Gravitational Field Strength Equation, Derivation & Rearrangements

Core idea

Overview

This equation defines gravitational field strength as the gravitational force exerted per unit mass on a small test object at a specific point. It demonstrates that the acceleration due to gravity depends solely on the mass of the source body and the square of the distance from its center, independent of the test object's mass.

When to use: Apply this formula when calculating the local acceleration of gravity on a planetary surface or at a specific altitude in space. It assumes the central body is a uniform sphere and requires the distance r to be measured from the center of mass, not the surface altitude.

Why it matters: This principle is fundamental for predicting orbital trajectories and ensuring the safety of satellite deployments. It also allows planetary scientists to compare physical conditions across different worlds, influencing how we design technology for lunar or Martian exploration.

Symbols

Variables

g = Field Strength, G = Grav Constant, M = Mass, r = Distance

g

Field Strength

N / k g

G

Grav Constant

V a r iab l e

M

Mass

k g

r

Distance

m

Walkthrough

Derivation

Understanding Gravitational Field Strength

The gravitational force per unit mass on a small test mass placed in the field.

The test mass is small enough not to alter the field.

1

Start with Newton's Law:

Force between mass M and test mass m at distance r.

F = \frac{GM m}{r ^{2}}

2

Use Definition g=F/m:

Divide by m to obtain gravitational field strength.

g = \frac{F}{m} = \frac{GM}{r ^{2}}

Result

g = \frac{F}{m} = \frac{GM}{r ^{2}}

Source: Edexcel A-Level Physics — Gravitational Fields

Free formulas

Rearrangements

Solve for $g$

Make g the subject

g = \frac{GM}{r ^{2}}

g is already the subject of the formula.

Difficulty: 1/5

Solve for $r$

Make r the subject

r = \frac{GM}{g}

Start from Gravitational Field Strength. To make r the subject, clear $r^{2}$ , then make $r^{2}$ the subject, then take the square root.

Difficulty: 4/5

Solve for $M$

Make M the subject

M = \frac{g r ^{2}}{G}

Start from Gravitational Field Strength. To make M the subject, clear $r^{2}$ , then divide by G.

Difficulty: 3/5

Solve for $G$

Make G the subject

G = \frac{g r ^{2}}{M}

Start from Gravitational Field Strength. To make G the subject, clear $r^{2}$ , then divide by M.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows an inverse-square relationship where g decreases as r increases. Because r is in the denominator and squared, the curve drops sharply near the y-axis and approaches zero as r grows. For a student of physics, this means that gravitational field strength is extremely intense at small distances near the center of mass but weakens rapidly as distance increases. The most important feature is that the curve never touches the axes, meaning that while the field strength becomes negligible at large distanc

Graph type: inverse

Why it behaves this way

Intuition

Visualize a massive central body as a point source, radiating an invisible 'pulling field' that diminishes in strength proportionally to the inverse square of the distance, like light spreading from a bulb.

g

Gravitational field strength (or acceleration due to gravity)

The acceleration an object experiences solely due to gravity at a specific location, representing the 'pull' per unit mass.

G

Universal Gravitational Constant

A fundamental constant that quantifies the intrinsic strength of the gravitational force between any two masses in the universe.

M

Mass of the central (source) body

The amount of matter in the object generating the gravitational field; more mass means a stronger field.

r

Distance from the center of the central body to the point where 'g' is calculated

How far away you are from the source of gravity; the further you are, the weaker its influence becomes.

Signs and relationships

r^2 in the denominator: This represents an inverse-square law, meaning the gravitational field strength diminishes rapidly (quadratically) with increasing distance.

Free study cues

Insight

Canonical usage

All quantities are typically expressed in SI units to ensure consistency and derive 'g' in meters per second squared.

Common confusion

A common mistake is confusing the gravitational field strength 'g' (acceleration due to gravity) with the universal gravitational constant 'G'.

Unit systems

$g$ m/s^2 · Represents the gravitational field strength or the acceleration experienced by an object due to gravity.

$G$ m^3 kg^-1 s^-2 · The universal gravitational constant, a fundamental physical constant.

$M$ kg · The mass of the central body (e.g., planet, star) creating the gravitational field.

$r$ m · The distance from the center of mass of the central body to the point where 'g' is being calculated.

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

Calculate the gravitational field strength on the surface of Mars, given its mass is 6.39 × 10²³ kg and its radius is 3.39 × 10⁶ m.

Grav Constant6.674e-11

Mass6.39e+23 kg

Distance3390000 m

Solve for: $g$

Hint: Plug the mass and radius into the formula g = GM/r² and ensure you square the radius.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating g at satellite altitude.

Study smarter

Tips

Always verify that r includes the radius of the planet plus any altitude above the surface.
Use the standard gravitational constant G as 6.674 × 10⁻¹¹ N·m²/kg².
Ensure the mass M is in kilograms and the distance r is in meters for the result to be in m/s².

Avoid these traps

Common Mistakes

Using r instead of r².
Mixing km and m.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The gravitational force per unit mass on a small test mass placed in the field.

Apply this formula when calculating the local acceleration of gravity on a planetary surface or at a specific altitude in space. It assumes the central body is a uniform sphere and requires the distance r to be measured from the center of mass, not the surface altitude.

This principle is fundamental for predicting orbital trajectories and ensuring the safety of satellite deployments. It also allows planetary scientists to compare physical conditions across different worlds, influencing how we design technology for lunar or Martian exploration.

Using r instead of r². Mixing km and m.