Gravitation is a force that attracts every body with mass or energy in the universe towards every other body. Gravitation was first theorized by Sir **Isaac Newton**. In this article, we will see some important gravitation formulas which will also help you in preparation for competitive exams like JEE and NEET.

Index

**Universal Law of Gravitation**

The universal law of gravitation is represented as

\(F\propto \frac{m_{1}m_{2}}{r^{2}}\;or\;\;F=G\frac{m_{1}m_{2}}{r^{2}}\)where,**G** = 6.67 × 10–11 Nm^{2}kg^{–2} is the **universal gravitational constant**.**m _{1}, m_{2}** are the mass of two objects

**r**is the distance between the two objects

Know about Sir Isaac Newton

**Newton’s Law of Gravitation in Vector Form**

The vector form of law of gravitation is,

\(\overrightarrow{F}_{12}=\frac{Gm_{1}m_{2}}{r^{2}}\hat{r}_{12}\;\;\&\;\;\overrightarrow{F}_{21}=\frac{Gm_{1}m_{2}}{r^{2}}\)Now \(\;\;\;\hat{r}_{12}=-\hat{r}_{21},\;\;\;Thus\;\vec{F}_{21}=\frac{-Gm{1}m_{2}}{r^{2}}\hat{r}_{12}\)

Comparing above, we get \(\;\;\hat{F}_{12}=-\hat{F}_{21}\)

**Gravitational Field** \(\;\;\;E=\frac{F}{m}=\frac{GM}{r^{2}}\)

**Gravitational Potential**

Gravitational potential,

\(V=-\frac{GM}{r}\;\;\;\;\;E=-\frac{dV}{dr}\)**Ring**

\(V=\frac{-GM}{x\;or(a^{2}+r^{2})^{\frac{1}{2}}}\;\;\&\;\;\;E=\frac{-GMr}{(a^{2}+r^{2})^{\frac{3}{2}}}\hat{r}\;\;\;or\;\;E=-\frac{GM\;\cos\theta}{x^{2}}\)The gravitational field is maximum at a distance,

\(r=\pm a\sqrt{2}\;\;\mathbf{and\;it\;is}\;-2GM/3\sqrt{3}a^{2}\)**Thin Circular Disc**

\(V=\frac{-2GM}{a^{2}}\left [ \left [ a^{2}+r^{2} \right ]^{\frac{1}{2}}-r \right ]\;\;\&\;\;\;E=-\frac{2GM}{a^{2}}\left [ 1-\frac{r}{\left [ r^{2}+a^{2} \right ]^{\frac{1}{2}}} \right ]=-\frac{2GM}{a^{2}}\left [ 1-\cos \theta \right ]\)**Non Conducting Solid Sphere**

**(a) Point P inside the sphere.** \(r\leq a,\;\mathbf{then}\) \(V=-\frac{GM}{2a^{3}}(3a^{2}-r^{2})\;\&\;\;E=-\frac{GMr}{a^{3}},\)

and at the center \(V=-\frac{3GM}{2a}\;\;\&\;E=0\)

**(b) Point P outside the sphere.**

**Uniform Thin Spherical Shell / Conducting Solid Sphere**

**(a) Point P inside the shell.**

**(b) Point P outside shell.**

**Variation of Acceleration Due To Gravity**

**Effect of Altitude**

\(g_{h}=\frac{GM_{e}}{(R_{e}+h)^{2}}=g\left ( 1+\frac{h}{R_{e}} \right )^{-2}\simeq g\left ( 1-\frac{2h}{R_{e}} \right )\mathbf{when}\;h<<R\)**Effects of Depth**

\(g_{d}=g\left ( 1-\frac{d}{R}_{e} \right )\)**Effect of the Surface of Earth**

The equatorial radius is about 21 km longer than its polar radius.

We know,\(\;\;g=\frac{GM_{e}}{R_{e}^{2}}\;\mathbf{Hence}\;\;g_{pole}>g_{equator}\)

**Satellite Velocity (or Orbital Velocity)**

\(V_{0}=\left [ \frac{GM_{e}}{(R_{e}+h)} \right ]^{\frac{1}{2}}=\left [ \frac{gR_{e}^{2}}{(R_{e}+h)} \right ]^{\frac{1}{2}}\)When h<<R_{e} than \(\sqrt{gR_{e}}\)

therefore, satellite velocity around earth will be \(v_{0}\sqrt{9.8\times6.4\times10^{6}}=7.92\times10^{3}\;ms^{-1}=7.92\;km\;s^{1}\)

**Time Period of Satellite**

\(T=\frac{2\pi(R_{e}+h)}{\left [ \frac{gR_{e}^{2}}{(R_{e}+h)} \right ]^{\frac{1}{2}}}=\frac{2\pi}{R_{e}}\left [ \frac{(R_{e}+h)^{3}}{g} \right ]^{\frac{1}{2}}\)## Escape Velocity

The velocity required to escape the gravitation force or pull of a mass is given by

V_{e} = \(\sqrt{\frac{2GM}{R}}\)

**Kepler’s Law**

Following are the three of Kepler’s laws of planetary motion

**First law: **It states that planets go around the sun in an elliptical orbit with the sun at one of the focii.

**Second Law**: It states that areal velocity or area swept by a planet in a constant time is constant (dA/dt =0)

**Third Law: **It gives the following relation between the orbital period and semimajor axis T^{2} ∝ R^{3}

So, those were some of the important gravitation formulas which would help you in your preparations for competitive examinations.