² / (R + h)²] = g [1 + h/R]⁻²
3. For h ≪ R, using Binomial Expansion:
   • gh = g (1 - 2h/R). (Proven)

II. Variation with Depth (d)

Let gd be acceleration at depth d.

1. Mass of Earth interior to depth d is M' = M [(R-d)/R]³.
2. gd = G M' / (R-d)² = G M (R-d) / R³
3. gd = (GM/R²) · [(R-d)/R]
4. gd = g (1 - d/R). (Proven) Conclusion: Gravity decreases whether you go up into the sky or down into a mine.

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3. Gravitational Potential Energy (U)

Derivation: The work done in bringing a mass m from infinity to a point r.

1. dW = F dr = (GMm / r²) dr.
2. Integrating from ∞ to r:
   • U = ∫ [∞ to r] (GMm / r²) dr
   • U = GMm [-1/r]∞ʳ
   • U = -GMm / r. (Proven) Note: The negative sign indicates that the force is attractive and the system is bound.

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4. Escape Velocity (v_e)

The minimum velocity required for an object to break free from Earth's gravitational pull. Derivation:

1. Using Conservation of Energy: Total Energy at Surface = Total Energy at Infinity.
2. (1/2 mv_e²) + (-GMm/R) = 0 + 0. (At infinity, both K and U are zero).
3. 1/2 mv_e² = GMm / R.
4. v_e = √(2GM/R) = √(2gR). Earth Value: v_e ≈ 11.2 km/s.

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5. Kepler’s Laws of Planetary Motion

1. Law of Orbits: Planets move in elliptical orbits with the Sun at one focus.
2. Law of Areas: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time. (Proves Conservation of Angular Momentum).
3. Law of Periods (T² ∝ R³): Derivation for Circular Orbits:
   • Centripetal Force = Gravitational Force
   • mv² / R = GmM / R² => v² = GM / R.
   • Since v = 2πR / T:
   • (4π²R² / T²) = GM / R
   • T² = (4π²/GM) R³. (Proven)

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Comprehensive Exam Strategy (Q&A)

Q1: Where is the weight of a body zero? Answer:

1. At the Centre of the Earth (where d = R, gd = 0).
2. In a freely falling elevator (Effective g = 0).
3. During space travel at a point where the gravitational pulls of different celestial bodies cancel each other out.

Q2: Does the escape velocity depend on the mass of the object being launched? Answer: No. The formula v_e = √(2gR) only depends on the mass and radius of the planet. A feather and a rocket need the same initial velocity to escape Earth's gravity (ignoring air resistance).

Q3: Why is the Gravitational Potential always negative? Answer: By convention, potential at infinity is zero. Since gravity is an attractive force, work is done by the field as an object moves closer, decreasing its potential below zero.

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Related Revision Notes

• Chapter 6: Rotational Motion (Angular Momentum)
• Chapter 2: Motion in a Straight Line (Free Fall)
• Advanced Orbital Mechanics Simulator

Conclusion

Gravitation is the foundational law of the macro-universe. By understanding how gravity changes with position and how energy is stored in gravitational fields, we can unlock the secrets of satellite technology, space exploration, and the very structure of the cosmos. Master the derivations of 'g' and escape velocity—these are the equations that humanity used to reach the Moon and beyond. Stay grounded, but keep looking up!

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Reference: NASA: Gravity and Orbits Guide