PhysicsGravitational FieldsA-Level

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Newton's Law of Gravitation

Force between two point masses.

Understand the formulaSee the free derivationOpen the full walkthrough

F = \frac{G m _{1} m _{2}}{r ^{2}}

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Core idea

Overview

Newton's Law of Gravitation describes the attractive force between any two objects with mass, establishing that the magnitude of this force is proportional to the masses and inversely proportional to the square of the distance between their centers. This principle governs the motion of celestial bodies and explains the force of weight experienced on a planet's surface.

When to use: Apply this formula when analyzing the gravitational interaction between two distinct bodies that can be treated as point masses or uniform spheres. It is the primary tool for determining orbital velocity, escape velocity, and surface gravity in classical physics scenarios where velocities are much lower than the speed of light.

Why it matters: This equation enabled scientists to calculate the masses of the Sun and planets and to understand the mechanics of the solar system. It remains essential for calculating the trajectories of satellites, probes, and human spacecraft in modern aerospace engineering.

Symbols

Variables

F = Force, G = Grav. Constant, m_1 = Mass 1, m_2 = Mass 2, r = Distance

F

Force

N

G

Grav. Constant

N m^{2} / k g^{2}

m_{1}

Mass 1

k g

m_{2}

Mass 2

k g

r

Distance

m

Walkthrough

Derivation

Formula: Newton's Law of Gravitation (Empirical)

Describes the attractive force between two point masses, stating it is directly proportional to the product of their masses and inversely proportional to the square of their separation.

The masses are point masses (or uniform spheres where mass acts from the centre).
Relativistic effects are negligible (weak gravitational fields and non-relativistic speeds).

State the Proportionality:

Newton observed that gravitational force depends on the masses of the objects and weakens with the square of the separation.

F \propto \frac{m _{1} m _{2}}{r ^{2}}

Introduce the Gravitational Constant:

G is the universal gravitational constant. This gives the magnitude of the attractive force between the two masses.

F = \frac{G m _{1} m _{2}}{r ^{2}}

Note: In vector form, the force points towards the other mass: $F = - \frac{G m _{1} m _{2}}{r ^{2}} \hat{r}$ .

Result

F = \frac{G m _{1} m _{2}}{r ^{2}}

Source: AQA A-Level Physics — Gravitational Fields

Free formulas

Rearrangements

Solve for $F$

Make F the subject

F = \frac{G m _{1} m _{2}}{r ^{2}}

F is already the subject of the formula.

Difficulty: 1/5

Solve for $G$

Newton's Law of Gravitation: Make G the subject

G = \frac{F r ^{2}}{m _{1} m _{2}}

To make G the subject of Newton's Law of Gravitation, first multiply both sides by $r^{2}$ to clear the denominator, then divide by $m_{1} m_{2}$ .

Difficulty: 2/5

Solve for $m_{1}$

Make m1 the subject

m_{1} = \frac{F r ^{2}}{G m _{2}}

Rearrange Newton's Law of Gravitation to solve for Mass 1 ( $m_{1}$ ).

Difficulty: 2/5

Solve for $m_{2}$

Make m2 the subject

m_{2} = \frac{F r ^{2}}{G m _{1}}

Start from Newton's Law of Gravitation. To make m2 the subject, clear $r^{2}$ , then divide by Gm1.

Difficulty: 2/5

Solve for $r$

Newton's Law of Gravitation: Make r the subject

r = \frac{G m _{1} m _{2}}{F}

Rearrange Newton's Law of Gravitation to solve for $r$ , the distance between the centers of two objects. This involves isolating $r^{2}$ and then taking the square root.

Difficulty: 2/5

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

Visual intuition

Graph

The graph is an inverse-square curve that approaches the axes as asymptotes, with the domain restricted to r greater than zero. For a physics student, this shape shows that the force is extremely strong when the distance between masses is small, but it weakens rapidly as the distance increases. The most important feature is that the curve never reaches zero, meaning that gravitational force exists between two masses regardless of how far apart they are.

Graph type: inverse

Why it behaves this way

Intuition

Imagine each mass creating an invisible 'gravitational field' around itself, like a web of attraction that pulls other masses towards its center, with the strength of the pull diminishing rapidly as you move further

The magnitude of the attractive gravitational force between two objects.

This is the strength of the pull that one object exerts on another due to their masses. A larger F means a stronger pull.

The universal gravitational constant, a fundamental constant of nature.

This constant determines the overall strength of gravity throughout the universe. It's a very small number, indicating that gravity is a relatively weak force unless masses are enormous.

m_{1}

The mass of the first interacting object.

This represents the 'amount of matter' in the first object, which is a source of its gravitational influence. More massive objects exert and experience stronger gravitational pulls.

m_{2}

The mass of the second interacting object.

This represents the 'amount of matter' in the second object, which is a source of its gravitational influence. More massive objects exert and experience stronger gravitational pulls.

The distance between the centers of mass of the two objects.

This is how far apart the objects are. The force of gravity diminishes rapidly as this distance increases.

Signs and relationships

r^2 in the denominator: The inverse square dependence means that the gravitational force weakens rapidly with increasing distance. This arises because the gravitational influence spreads out over the surface area of a sphere centered on the

Free study cues

Insight

Canonical usage

This equation is typically used in the International System of Units (SI) to calculate the gravitational force between two masses.

Common confusion

A common mistake is using units other than meters for distance (e.g., kilometers or centimeters) or grams for mass without proper conversion to SI base units before calculation, leading to incorrect force values.

Unit systems

$F$ N · Gravitational force is expressed in Newtons (N).

$G$ N m^2 kg^-2 · The gravitational constant, also expressible as m^3 kg^-1 s^-2.

$m_{1}$ kg · Mass of the first object, in kilograms.

$m_{2}$ kg · Mass of the second object, in kilograms.

$r$ m · Distance between the centers of the two masses, in meters.

One free problem

Practice Problem

Calculate the gravitational force of attraction between the Earth and the Moon. Use the following values: Earth's mass is 5.972 × 10²⁴ kg, the Moon's mass is 7.348 × 10²² kg, and the average distance between their centers is 3.844 × 10⁸ meters.

Mass 15.972e+24 kg

Mass 27.348e+22 kg

Distance384400000 m

Grav. Constant6.674e-11 N m^2/kg^2

Solve for: $F$

Hint: Multiply the masses and the gravitational constant first, then divide by the square of the distance.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Estimating gravitational force between two satellites.

Study smarter

Tips

Always measure the distance 'r' from the center of mass of the objects, not from their surfaces.
Ensure all masses are in kilograms and distances are in meters to maintain consistency with the gravitational constant G.
Remember that gravity is an inverse-square law, so doubling the distance reduces the force to one-fourth.

Avoid these traps

Common Mistakes

Forgetting r is squared.
Using km without converting to m.

Common questions

Frequently Asked Questions

Describes the attractive force between two point masses, stating it is directly proportional to the product of their masses and inversely proportional to the square of their separation.

Apply this formula when analyzing the gravitational interaction between two distinct bodies that can be treated as point masses or uniform spheres. It is the primary tool for determining orbital velocity, escape velocity, and surface gravity in classical physics scenarios where velocities are much lower than the speed of light.

This equation enabled scientists to calculate the masses of the Sun and planets and to understand the mechanics of the solar system. It remains essential for calculating the trajectories of satellites, probes, and human spacecraft in modern aerospace engineering.

Forgetting r is squared. Using km without converting to m.

Estimating gravitational force between two satellites.

Always measure the distance 'r' from the center of mass of the objects, not from their surfaces. Ensure all masses are in kilograms and distances are in meters to maintain consistency with the gravitational constant G. Remember that gravity is an inverse-square law, so doubling the distance reduces the force to one-fourth.

References

Sources

Fundamentals of Physics by Halliday, Resnick, and Walker
Wikipedia: Newton's law of universal gravitation
Britannica: Newton's law of universal gravitation
NIST CODATA (2018 CODATA Recommended Values of the Fundamental Physical Constants)
Halliday, Resnick, Walker. Fundamentals of Physics. (Any recent edition, e.g., 10th or 11th edition)
Halliday, Resnick, Walker - Fundamentals of Physics
Wikipedia: General relativity
Wikipedia: Quantum gravity

Newton's Law of Gravitation

Overview

Variables

Derivation

State the Proportionality:

Introduce the Gravitational Constant:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Electric Field Strength (Point charge)

Frequently Asked Questions

Sources