1. What Is a Gravitational Field?

Think of a gravitational field as an invisible "zone of influence" that surrounds anything with mass. Step into that zone with your own mass, and you'll feel a pull — even if nothing is touching you. The Earth's field is why you stay glued to the ground; the Sun's field is why the Earth doesn't go flying off into deep space.

The direction of a gravitational field at any point is shown using field lines (arrows). Because gravity only ever attracts — it never pushes things apart — every field line points towards the centre of the mass creating the field. If you dropped a test mass at any point in the field, the field line tells you which way it would accelerate.

Radial fields

Around a single point mass (or any spherical object, like a planet), the field is radial — the lines spread out symmetrically in all directions, like the spokes of a wheel, all converging on the centre.

2. Gravitational Field Strength (g)

Gravitational field strength answers a very specific question: "If I put a 1 kg mass at this exact point, how much force would it feel?" That's literally what the formula says — force per unit mass.

A crucial thing to internalise: an object's mass never changes, no matter where it is in the universe. What changes from planet to planet is g — and since weight = mg, your weight changes even though your mass doesn't. Stand on Earth (g = 9.81 N kg⁻¹) and you weigh a certain amount; stand on Jupiter (g ≈ 25 N kg⁻¹) and that same mass would feel roughly 2.5× heavier — potentially not even able to stand upright!

What affects g at a planet's surface?

Two things only: the planet's mass (or density) and its radius. A bigger or denser planet has stronger gravity; a planet with a larger radius (pushing you further from its centre) has weaker surface gravity.

3. Newton's Law of Universal Gravitation

This is the master equation of the whole chapter. It tells you the actual force between any two masses anywhere in the universe — from two atoms to two galaxies.

In words: The gravitational force between two masses is proportional to the product of their masses, and inversely proportional to the square of the distance between their centres.

The inverse-square law, unpacked

The "1/r²" part of the formula is called an inverse square law, and it has a dramatic effect: if you double the distance between two masses, the force doesn't halve — it drops to a quarter. Triple the distance, and the force falls to a ninth. Distance is brutally punishing to gravitational force.

4. Gravitational Field Due to a Point Mass

Here's a neat trick: you can actually derive the g = GM/r² formula just by combining the two equations you already know. Watch how it flows:

This cancelling of m is actually profound — it tells you that g at a given point doesn't depend on what you're measuring it with. A feather and a bowling ball dropped from the same height experience the exact same g (which is why, ignoring air resistance, they fall at the same rate — Galileo's famous result).

5. Gravitational Potential in a Radial Field

You've probably used G.P.E = mgΔh before — but that formula has a hidden catch: it only works near Earth's surface, where the field is approximately uniform (parallel field lines, constant g). Once you zoom out far enough that Earth's curvature matters, the field becomes radial, and you need a new tool: gravitational potential.

Gravitational potential (V) is defined as the work done per unit mass to bring a test mass from infinity (where the field's influence is zero) to a specific point in the field.

Why is potential always negative?

Here's the logic: gravity is always attractive, so it always pulls — it never needs help pulling a mass in. But to move a mass away from another mass (against that pull), you have to do work against gravity. Since "infinity" (zero potential, by definition) is the point of maximum separation, and everywhere else requires less energy than getting to infinity, every real point in a gravitational field sits at a potential below zero — hence negative.

6. Comparing Electric & Gravitational Fields

Gravity and electric fields are mathematical cousins — both follow inverse-square laws, both have radial fields around point sources, and both have near-identical-looking formulas. But they come from fundamentally different sources and behave differently in one key way: direction.

7. Orbital Motion

This is where the whole chapter comes together. A satellite (or planet) in a stable circular orbit isn't "floating" — it's actually constantly falling toward the object it orbits, but moving sideways fast enough that it keeps missing! The gravitational force is exactly what's needed to act as the centripetal force that keeps it curving around in a circle instead of flying off in a straight line.

Deriving orbital speed

Simply set the gravitational force equal to the centripetal force needed for circular motion, then solve:

Deriving the time period relationship

Since orbital speed can also be written as distance ÷ time (circumference ÷ period), i.e. v = 2πr/T, you can combine this with v² = GM/r to link time period directly to orbital radius:

What to Memorise

Concepts Checklist

• I can explain what a gravitational field is and why field lines point toward the mass
• I can define and calculate gravitational field strength g = F/m
• I understand the difference between g (little g) and G (big G)
• I can state and apply Newton's Law of Universal Gravitation, F = Gm₁m₂/r²
• I remember to always measure r between the centres of both masses
• I can derive g = GM/r² from F = Gm₁m₂/r² and g = F/m
• I understand why gravitational potential V is always negative
• I know V ∝ 1/r while g ∝ 1/r² — and I don't mix these up
• I can compare gravitational and electric fields (similarities & differences)
• I can derive v² = GM/r by equating gravitational and centripetal force
• I can derive and apply T² = 4π²r³/GM (Kepler's Third Law)
• I understand why a satellite in circular orbit is still "accelerating"
• I can solve binary star / two-orbiting-body problems involving R₁ + R₂

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