Class 11 · Chapter 8

Gravitation

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Gravitation Formula Sheet

10 formulas across 3 topics in Gravitation.

1 min read

Updated 2026-07-03 · v1.0.1

Newton's Law of Gravitation & Field

Every mass attracts every other mass. The inverse-square law plus superposition is the entire foundation of this chapter.

F = G m₁m₂ / r²

Newton's law of gravitation

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
G	universal gravitational constantG = 6.67 × 10⁻¹¹ N·m²/kg²	N·m²/kg²	[M⁻¹L³T⁻²]
m₁, m₂	the two point masses	kg	[M]
r	separation between their centres	m	[L]

Valid when

Point masses, or spheres with uniform (radial) density measured centre-to-centre

Common mistakes

Gravitational force is ALWAYS attractive — no gravitational shielding exists

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g = GM / R²

Acceleration due to gravity at a planet's surface

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mg = GMm/R² on a surface mass

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
M	mass of the planet	kg	[M]
R	radius of the planet	m	[L]

Common mistakes

g is independent of the falling body's mass — that is the equivalence of inertial and gravitational mass

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g(h) = g (1 − 2h/R) for h ≪ R; g(h) = gR²/(R+h)² exactly

Variation of g with height

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Binomial expansion of gR²/(R+h)² for small h/R

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
h	height above the surface	m	[L]
R	planet radius	m	[L]

Valid when

The (1 − 2h/R) form is an approximation valid only for h ≪ R

Common mistakes

Depth variation is DIFFERENT: g(d) = g(1 − d/R) — factor 1, not 2; g falls linearly to zero at the centre

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Worth remembering

Inside a uniform spherical SHELL, the gravitational field is exactly zero everywhere
At depth d, only the inner sphere of radius (R − d) attracts — the outer shell contributes nothing

Gravitational Potential & Energy

With the zero of potential energy at infinity, all bound systems have negative energy — the sign carries the physics.

U = − G M m / r

Gravitational potential energy (zero at infinity)

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Work done by gravity bringing m from infinity to r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
U	potential energy of the pair	J	[ML²T⁻²]
r	centre-to-centre separation	m	[L]

Valid when

U = mgh is this formula's near-surface approximation for the CHANGE in U

Common mistakes

U is negative and INCREASES (toward zero) as r increases — 'less negative' means higher energy

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V = − G M / r

Gravitational potential

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
V	potential (PE per unit mass)	J/kg	[L²T⁻²]

Valid when

Inside a uniform solid sphere the potential at the CENTRE is −3GM/2R (1.5× the surface value)

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v(e) = √(2GM/R) = √(2gR)

Escape velocity

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Total energy zero at launch: ½mv² − GMm/R = 0

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v(e)	escape speed from the surface11.2 km/s for Earth	m/s	[LT⁻¹]

Valid when

Independent of the projectile's mass AND of launch direction (ignoring rotation and atmosphere)

Common mistakes

v(e) = √2 × v(orbital, low orbit) — remember the √2 bridge between the two speeds

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Worth remembering

Total energy decides fate: E < 0 bound (ellipse/circle), E = 0 parabola (barely escapes), E > 0 hyperbola

Satellites & Kepler's Laws

A satellite is perpetual free fall — gravity supplies exactly the centripetal force. Every orbital formula falls out of that single equation.

v(o) = √(GM/r); near surface v(o) = √(gR) ≈ 7.9 km/s

Orbital velocity

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Gravity as centripetal force: GMm/r² = mv²/r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v(o)	orbital speed for a circular orbit of radius r	m/s	[LT⁻¹]
r	orbit radius from the planet's CENTRE (r = R + h)	m	[L]

Valid when

Circular orbit
Higher orbit → SLOWER speed

Common mistakes

r is measured from the planet's centre, not from the surface — forgetting to add R to the height is the classic slip

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T = 2π √(r³/GM)

Time period of a satellite

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T = 2πr / v(o)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	orbital period	s	[T]

Valid when

This IS Kepler's third law: T² ∝ r³
Geostationary orbit: T = 24 h, r ≈ 42000 km, equatorial plane, west-to-east

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E = −GMm/2r; K = +GMm/2r; U = −GMm/r

Energies of an orbiting satellite

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K = ½mv(o)² with v(o)² = GM/r; E = K + U

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
E	total mechanical energy of the orbit	J	[ML²T⁻²]

Valid when

The memory pattern: U = 2E, K = −E, |U| = 2K

Common mistakes

Binding energy = +GMm/2r — the energy you must SUPPLY to free the satellite

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T² ∝ a³ (a = semi-major axis)

Kepler's three laws

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
a	semi-major axis of the elliptical orbit	m	[L]

Valid when

1st law: orbits are ellipses with the Sun at one focus
2nd law: equal areas in equal times (angular momentum conservation)
3rd law: T² ∝ a³

Common mistakes

2nd law consequence: a planet moves FASTEST at perihelion (closest approach), slowest at aphelion

Read the concept →

Worth remembering

A satellite needs no fuel to stay in orbit but plenty to CHANGE orbit — raising the orbit costs energy even though the speed decreases
Weightlessness in orbit is free fall, not absence of gravity — g at ISS altitude is still ≈ 90% of surface g

NEET repeats escape/orbital velocity ratios yearly; JEE Advanced uses elliptical orbits where only L and E are conserved — apply both at perigee and apogee.

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Previous chapter7. System of Particles & Rotational Motion Next chapter 9. Mechanical Properties of Solids

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