Gravitation Short Notes for JEE Main: Revision Notes PDF

Newton's law and superposition

Newton's law of gravitation

4× asked, last 2025

F = \frac{G m_1 m_2}{r^2},\qquad G = 6.67\times10^{-11}\ \text{N m}^2\,\text{kg}^{-2}

Point or spherically symmetric masses,

r

between centres; attractive, along that line. Other shapes attract too: integrate.

Rod of mass $M$ , length $L$ , with $m$ on its axis at distance $d$ from the near end: integrating gives $F=GMm/[d(d+L)]$ , not $GMm/(d+L/2)^2$ .
Not unchanged when mass moves from one body to the other. $F\propto m_1m_2$ , not the sum. For a fixed total the product is largest for equal masses: a transfer that narrows the gap raises $F$ , one that widens it lowers $F$ .

Superposition of point masses

5× asked, last 2026

\vec F_{\text{net}} = \sum_i \frac{G m\,m_i}{r_i^2}\,\hat r_i

\hat r_i

points from

m

towards

m_i

. Add as vectors; other masses do not change any pair's force.

Not only the nearest mass. Include every mass. With masses placed symmetrically, cancel the sideways components first, then add the components along the line of symmetry.
Quick method. Pair masses first: the field at the centre from opposite corners is $G|m_1-m_2|/r^2$ towards the heavier; equal masses on a regular polygon give zero. Square corner: $(\sqrt2+1/2)\,Gm^2/a^2$ along the diagonal.

Masses circling under mutual attraction

5× asked, last 2023

\frac{G m^2}{(2r)^2} = \frac{m v^2}{r}\;\Rightarrow\; v=\sqrt{\frac{Gm}{4r}},\qquad \omega=\sqrt{\frac{Gm}{4r^3}}

Two equal masses on a circle of radius

r

. Four equal masses

M

(a square):

v^2=(2\sqrt2+1)GM/4R

Three equal masses $m$ at the corners of an equilateral triangle of side $a$ : each is pulled $\sqrt3\,Gm^2/a^2$ to the centre and circles at radius $a/\sqrt3$ , so $v=\sqrt{Gm/a}$ .
Not $r$ as the separation. The masses are a diameter apart, so Newton's law uses $(2r)^2$ while $mv^2/r$ uses $r$ . On a circle of radius $R$ , a square's side is $\sqrt2R$ and its diagonal $2R$ .

Binary stars

2× asked, last 2021

\omega^2=\frac{G(m_1+m_2)}{d^3},\qquad r_1=\frac{m_2\,d}{m_1+m_2}

Separation

d

. Both circle the centre of mass with the same

\omega

and

T

; the heavier star has the smaller circle.

COM

2m

m

Both circle the common COM with one

\omega

; the heavier star on the smaller circle

Not $\omega/2\pi$ or $1/2\pi\omega$ for the period. $T=2\pi/\omega$ . Check units: $\sqrt{Gm/d^3}$ is already in s⁻¹, so it is $\omega$ , not a time.

Ring on its axis

2× asked, last 2021

E=\frac{GMx}{(R^2+x^2)^{3/2}},\qquad V=-\frac{GM}{\sqrt{R^2+x^2}}

Ring radius

R

x

on the axis from its centre.

E

points to the centre, is zero there and largest at

x=R/\sqrt2

x

E

\frac{R}{\sqrt2}

2R

largest

Zero at the centre, largest at

x=R/\sqrt2

, then falls off

A uniform sphere on the ring's axis acts as a point mass at the sphere's centre.

Arc at its centre

1× asked, last 2024

F=\frac{2Gm\lambda}{R}\sin\frac{\phi}{2},\qquad \lambda=\frac{M}{R\phi}

Arc angle

\phi

in radians. Semicircle:

2Gm\lambda/R

; full ring: zero at the centre.

Not $GMm/R^2$ for a semicircle. The pieces pull at different angles and the sideways parts cancel; only $2/\pi$ of it survives. A wire of length $L$ bent into a semicircle has $R=L/\pi$ .

Sphere with a cavity

2× asked, last 2025

F = F_{\text{full}} - F_{\text{removed}},\qquad M_{\text{removed}} = M\left(\frac{r}{R}\right)^3

Uniform sphere: radius

R

, uncut mass

M

. Cavity radius

r

: its mass acts as a point at its centre, at its distance from

m

Inside the cavity the field is uniform: size $(4/3)\pi G\rho d$ , pointing from the cavity's centre towards the sphere's centre; $d$ = distance between the centres.
Not $M(r/R)^3$ when the given $M$ is the mass left after cutting. Then $M_{\text{removed}}=Mr^3/(R^3-r^3)$ , e.g. $M/7$ , not $M/8$ , for $r=R/2$ .

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Practise Gravitation

26 more concepts in 5 topics

Acceleration due to gravity and its variation, Gravitational field and potential, Gravitational PE and escape speed, Satellites and orbital motion and Kepler's laws. Read the first part free. Sign in free with Google to read the rest and download the PDF.

Acceleration due to gravity and its variation

g at the surface10×
Variation with height15×
Variation with depth15×
Latitude and earth's rotation6×
Tunnel through the earth1×

Gravitational field and potential

Point mass: field and potential4×
Thin spherical shell2×
Uniform solid sphere5×
Non-uniform density2×

Gravitational PE and escape speed

PE of a system of masses4×
Change in PE with height6×
Escape speed18×
Escape from a height or between bodies3×
Launched below escape speed4×
Neutral point between two bodies3×

Satellites and orbital motion

Orbital speed8×
Orbit shape and launch speed
Time period10×
Angular momentum of a satellite4×
Energy of a satellite4×
Energy to launch or change orbit3×
Geostationary and polar satellites3×
Weightlessness in orbit

Kepler's laws

First and second laws4×
Perihelion and aphelion speeds5×
Third law16×

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Gravitation revision notes

Newton's law and superposition

26 more concepts in 5 topics

Acceleration due to gravity and its variation

Gravitational field and potential

Gravitational PE and escape speed

Satellites and orbital motion

Kepler's laws

Practise Gravitation next

Also free on PrepWiser

Downloading Gravitation revision notes (PDF)

Gravitation revision notes

1Newton's law and superposition7 concepts

26 more concepts in 5 topics

2Acceleration due to gravity and its variation5 concepts

3Gravitational field and potential4 concepts

4Gravitational PE and escape speed6 concepts

5Satellites and orbital motion8 concepts

6Kepler's laws3 concepts

Practise Gravitation next

Also free on PrepWiser

Newton's law and superposition

Acceleration due to gravity and its variation

Gravitational field and potential

Gravitational PE and escape speed

Satellites and orbital motion

Kepler's laws