PiTheory

Gravitation formulas

Class 11 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 7, the special cases questions are built on, and diagrams where they help.

23 formulas5 sectionsClass 11 · Chapter 72 of 5 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Kepler's laws

  • Orbits: ellipse with the Sun at a focus. $r_p=a(1-e)$ (perihelion), $r_a=a(1+e)$ (aphelion), $a=\dfrac{r_p+r_a}2$ (mean distance).
  • Areas: equal areas in equal times (angular momentum conserved, central force).
$$\frac{dA}{dt}=\frac L{2m}=\frac{r^2\omega}2=\text{const},\qquad \frac{v_p}{v_a}=\frac{r_a}{r_p}=\frac{1+e}{1-e}$$
$$\begin{array}{l}\displaystyle \frac{dA}{dt}=\frac L{2m}=\frac{r^2\omega}2=\text{const}\\[6pt]\displaystyle \frac{v_p}{v_a}=\frac{r_a}{r_p}=\frac{1+e}{1-e}\end{array}$$

$v_p=\sqrt{\dfrac{GM}a\dfrac{1+e}{1-e}}$, $v_a=\sqrt{\dfrac{GM}a\dfrac{1-e}{1+e}}$. Longer time on the far half of the orbit.

$$T^2=\frac{4\pi^2}{GM}a^3,\qquad \frac{T_1^2}{T_2^2}=\frac{a_1^3}{a_2^3}$$

Periods: $\dfrac{T_1}{T_2}=\left(\dfrac{r_1}{r_2}\right)^{3/2}$. Mass of Sun: $M=\dfrac{4\pi^2r^3}{GT^2}$. If force $\propto r^{-n}$: $T\propto r^{(n+1)/2}$, $v\propto r^{(1-n)/2}$. Speed in ellipse at distance $r$: $v^2=GM\left(\dfrac2r-\dfrac1a\right)$.

  • Orbit shape by total energy: $E<0$ ellipse ($e=0$ circle), $E=0$ parabola, $E>0$ hyperbola.

2Newton's law of gravitation

$$F=\frac{Gm_1m_2}{r^2},\qquad G=6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2}$$
$$\begin{array}{l}\displaystyle F=\frac{Gm_1m_2}{r^2}\\[6pt]\displaystyle G=6.67\times10^{-11}\ \text{N m}^2\text{kg}^{-2}\end{array}$$

Always attractive, central, conservative, independent of medium; action–reaction pair. Weakest force (gravity : weak : EM : strong ≈ $1:10^{25}:10^{36}:10^{38}$). Cavendish: $G=\dfrac{k\theta r^2}{Mml}$.

ArrangementResult
Two masses $m$ at corners of equilateral triangle (side $a$), $m$ at third$\sqrt3\dfrac{Gm^2}{a^2}$
Four $m$ at square corners, force on one$\dfrac{Gm^2}{a^2}\left(\sqrt2+\dfrac12\right)$
$m_1..m_4$ at square corners, $m$ at centre$\dfrac{2Gm}{a^2}\sqrt{(m_1-m_3)^2+(m_2-m_4)^2}$
Rod bent into semicircle (mass $M$, length $L$), $m$ at centre$\dfrac{2\pi GMm}{L^2}$; full circle: 0
Rod (length $L$), $m$ at distance $d$ from one end, on the axis$\dfrac{GMm}{d(d+L)}$
$m$ at 1, 2, 4, 8 … m from origin$\dfrac43Gm^2$
Mass $M$ split into $m$, $M-m$force maximum when $m=M/2$
Two masses $m$ circling each other (radius $r$)$v=\sqrt{\dfrac{Gm}{4r}}$; four at square corners: $v=\sqrt{\dfrac{GM(1+2\sqrt2)}{4R}}$
Binary stars $m_1$, $m_2$ at separation $x$$\omega=\sqrt{\dfrac{G(m_1+m_2)}{x^3}}$
Sphere with spherical cavity (radius $R/2$ touching surface), $m$ at $3R$$F_1:F_2=50:41$

3 more sections and 17 formulas in the full chapter

  1. 3Acceleration due to gravity4 formulas · 1 diagram
  2. 4Field and potential7 formulas · 1 case table
  3. 5Escape and orbital motion6 formulas · 2 case tables · 1 diagram

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