Gravitation Formula Table

Master the cosmos: Newton's Law, Kepler's orbital mechanics, and escape velocity.

Newton

Universal Gravitation Law

F = G m₁m₂ / r²
UnitNewton (N)
Attractive force between any two point masses.
Ex: Force between Earth and Moon.
Newton

Vector Form

F_12 = -G m₁m₂/r² · r^
UnitNewton (N)
Force is always attractive (negative sign).
Ex: Directional force analysis.
Newton

Acceleration due to Gravity

g = GM / R²
Unitm/s²
Gravity on the surface of a planet mass M, radius R.
Ex: Earth g ≈ 9.8 m/s².
Newton

Gravity at Height h

g_h = g (R / (R+h))²
Unitm/s²
Gravity decreases as you go higher above surface.
Ex: Gravity at ISS altitude.
Newton

Gravity at Depth d

g_d = g (1 - d/R)
Unitm/s²
Gravity decreases linearly as you go deep underground.
Ex: Gravity in a deep mine shaft.
Kepler

Law of Periods (3rd Law)

T² = (4π²/GM) r³
Units²
Square of period is proportional to cube of radius.
Ex: Comparing Earth and Mars years.
Kepler

Areal Velocity (2nd Law)

dA/dt = L / 2m = Constant
Unitm²/s
Planets sweep equal areas in equal times.
Ex: Earth moves faster when closer to Sun.
Orbits

Orbital Velocity

v_o = √(GM / r)
Unitm/s
Speed required to maintain a circular orbit.
Ex: Low Earth Orbit satellite speed.
Orbits

Escape Velocity

v_e = √(2GM / R)
Unitm/s
Speed to break free from a planet's gravity forever.
Ex: Rocket leaving Earth (~11.2 km/s).
Orbits

Time Period

T = 2π √(r³ / GM)
UnitSecond (s)
Time to complete one full revolution.
Ex: Geostationary satellite (24 hours).
Energy

Gravitational Potential Energy

U = -G Mm / r
UnitJoule (J)
Energy due to position. Zero at infinity.
Ex: Energy binding Moon to Earth.
Energy

Gravitational Potential

V = -GM / r
UnitJ/kg
Potential energy per unit mass at a point.
Ex: Field potential map.
Energy

Total Satellite Energy

E = -GMm / 2r
UnitJoule (J)
Sum of Kinetic and Potential energy for circular orbit.
Ex: Bound system energy is negative.

Universal Constant (G)

The essential constant for all gravitational calculations.

Value of G6.674 × 10⁻¹¹
N · m² / kg²

The Clockwork of the Cosmos

Gravitation is the architect of the universe. It clumps dust into stars, binds galaxies together, and keeps our feet on the ground. Historically, it was the first force to be understood mathematically, bridging the gap between apples falling on Earth and moons orbiting Jupiter.

Escape Velocity

"What comes up must come down"—unless you go fast enough. Escape velocity is the speed needed to break free from a planet's gravity forever. For Earth, it's 11.2 km/s. For a Black Hole, it's faster than light!

Kepler's Revolution

Before Newton, Johannes Kepler discovered that planets don't move in perfect circles, but in **ellipses**. His laws are vital for planning spacecraft trajectories to Mars and beyond.

Mass vs Weight: The Big G vs Little g

This is the #1 mistake students make.

Fundamental G ($6.67 \times 10^-11$)

"The Universal Constant"
This number never changes. It represents the inherent strength of gravity in our universe. It is incredibly small, which is why you don't stick to the person sitting next to you.

Local g ($9.8\ m/s^2$)

"Acceleration due to Gravity"
This depends on where you are! It is 9.8 on Earth, 1.6 on the Moon, and 24.8 on Jupiter. It is calculated using Big G: $g = GM/R^2$.

Satellite Physics

Satellites are effectively usually falling forever. To stay in orbit, they must travel sideways fast enough that as they fall, the Earth curves away beneath them.

v_o = √(GM / r)

"Orbital velocity depends only on Mass of the planet (M) and distance (r), not the satellite's mass."

Frequently Asked Questions

What is the value of G (Universal Gravitational Constant)?

The value of G is approximately $6.674 \times 10^{-11} , N \cdot m^2/kg^2$. It is a universal constant, meaning it is the same everywhere in the universe, unlike little 'g' which changes depending on the planet.

What are Kepler's Three Laws?
  1. Law of Orbits: Planets orbit in ellipses with the Sun at one focus.
  2. Law of Areas: Planets sweep equal areas in equal times (moving faster when closer).
  3. Law of Periods: The square of the time period is proportional to the cube of the radius ($T^2 \propto r^3$).
Why is gravitational potential energy negative?

Because we define the potential energy to be zero at infinity (very far away). Since gravity is an attractive force, you have to do work to pull objects apart. This " debt " of energy creates a negative value for bound systems.

What is Escape Velocity for Earth?

It is approximately 11.2 km/s (about 25,000 mph). This is the speed an object needs to break free from Earth's gravitational pull without any further propulsion.

What is a Geostationary Satellite?

A satellite with an orbital period of exactly 24 hours. This matches Earth's rotation, so it appears to stay fixed above a single point on the equator. They are used for communications and weather.

Does weight change on different planets?

Yes! Mass (kg) stays the same, but Weight (N) changes because gravity ($g$) is different. On the Moon, gravity is 1/6th of Earth's, so you would weigh 6 times less.

What happens to gravity inside the Earth?

Gravity actually DECREASES as you go deeper underground. At the exact center of the Earth, the gravitational force is zero because the mass pulling you in all directions cancels out.

Why don't astronauts simply float away from the ISS?

Because gravity is still very strong there (about 90% of surface gravity)! They "float" because they are in freefall—constantly falling towards Earth but missing it because of their horizontal speed.

Is gravity a force?

In Newton's view, yes. In Einstein's General Relativity, gravity is not a force but the curvature of spacetime caused by mass. But for most engineering calculations, treating it as a force works perfectly.

How does altitude affect gravity?

Gravity follows an inverse square law. If you double your distance from the Earth'center ($2R$), gravity becomes 4 times weaker ($1/2^2$). It drops off rapidly as you go higher.