1. Introduction

Newton’s Law of Universal Gravitation describes the gravitational force between any two objects with mass.

Proposed by Isaac Newton, this law explains:

• planetary motion
• satellite orbits
• falling objects
• tides
• the structure of the solar system

The law states that every mass attracts every other mass through a gravitational force.

2. Newton’s Law of Universal Gravitation

Statement of the Law

Any two objects with mass exert an attractive gravitational force on each other.

The magnitude of the force is:

FG=Gm1m2r2F_G = \frac{Gm_1m_2}{r^2}FG​=r2Gm1​m2​​

where:

• FGF_GFG​ = gravitational force
• $G$ = universal gravitational constant
• m1m_1m1​ = first mass
• m2m_2m2​ = second mass
• $r$ = distance between the centers of the masses

3. Universal Gravitational Constant

The constant $G$ has the value:

G=6.67×10−11 N⋅m2/kg2G = 6.67\times10^{-11} \ \text{N}\cdot\text{m}^2/\text{kg}^2G=6.67×10−11 N⋅m2/kg2

This extremely small number explains why gravitational forces between everyday objects are usually negligible.

4. Characteristics of Gravitational Force

Attractive Force

Gravity is always attractive.

Objects pull toward one another.

There is no known gravitational repulsion between ordinary masses.

Acts at a Distance

Objects do not need to touch.

Gravity acts through space.

Mutual Force

According to Newton’s Third Law:

F12=F21F_{12} = F_{21}F12​=F21​

The forces on the two masses are equal in magnitude and opposite in direction.

5. Inverse-Square Relationship

The gravitational force varies inversely with the square of the distance.

FG∝1r2F_G \propto \frac{1}{r^2}FG​∝r21​

Example

If the distance doubles:

$$r\to 2r$$

then:

FG=14FF_G = \frac{1}{4}FFG​=41​F

The force becomes one-fourth as large.

Example

If the distance triples:

$$r\to 3r$$

then:

FG=19FF_G = \frac{1}{9}FFG​=91​F

The force becomes one-ninth as large.

6. Dependence on Mass

The gravitational force is directly proportional to both masses.

FG∝m1m2F_G \propto m_1m_2FG​∝m1​m2​

Example

If one mass doubles:

m1→2m1m_1 \rightarrow 2m_1m1​→2m1​

then:

FG→2FGF_G \rightarrow 2F_GFG​→2FG​

The force doubles.

Example

If both masses double:

FG→4FGF_G \rightarrow 4F_GFG​→4FG​

The force quadruples.

7. Gravitational Force as a Vector

The force acts along the line connecting the centers of the two objects.

The direction is always toward the other mass.

Because gravity is attractive, both objects pull inward toward one another.

8. Gravitational Field

A mass creates a gravitational field around itself.

The gravitational field strength is:

g=GMr2g = \frac{GM}{r^2}g=r2GM​

where:

• $M$ = source mass
• $r$ = distance from the center of the mass

This field describes the gravitational force per unit mass.

9. Weight as a Gravitational Force

The weight of an object is the gravitational force exerted by Earth.

Near Earth’s surface:

FG=mgF_G = mgFG​=mg

where:

• $m$ = object’s mass
• $g$ = gravitational field strength

10. Deriving the Value of $g$

Applying Newton’s Law of Gravitation to Earth:

FG=GMEmRE2F_G = \frac{GM_E m}{R_E^2}FG​=RE2​GME​m​

Since:

FG=mgF_G=mgFG​=mg

we obtain:

g=GMERE2g = \frac{GM_E}{R_E^2}g=RE2​GME​​

where:

• MEM_EME​ = Earth’s mass
• RER_ERE​ = Earth’s radius

11. Variation of Gravity with Altitude

Because:

g=GMr2g = \frac{GM}{r^2}g=r2GM​

gravity decreases as distance from Earth’s center increases.

At higher altitudes:

$$r\uparrow$$

therefore:

$$g\downarrow$$

Astronauts in orbit still experience gravity; they are simply in continuous free fall.

12. Gravitational Force and Circular Orbits

For a satellite orbiting Earth:

Gravity provides the required centripetal force.

FG=FCF_G = F_CFG​=FC​ GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r}r2GMm​=rmv2​

The satellite remains in orbit because gravity continuously changes the direction of its velocity.

13. Orbital Speed

From:

GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r}r2GMm​=rmv2​

the satellite mass cancels.

Solving for speed:

v=GMrv = \sqrt{\frac{GM}{r}}v=rGM​​

This equation gives the orbital speed for a circular orbit.

14. Orbital Period

Using:

v=2πrTv = \frac{2\pi r}{T}v=T2πr​

and the orbital-speed equation:

T=2πr3GMT = 2\pi \sqrt{\frac{r^3}{GM}}T=2πGMr3​​

This result leads directly to Kepler’s Third Law.

15. Example Problem

Two masses:

m1=5 kgm_1=5\,kgm1​=5kg m2=10 kgm_2=10\,kgm2​=10kg

are separated by:

$$r=2m$$

Find the gravitational force between them.

Solution

Using:

FG=Gm1m2r2F_G = \frac{Gm_1m_2}{r^2}FG​=r2Gm1​m2​​

Substitute values:

FG=(6.67×10−11)(5)(10)(2)2F_G = \frac{(6.67\times10^{-11})(5)(10)} {(2)^2}FG​=(2)2(6.67×10−11)(5)(10)​ FG=8.34×10−10 NF_G = 8.34\times10^{-10}\,NFG​=8.34×10−10N

The force is extremely small.

16. Superposition Principle

When multiple masses are present, gravitational forces add vectorially.

The net gravitational force is:

F⃗net=∑F⃗\vec{F}_{\text{net}} = \sum \vec{F}Fnet​=∑F

Each force must be calculated separately before combining them as vectors.

17. Common AP Physics C Mistakes

Mistake 1

Using surface gravity equations when the object is far from Earth.

Use:

FG=GMmr2F_G = \frac{GMm}{r^2}FG​=r2GMm​

for general situations.

Mistake 2

Using the radius of Earth instead of the distance between centers.

Always measure $r$ from center to center.

Mistake 3

Assuming gravity disappears in orbit.

Gravity remains significant and provides the centripetal force for orbital motion.

18. Relationship to Kepler’s Laws

Newton’s Law of Gravitation explains all three of Kepler’s Laws.

It provides the physical reason why:

• planets move in ellipses
• orbital speeds vary
• orbital period depends on orbital radius

Summary

Newton’s Law of Universal Gravitation:

FG=Gm1m2r2F_G = \frac{Gm_1m_2}{r^2}FG​=r2Gm1​m2​​

Gravitational field strength:

g=GMr2g = \frac{GM}{r^2}g=r2GM​

Weight near Earth’s surface:

FG=mgF_G=mgFG​=mg

Orbital speed:

v=GMrv = \sqrt{\frac{GM}{r}}v=rGM​​

Orbital period:

T=2πr3GMT = 2\pi \sqrt{\frac{r^3}{GM}}T=2πGMr3​​

Key ideas:

• every mass attracts every other mass
• gravitational force follows an inverse-square law
• gravity provides the centripetal force for orbits
• Newton’s Law of Gravitation explains Kepler’s Laws and orbital motion

Newton’s Law of Gravitation is the foundation of celestial mechanics and connects forces, orbits, satellites, and planetary motion within a single universal framework.