Newton's Law of Gravitation

• Newton's law: F = Gm₁m₂/r² — every mass attracts every other mass; force along the line joining them.
• Value of G = 6.67 × 10⁻¹¹ N·m²·kg⁻² — the universal gravitational constant.
• Superposition principle: total gravitational force on a body = vector sum of forces from all other bodies.
• How surface gravity g = GM/R² is derived from the universal law.
• Distinction between G (universal constant) and g (varies with location).

Level 1 — Foundations

Verbal: Every object with mass attracts every other object with mass; the force is proportional to both masses and inversely proportional to the square of distance between them.

Formula: F = Gm₁m₂/r², where r = distance between centres of mass.

G = 6.67 × 10⁻¹¹ N·m²·kg⁻² — extremely small; gravity is the weakest fundamental force but acts over infinite range.

Direction: Force is attractive, directed along line joining the two masses (Newton's 3rd law: equal and opposite pair).

Relation to g: At Earth's surface, F = mg and F = GMm/R² → g = GM/R². With M = 6 × 10²⁴ kg, R = 6.4 × 10⁶ m → g ≈ 9.8 m/s².

Units check: [G] = [F][r²]/[m²] = N·m²·kg⁻².

Level 2 — JEE / NEET depth

Superposition principle: F_total on mass m₁ = ΣFᵢ where Fᵢ = force from each other mass. Vector addition required — cannot simply add magnitudes unless forces are collinear.

Variation of g with height h above Earth's surface: g_h = GM/(R+h)² = g(1 + h/R)⁻² ≈ g(1 − 2h/R) for h << R.

Variation of g with depth d below Earth's surface (uniform density): g_d = g(1 − d/R) — g decreases linearly to zero at centre.

At poles vs equator: g_pole > g_equator (Earth is oblate: R_pole < R_equator; also centrifugal effect at equator reduces effective g).

Shell theorem (for spherically symmetric bodies):

• Outside shell: acts as point mass at centre.
• Inside shell: net gravitational force = 0.

g on another planet/moon: g' = GM'/R'² — use this to scale problems.

Cavendish experiment (1798): First measured G using torsion balance with lead spheres.

Gravitational force between two spheres

Find the gravitational force between two uniform spheres:
m₁ = 50 kg, m₂ = 30 kg, separated by r = 0.5 m (centre to centre).
G = 6.67 × 10⁻¹¹ N·m²·kg⁻².

Step 1 — Apply Newton's law:
F = Gm₁m₂/r²
F = (6.67 × 10⁻¹¹ × 50 × 30) / (0.5)²

Step 2 — Numerator:
6.67 × 10⁻¹¹ × 1500 = 1.0005 × 10⁻⁷ N·m²

Step 3 — Denominator: (0.5)² = 0.25 m²

Step 4 — Force: F = 1.0005 × 10⁻⁷ / 0.25 = 4.0 × 10⁻⁷ N

Step 5 — Direction: attractive, each sphere pulled toward the other.
(Compare to weight of m₁: 50 × 10 = 500 N — gravitational attraction between
everyday objects is negligible!)

g at height and at depth

Find g at (a) height h = R/2 above Earth surface, (b) depth d = R/2 below surface.
g at surface = 10 m/s².

Part (a) — At height h = R/2:
g_h = g × R² / (R+h)² = g × R² / (R + R/2)² = g × R² / (3R/2)²
    = g × R² / (9R²/4) = g × 4/9 = 10 × 4/9 ≈ 4.44 m/s²

Part (b) — At depth d = R/2 (inside Earth, uniform density):
g_d = g(1 − d/R) = 10 × (1 − 1/2) = 10 × 0.5 = 5 m/s²

Conclusion: At depth R/2, g = 5 m/s² (linear decrease).
At height R/2, g = 4.44 m/s² (inverse square — less reduction than depth for same geometric fraction).

• State Newton's law of gravitation in words and formula.
• Calculate g on Mars: M_Mars = 6.4×10²³ kg, R_Mars = 3.4×10⁶ m.
• Does the gravitational force between two objects depend on the medium between them?
• At what height above Earth is g reduced to g/4? (Express in terms of R.)
• Stretch: Three equal masses m at vertices of equilateral triangle (side a). Find net force on one mass.

NCERT Chapter 8 link: Newton's law is the foundation for all gravitation problems. Always check whether height/depth is small or large compared to R — different approximations apply. The shell theorem simplifies force calculations for spherically symmetric bodies.

Exam connections: JEE tests: force between two masses (numerical G calculation); variation of g with h and d (especially h = nR cases); superposition with 2-3 masses; g on moon/planet using ratio method (avoid computing from scratch). Ratio method: g'/g = (M'/M)(R/R')².

Study strategy: Memorise g = GM/R² and its approximate forms for height and depth. For multi-mass problems, place masses on coordinate axes and resolve forces into x and y components before finding magnitude.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.