NDA Physics · Formula sheet

Gravitation formulas

13 formulas, 1 reference table and 17 common traps for NDA Physics Gravitation, grouped by subtopic.

Newton's Law of Gravitation

Newton's law of gravitation — the inverse-square law

Newton's law of gravitation

F = \dfrac{G\,m_1 m_2}{r^2}

$F$ gravitational force between the masses
$m_1, m_2$ the two point masses
$r$ distance between their centres
$G$ universal gravitational constant

Scaling the force — changing masses and distance together

Force scaling factor

\dfrac{F'}{F} = \dfrac{a \cdot b}{c^2}

$a, b$ factors by which the two masses change
$c$ factor by which the distance changes
$F'/F$ ratio of new force to original force

Gravitational force is action-reaction — equal and opposite

Action-reaction pair

\vec{F}_{12} = -\,\vec{F}_{21}, \qquad |\vec{F}_{12}| = |\vec{F}_{21}| = \dfrac{G m_1 m_2}{r^2}

$\vec{F}_{12}$ force on body 1 due to body 2
$\vec{F}_{21}$ force on body 2 due to body 1

The universal gravitational constant G

Property	Value / Statement
SI unit	N·m²/kg² (newton metre-squared per kilogram-squared)Q NDA 2025 — the unit of G is N-m²/kg², derived from G = Fr²/(m₁m₂).
Dimensional formula	M⁻¹L³T⁻²
Approximate value	6.674 × 10⁻¹¹ N·m²/kg²
Universality	Same for ALL pairs of bodies, everywhere; independent of mass, distance, location, or local gQ NDA 2017 — G is a universal constant; it does NOT depend on the local value of g.
Force, in contrast, is NOT universal	F itself depends on the masses and separation, so it differs for every pair of bodiesQ NDA 2018 — the false statement is 'gravitational force is the same for all pairs of bodies'. The force varies; only G is constant.

G is the constant; the FORCE is not. Don't confuse 'G is universal' with 'the gravitational force is the same for all bodies' — the latter is false.

Common traps

Distance enters as a square, masses do not

The force is linear in each mass but inverse-SQUARE in the distance. Doubling a mass doubles the force; doubling the distance quarters it. Confusing the two powers is the most common scaling error in this chapter.

G is universal, but the FORCE is not

A favourite distractor states 'gravitational force is the same for all pairs of bodies in the universe' — this is FALSE. The constant G is universal; the force F = Gm₁m₂/r² depends on the specific masses and their separation, so it differs for every pair.

Don't confuse G with g

G (capital) is the universal constant 6.674 × 10⁻¹¹ N·m²/kg², the same everywhere. g (small) is the acceleration due to gravity ≈ 9.8 m/s² at Earth's surface, and it changes with planet, altitude and location.

Square only the distance factor

When the distance changes by a factor c, the force changes by 1/c² — but the mass factors are NOT squared. For masses 2M each at R/2: the masses give 2 × 2 = 4, the distance gives 1/(½)² = 4, so the force is 16F, not 4F.

The bigger mass does NOT exert the bigger force

Students often assume the more massive body pulls harder. It does not: the gravitational force is a single action-reaction pair, so the Earth pulls the Moon with exactly the magnitude the Moon pulls the Earth. Unequal masses, equal forces.

Gravitational Field and Potential

Learn this subtopic in the notes

Surface gravity — g = GM/R²

Surface gravity

g = \dfrac{GM}{R^2}

$g$ acceleration due to gravity at the surface
$M$ mass of the planet
$R$ radius of the planet

Surface gravity from density — g = (4/3)πGρR

Surface gravity from density

g = \dfrac{4}{3}\pi G \rho R

$\rho$ mean density of the planet
$R$ radius of the planet
$g$ surface gravity (∝ ρR)

Average density of a composite body

\bar{\rho} = \dfrac{m_1 + m_2}{V_{\text{total}}} = \dfrac{\rho_1 V_1 + \rho_2 V_2}{V_1 + V_2}

$\rho_i, V_i$ density and volume of part i
$\bar{\rho}$ average density of the whole body

Gravitational field versus potential

Work and equal potential

W = -\,m\,(V_B - V_A); \qquad V_A = V_B \Rightarrow W = 0

$W$ work done by gravity, A → B
$V_A, V_B$ gravitational potential at A and B
$m$ mass moved

Weightlessness in orbit — zero normal reaction

Apparent weight = normal reaction

W_{\text{apparent}} = N = 0 \quad (\text{free fall}); \qquad F_{\text{gravity}} \neq 0

$N$ normal (contact) reaction from the floor
$W_{\text{apparent}}$ apparent weight (= N)
$F_{\text{gravity}}$ actual gravitational pull (non-zero)

g is the same for all bodies — free fall and weight

Weight and spring extension scale with g

W = mg, \qquad x = \dfrac{mg}{k} \propto g

$g$ acceleration due to gravity (independent of the body's mass)
$W$ weight of the body
$x$ spring extension; k = spring constant

Common traps

Radius enters as a square in g, just like in F

g = GM/R². When both mass and radius double, g does NOT stay the same — the mass factor 2 is divided by the radius factor squared (2² = 4), giving g/2. Always square the radius factor.

Same density does NOT mean same gravity

When two planets share a density, g = (4/3)πGρR makes g proportional to R, so the larger planet has the stronger surface gravity. Don't assume equal density gives equal g — only same density AND same radius would.

Average density is volume-weighted, not the mean of densities

For the shell-and-core sphere, averaging ρ and ρ/2 to get 3ρ/4 is wrong — that ignores that the denser core is the smaller part. The correct answer 9ρ/16 comes from total mass ÷ total volume, where the shell (the larger volume) pulls the average down.

Equal potential, not equal field, decides the work

Work by gravity depends on the potential DIFFERENCE, not on the field. If A and B are at the same potential, the work is zero even when the field strength differs between them. A different field never makes gravity non-conservative.

Weightless does NOT mean gravity-free

The common wrong choice says the gravitational pull on the astronaut is zero. It isn't — that pull is precisely what keeps the station in orbit. Weightlessness means the NORMAL REACTION is zero because everything is in free fall together.

In vacuum, mass and shape don't change the fall time

Without air resistance a coin, a feather and a mango fall with the SAME g and reach the bottom together (t₁ = t₂ = t₃). The 'heavier falls faster' intuition only holds when air drag is present.

Spring extension follows g, not just the mass

Extension x = mg/k is proportional to g. The same hanging mass stretches the spring less on the Moon (g/6 → extension/6). Don't leave the extension unchanged just because the mass is unchanged.

Orbits, Kepler and Escape

Learn this subtopic in the notes

Kepler's third law — T² ∝ a³

Kepler's third law

T^2 \propto a^3 \qquad\Longleftrightarrow\qquad \dfrac{T_1}{T_2} = \left(\dfrac{a_1}{a_2}\right)^{3/2}

$T$ orbital period (the 'year')
$a$ semi-major axis (orbit radius for a circle)

Orbital velocity — v_o = √(GM/R)

Orbital velocity

v_o = \sqrt{\dfrac{GM}{R}}

$v_o$ circular orbital speed at radius R
$M$ mass of the central body
$R$ orbit radius (from the centre)

Escape velocity — v_e = √(2GM/R) and how it scales

Escape velocity and its density scaling

v_e = \sqrt{\dfrac{2GM}{R}} = \sqrt{2gR}; \qquad v_e \propto R\sqrt{\rho}

$v_e$ escape velocity from the surface
$M, R$ planet's mass and radius
$\rho$ planet's mean density

What keeps a satellite up — no fuel required

Orbit is sustained by gravity alone

F_{\text{gravity}} = \dfrac{GMm}{R^2} = \dfrac{mv_o^2}{R} \;\Rightarrow\; \text{no propulsion needed}

$F_{\text{gravity}}$ gravitational pull = the centripetal force
$v_o$ orbital speed

Common traps

It's T² ∝ a³, not T ∝ a

Don't read Kepler's law as 'period proportional to radius'. The square of the period goes as the cube of the radius. A 4× larger orbit gives an 8× longer period (4^(3/2)), not a 4× one.

Orbital speed is set by the orbit, not the satellite

The satellite's own mass cancels out of v_o = √(GM/R). A heavy and a light satellite at the same radius orbit at exactly the same speed; only the radius (and the planet's mass) sets it.

Halving R while quadrupling ρ leaves v_e UNCHANGED

With v_e ∝ R√ρ, a radius factor of ½ and a density factor of 4 give ½ × √4 = ½ × 2 = 1. The escape speed does not change — it stays about 11.2 km/s. The frequent slip is to combine the factors as √(½ × 2) = √1 incorrectly, or to forget that radius enters linearly while density enters as a square root.

Escape velocity is independent of the projectile's mass and launch angle

v_e = √(2GM/R) contains no reference to the escaping body's mass or the direction of launch — it is the same minimum speed for a pebble or a rocket, fired in any direction (ignoring air and obstacles).

Orbiting needs no fuel — gravity does the work

A satellite is not 'held up' by rockets, remote control or solar power. It is in free fall, and gravity supplies the centripetal force. With no atmosphere to slow it, it coasts without any energy input.

Gravitation formulas

Newton's Law of Gravitation

Newton's law of gravitation — the inverse-square law

Scaling the force — changing masses and distance together

Gravitational force is action-reaction — equal and opposite

The universal gravitational constant G

Common traps

Gravitational Field and Potential

Surface gravity — g = GM/R²

Surface gravity from density — g = (4/3)πGρR

Gravitational field versus potential

Weightlessness in orbit — zero normal reaction

g is the same for all bodies — free fall and weight

Common traps

Orbits, Kepler and Escape

Kepler's third law — T² ∝ a³

Orbital velocity — v_o = √(GM/R)

Escape velocity — v_e = √(2GM/R) and how it scales

What keeps a satellite up — no fuel required

Common traps

More NDA Physics formula sheets