Unit 7 · Topic 01

Universal gravitation

Newtonian gravitation is a mutual, attractive, vector interaction of unlimited range.

Unit 7Universal gravitationOpen navigation

Concept 01

A mutual force

Essential The minimum you should retain

Two point masses attract with equal magnitudes |F|=Gm₁m₂/r² and opposite directions along their center line.

UnderstandInterpret and connect

Doubling r reduces the magnitude to F/4; accelerations may differ although the forces do not.

DeepenFormulation and conditions

The vector form on m due to M is F=−GMm/r² r_hat: the sign expresses attraction.

ExploreConnections for further study

The small value of G makes the interaction between many everyday objects imperceptible.

Worked example

Attraction between two masses

Masses of 1000 kg and 1500 kg are 2.0 m apart; G=6.674×10^-11 N·m²/kg².

Given
  • m₁=1000 kg
  • m₂=1500 kg
  • r=2.0 m
Target

Find |F|.

  1. Model

    F=Gm₁m₂/r2 F=Gm_1m_2/r^2 .

  2. Substitution

    F=(6.674×10^-11)(1000)(1500)/(2.0)².

  3. Result

    F=2.50×10^-5 N.

  4. Direction

    Each force points toward the other mass.

Conclusion

Each mass feels 2.50×10^-5 N in opposite directions.

Mathematical relation

Universal gravitation

|F|=Gm1m2r2,F→=−GMmr2r_hat |F|=Gm_1m_2/r^2,\quad \vec F_m=-GMm\hat r/r^2
Represents

Mutual attractive gravitational force.

Physical interpretation

Direction is toward the source and magnitude falls as 1/r².

DeepenVariables, conditions, and checks

Variables

G
gravitational constant; usual unit: N·m²/kg²
r
center-to-center distance; usual unit: m

Conditions of application

  • Point masses or spherical bodies in the appropriate exterior region; r>0.

Dimensional check

N.

Two masses show the gravitational pair and a normalized 1/r² curve.F₁₂F₂₁rm₁m₂

The pair forces have equal magnitude and opposite directions; doubling r reduces F to F₀/4.

Concept 02

Equal magnitude, different acceleration

Essential The minimum you should retain

Each body exerts a gravitational force of equal magnitude and opposite direction on the other.

UnderstandInterpret and connect

The larger mass usually accelerates less because a=F/m, not because it receives less force.

DeepenFormulation and conditions

The pair acts on different bodies and does not cancel when analyzing either body alone.

ExploreConnections for further study

Linear-momentum conservation of the system is consistent with this internal pair.

Concept 03

Adding contributions

Essential The minimum you should retain

The effect of several masses is the vector sum of their forces or fields.

UnderstandInterpret and connect

Contributions in different directions must not be added as magnitudes.

DeepenFormulation and conditions

Each vector uses its own distance and direction before being resolved into components.

ExploreConnections for further study

Zero-field points can arise by cancellation even though every contribution is nonzero.

Two source masses attract a test mass and produce an oblique resultant.g₁g₂g_netM₁M₂m

Each contribution is calculated with its own direction, and the resultant is a vector sum.

Concept 04

When the point-mass model works

Essential The minimum you should retain

Outside a spherically symmetric distribution, the field equals that of a central point mass.

UnderstandInterpret and connect

The equivalence depends on symmetry and on being in the exterior region.

DeepenFormulation and conditions

It cannot be extended to arbitrary distributions without integrating their density.

ExploreConnections for further study

Newton's law is highly accurate in its regime but is not the fundamental theory of spacetime.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Believing the larger body exerts a larger gravitational force on the smaller one.

Force magnitudes are equal; accelerations differ because masses differ.

Doubling r and halving F.

F∝1/r², so doubling r divides F by four.