Gravitation — Concept to Mastery
Build every concept from the ground up. If your foundation is weak, no formula saves you in JEE.
Newton's Universal Law of Gravitation
Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
m₁, m₂ = Masses of two particles (kg)
r = Distance between their centres (m)
Exam Insight
This is an inverse square law. If distance doubles, force becomes 1/4. If masses double, force doubles. JEE always tests this proportionality.
Common Mistake Alert
"r" is measured centre to centre, NOT surface to surface. For spheres, use the full distance between centres. Most students lose marks here in numericals.
Key Properties of Gravitational Force
Always Attractive
Unlike electric force, gravitational force is always attractive. There is no "gravitational repulsion" in classical physics.
Long Range Force
Acts over infinite distances (theoretically). Even across galaxies — the force just becomes negligibly small.
Superposition Principle
Net gravitational force on a body = vector sum of forces due to all other bodies individually. F_net = F₁ + F₂ + F₃ + ...
Medium-Independent
Does NOT depend on the medium between the masses. It works the same in vacuum, water, or rock.
Thinking Step
Why does G have such a tiny value (10⁻¹¹)? Because everyday masses (kg-scale) produce negligible gravitational forces. Only astronomical masses (10²⁴ kg) create detectable gravity. This is why you don't fall toward buildings!
🔬 Derivation: Value of g from Newton's Law
Kepler's Three Laws of Planetary Motion
These were empirically discovered by Kepler before Newton derived them theoretically. JEE loves mixing all three laws in one problem.
Sun at focus F₁
= L/2m
T²/a³ = const
M = Mass of central body (Sun/Earth)
Exam Insight
Kepler's 2nd law is a consequence of conservation of angular momentum. When planet is closest (perihelion) → speed is maximum. When farthest (aphelion) → speed is minimum. JEE Advanced directly tests this.
Common Mistake Alert
In Kepler's 3rd law: T² ∝ r³ holds for circular orbits (r = radius). For elliptical orbits, replace r with semi-major axis 'a'. Many students confuse r with the orbital radius in elliptical problems.
🔬 Derivation: Kepler's 3rd Law from Newton's Law
GMm/r² = mv²/r
Thinking Step
The ratio T₁²/r₁³ = T₂²/r₂³ is constant for all planets orbiting the same star. Use this directly in ratio problems without finding GM — saves time in NEET/JEE Main.
🔭 Live Kepler Orbit Visualization
Live animation: Planet sweeping equal areas in equal times (Kepler's 2nd Law)
Variation of Acceleration due to Gravity (g)
This is one of the most tested topics in NEET and JEE Main. Four distinct situations — each with a different formula.
📍 On Earth's Surface
R = Radius of Earth = 6400 km
🏔️ At Height h
🕳️ At Depth d
At Earth's centre (d=R): g = 0
🌀 Due to Earth's Rotation
Exam Insight – Critical Comparison
Rate of decrease of g: At the same small distance from surface — g decreases faster with altitude than with depth. This is a classic NEET/JEE MCQ trap. The formula shows g_h ∝ (1-2h/R) vs g_d ∝ (1-d/R).
Common Mistake Alert
At the centre of Earth, g = 0 — so your weight = 0. But your mass does NOT change. Also, the approximation formula g_h = g₀(1-2h/R) is only valid when h << R. For h comparable to R, use exact formula.
Escape Velocity
The minimum velocity required to escape a gravitational field — to reach infinity with zero kinetic energy.
For Moon: Vₑ ≈ 2.4 km/s
Independent of mass & direction of projection
🔬 Derivation from Energy Conservation
Thinking Step
Escape velocity is independent of: (a) the mass of the escaping object (m cancels), (b) direction of projection, (c) the shape of path. It's purely about energy, not direction!
Exam Insight: JEE Twist
JEE asks: "If a planet has double the radius but same density as Earth, what is its escape velocity?" — Using Vₑ = √(8πGρ/3)·R, Vₑ doubles. This form is key for JEE Advanced.
Common Mistake Alert
Escape velocity ≠ orbital velocity at the surface. Escape velocity = √2 × orbital velocity. Students often confuse the two. Orbital velocity at surface = √(GM/R) = 7.9 km/s for Earth.
Satellites & Orbital Mechanics
Everything from orbital speed to geostationary satellites — a favourite in NEET and JEE Main.
Total energy is always negative (bound system)
Geostationary Satellite
• T = 24 hours (matches Earth's rotation)
• Appears stationary from Earth
• Orbits at h ≈ 36,000 km above equator
• Must orbit in equatorial plane
• Used for: communication, GPS, weather
Exam Insight
Satellite energy relations: KE = -E, PE = 2E, KE = -½PE. When orbit radius increases: KE decreases, PE increases, Total Energy increases (becomes less negative). This seems paradoxical — it's a classic JEE trap!
Common Mistake Alert
In weightlessness (free fall), everything inside the satellite appears weightless. BUT the satellite is NOT outside Earth's gravitational field — g is still ~8.7 m/s² at 400 km altitude.
Gravitational Potential Energy & Binding Energy
The most conceptually tricky part — negative potential energy, reference at infinity, and binding energy. This is where students lose JEE marks.
Always negative (attractive force)
Increases (becomes less negative) as r increases
Thinking Step: Why Negative?
We define PE = 0 at infinity (reference). Work done BY gravity when two masses come from ∞ to r is positive → PE at r must be negative. You gain PE by moving away → need to do positive work to escape.
= -E_total (since E_total is negative)
Gravitational Potential (V)
Units: J/kg = m²/s²
Always negative, zero at infinity
Exam Insight: JEE Level
Potential inside a uniform solid sphere: V = -GM(3R²-r²)/2R³ (not tested in CBSE but JEE Advanced may ask this).
Common Mistake Alert
Gravitational PE = mV (scalar). Gravitational Field (g) is a vector. Don't mix up their properties. PE can be added algebraically; field must be added vectorially.