Physics · Mechanics and Waves

Gravitation formulas for JEE

Every Gravitation formula you need for JEE, grouped by concept.

FormulasRevision notes
29 formulas3 concepts
01Gravitation Laws02Acceleration due to Gravity03Gravitational Potential and Energy
01

Gravitation Laws

9 formulas

Angular Momentum Conservation in Orbit

mvprp=mvaram v_p r_p = m v_a r_a

Relationship between speeds and distances at aphelion and perihelion.

applies whenElliptical orbits around a central mass.
keplerangular momentum

Areal Velocity

ΔAΔt=L2m\frac{\Delta A}{\Delta t} = \frac{L}{2m}

Kepler's second law stating the rate at which area is swept by a planet's radius vector.

applies whenValid for any central force field.
keplerareal velocitymomentum

Cavendish Experiment Torque

τθ=GMmd2L\tau\theta = G\frac{Mm}{d^2}L

Balance of gravitational torque and wire restoring torque.

applies whenEquilibrium state in a torsional balance.
cavendishtorque

Gravitational Field of Disc

Eg=−2GMR2[1−xR2+x2]E_g = -\frac{2GM}{R^2} \left[ 1 - \frac{x}{\sqrt{R^2 + x^2}} \right]

Field on the axis of a uniform mass disc.

applies whenAlong the axis at distance x from the center.
fielddiscjee-advanced

Gravitational Field of Ring

Eg=−GMx(R2+x2)3/2E_g = -\frac{GMx}{(R^2 + x^2)^{3/2}}

Field on the axis of a uniform mass ring.

applies whenAlong the axis at distance x from the center.
fieldringjee-advanced

Universal Gravitation (Scalar)

F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}

Magnitude of the gravitational force between two point masses.

applies whenPoint masses or outside spherically symmetric uniform bodies.
newtongravitationforce

Universal Gravitation (Vector)

F12=−Gm1m2r3r12\mathbf{F}_{12} = -G \frac{m_1 m_2}{r^3} \mathbf{r}_{12}

Vector form of Newton's law of gravitation.

applies whenPoint masses; r12\mathbf{r}_{12} is the position vector from m1m_1 to m2m_2.
newtongravitationvector

Kepler's Third Law (Period)

T2=4π2GMa3T^2 = \frac{4\pi^2}{GM} a^3

Relation between the time period and the semi-major axis.

applies whenCentral mass M must be much larger than orbiting mass.
keplerperiodorbit

Neutral Point Distance

r1=d1+M2M1r_1 = \frac{d}{1 + \sqrt{\frac{M_2}{M_1}}}

Location between two masses where the net gravitational field is zero.

applies whenDistance r1r_1 from M1M_1, total separation dd.
neutral pointfieldjee-advanced
02

Acceleration due to Gravity

6 formulas

Acceleration due to Gravity

g=GMERE2g = \frac{GM_E}{R_E^2}

Gravity on the surface of a spherically symmetric body.

applies whenOn the surface of the planet (r=REr = R_E).
gravitysurface

Gravitational Force Inside Solid Sphere

F=GmMErRE3F = \frac{G m M_E r}{R_E^3}

Force on a mass m at distance r inside a uniform solid sphere.

applies whenInside a uniform solid sphere (r<REr < R_E).
gravityinside sphere

Gravity at Altitude (Approximation)

g(h)≈g(1−2hRE)g(h) \approx g \left(1 - \frac{2h}{R_E}\right)

Binomial approximation for g at small heights.

applies whenHeight is much smaller than the planet's radius (h≪REh \ll R_E).
gravityaltitudeapproximation

Gravity at Altitude (Exact)

g(h)=GME(RE+h)2g(h) = \frac{GM_E}{(R_E + h)^2}

Exact variation of g at a height h above the surface.

applies whenValid for any height h≥0h \ge 0.
gravityaltitude

Gravity at Depth

g(d)=g(1−dRE)g(d) = g \left(1 - \frac{d}{R_E}\right)

Variation of g at a depth d below the surface.

applies whenUniform density solid sphere.
gravitydepth

Variation of g with Latitude

g′=g−Rω2cos⁡2λg' = g - R\omega^2 \cos^2\lambda

Effective gravity due to the rotation of the Earth.

applies whenLatitude λ\lambda, angular velocity ω\omega.
gravityrotationlatitudejee-advanced
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03

Gravitational Potential and Energy

14 formulas

Total Orbital Energy

E=−GMm2rE = -\frac{GMm}{2r}

Total mechanical energy of a satellite in a circular orbit.

applies whenCircular orbit of radius r (r=RE+hr = R_E + h).
energyorbitsatellite

Escape Velocity

ve=2GMERE=2gREv_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E}

Minimum speed required to escape the gravitational pull from the surface.

applies whenLaunched from the surface, neglecting air resistance.
escape velocityenergy

Escape Velocity from Altitude

vi=2GMERE+hv_i = \sqrt{\frac{2GM_E}{R_E+h}}

Minimum escape speed from a height h above the surface.

applies whenLaunched from an altitude h.
escape velocityaltitude

Orbital Kinetic Energy

K=GMm2rK = \frac{GMm}{2r}

Kinetic energy of a satellite in a circular orbit.

applies whenCircular orbit of radius r.
kineticorbitsatellite

Orbital Velocity

V=GMERE+hV = \sqrt{\frac{GM_E}{R_E + h}}

Speed required to maintain a circular orbit at altitude h.

applies whenCircular orbit.
orbitalsatellitevelocity

Work to Shift Orbit

W=GMm2(1r1−1r2)W = \frac{GMm}{2} \left(\frac{1}{r_1} - \frac{1}{r_2}\right)

Energy required to transfer a satellite between two circular orbits.

applies whenTransfer from radius r1r_1 to radius r2r_2.
workorbit transferenergy

Gravitational Potential Energy

U=−Gm1m2rU = -\frac{G m_1 m_2}{r}

Potential energy of a two-particle system separated by distance r.

applies whenReference point U=0U=0 at r→∞r \to \infty.
energypotential

Satellite Period (Near Surface)

T0=2πREgT_0 = 2\pi\sqrt{\frac{R_E}{g}}

Time period of a satellite orbiting very close to the Earth's surface.

applies whenAltitude h≈0h \approx 0.
periodsatellitesurface

Gravitational Potential

V=−GMrV = -\frac{GM}{r}

Gravitational potential due to a point mass M at distance r.

applies whenReference V=0V=0 at r→∞r \to \infty.
potentialfield

Gravitational Potential of Ring

V(r)=−GMR2+x2V(r) = -\frac{GM}{\sqrt{R^2 + x^2}}

Potential on the axis of a uniform mass ring.

applies whenAlong the axis at distance x from the center.
potentialringjee-advanced

Potential Inside Spherical Shell

V(r)=−GMRV(r) = -\frac{GM}{R}

Gravitational potential everywhere inside a uniform hollow shell.

applies whenInside the shell (r<Rr < R).
potentialshelljee-advanced

Potential Inside Solid Sphere

V(r)=−GM2R3(3R2−r2)V(r) = -\frac{GM}{2R^3}(3R^2 - r^2)

Gravitational potential inside a uniform solid sphere.

applies whenInside the sphere (r≤Rr \le R).
potentialsolid spherejee-advanced

Self-Energy of Spherical Shell

Uself=−GM22RU_{self} = -\frac{GM^2}{2R}

Gravitational self-energy of assembling a uniform thin spherical shell.

applies whenThin uniform mass shell.
self energyshelljee-advanced

Self-Energy of Solid Sphere

Uself=−3GM25RU_{self} = -\frac{3GM^2}{5R}

Gravitational self-energy of assembling a uniform solid sphere.

applies whenUniform density solid sphere.
self energysolid spherejee-advanced
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