🎯⭐ INTERACTIVE LESSON

Newton's Law of Universal Gravitation

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Newton's Law of Universal Gravitation - Complete Interactive Lesson

Part 1: Newton\'s Law of Gravitation

🌍 Newton's Law of Universal Gravitation

Part 1 of 7 — Universal Gravitation

Newton's great insight was that the same force that makes an apple fall from a tree also keeps the Moon in orbit around Earth. Every object with mass attracts every other object with mass — this is universal gravitation.

In this lesson you will learn:

Newton's Law of Universal Gravitation: $F = Gm_1m_2/r^2$
The gravitational constant $G$
How gravitational force depends on mass and distance
Calculating gravitational force between objects

The Law of Universal Gravitation

Every two objects with mass attract each other with a force:

$F = \frac{Gm_1 m_2}{r^2}$

Gravitational Force Concepts 🎯

Gravitational Force Calculations 🧮

Use $G = 6.67 \times 10^{-11}$ N·m²/kg².

Find the gravitational force between Earth ( $5.97 \times 10^{24}$ kg) and the Moon ( kg) separated by m. Express in scientific notation as N. What is (round to 3 significant figures)?

Proportional Reasoning 🔍

Exit Quiz — Universal Gravitation ✅

Part 2: Gravitational Field Strength

🌐 Gravitational Field: $g = GM/r^2$

Part 2 of 7 — Universal Gravitation

The gravitational field describes how massive objects modify the space around them, creating a region where other masses experience a gravitational force. This is the concept behind $g$ — and it's not always $9.8$ m/s²!

In this lesson you will learn:

Part 3: Orbits & Satellite Motion

🛰️ Orbital Motion: Gravity as Centripetal Force

Part 3 of 7 — Universal Gravitation

The most beautiful connection in physics: the gravitational force provides the centripetal force for orbital motion. This insight unifies terrestrial and celestial physics.

In this lesson you will learn:

How gravity acts as centripetal force for orbiting objects
Derive orbital velocity: $v = \sqrt{GM/r}$

Part 4: Kepler\'s Laws

🪐 Kepler's Third Law: $T^2 \propto r^3$

Part 4 of 7 — Universal Gravitation

Kepler discovered three laws of planetary motion decades before Newton explained them. The Third Law — the relationship between orbital period and radius — is one of the most powerful tools in astrophysics.

In this lesson you will learn:

Kepler's Third Law: $T^2 \propto r^3$

Part 5: Gravitational PE (Universal)

🚀 Satellite Problems & Orbital Velocity

Part 5 of 7 — Universal Gravitation

Satellites are a cornerstone of AP Physics 1 gravitation problems. In this lesson, we'll tackle practical satellite problems, escape velocity, and the physics of different orbit types.

In this lesson you will learn:

Low Earth orbit (LEO) vs. geosynchronous orbit (GEO)
Escape velocity: $v_{esc} = \sqrt{2GM/r}$

Part 6: Problem-Solving Workshop

🔧 Problem-Solving Workshop

Part 6 of 7 — Universal Gravitation

Time to tackle complex gravitation problems that combine multiple concepts: Newton's Law, gravitational field, orbital motion, Kepler's Law, and energy. These multi-step problems mirror what you'll see on the AP exam.

In this lesson you will:

Solve complex multi-step gravitation problems
Combine gravitation with circular motion concepts
Practice AP-level free response problems
Master the "which formula to use" decision process

Formula Decision Guide

I Want to Find...	I Should Use...
Gravitational force between two objects	$F = GMm/r^2$

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Part 7 of 7 — Universal Gravitation

This final lesson brings together everything from universal gravitation for AP exam preparation. We'll cover the most tested question types, common mistakes, and strategies for both multiple choice and free response.

In this lesson you will:

Tackle AP-style multiple choice questions
Practice FRQ-style derivations
Review the complete gravitation toolkit
Master proportional reasoning — the #1 AP skill for gravitation

Complete Gravitation Toolkit

Essential Formulas

Formula	Name
$F = GMm/r^2$

Symbol	Meaning	Value/Units
$F$	Gravitational force	Newtons (N)
$G$	Universal gravitational constant	$6.67 \times 10^{-11}$ N·m²/kg²
$m_1, m_2$	Masses of the two objects	kg
$r$	Distance between centers of the objects	m

Inverse-square law: $F \propto 1/r^2$
- Double the distance → force is $1/4$ as strong
- Triple the distance → force is $1/9$ as strong
Proportional to both masses: $F \propto m_1 m_2$
- Double one mass → force doubles
- Double both masses → force quadruples
Always attractive: gravity only pulls, never pushes
Newton's 3rd Law applies: the force on $m_1$ due to $m_2$ equals the force on $m_2$ due to (same magnitude, opposite direction)

Why Don't We Feel Gravity Between Everyday Objects?

$G$ is incredibly small: $6.67 \times 10^{-11}$ . Two 100 kg people standing 1 m apart attract each other with only $F = 6.67 \times 10^{-7}$ N — about the weight of a grain of sand!

7.35 \times 10^{22}

Two 70 kg people stand 2 m apart. What is the gravitational force between them (in N, express in scientific notation, give the coefficient to 3 significant figures, e.g., "8.17" for $8.17 \times 10^{-8}$ N)?

At Earth's surface, $r = 6.37 \times 10^6$ m. Calculate $g = GM/r^2$ where $M = 5.97 \times 10^{24}$ kg. What value do you get (in m/s², round to 3 significant figures)?

The gravitational field concept: $g = GM/r^2$
How $g$ varies with altitude and location
Surface gravity on different planets
The relationship between weight and gravitational field

The Gravitational Field

Definition

The gravitational field $\vec{g}$ at a point in space tells you the gravitational force per unit mass that would be experienced by a test mass placed there:

$\vec{g} = \frac{\vec{F}}{m_{test}} \quad \Rightarrow \quad g = \frac{GM}{r^2}$

Direction: toward the center of the mass $M$ creating the field
Units: N/kg (equivalent to m/s²)
$g$ is a property of space — it exists whether or not a test mass is there

Weight and Gravitational Field

$W = mg$

where $g$ is the local gravitational field strength. On Earth's surface: $g \approx 9.8$ m/s².

How $g$ Varies with Distance

$g(r) = \frac{GM}{r^2}$

At Earth's surface ( $r = R_E$ ): $g_0 = GM_E/R_E^2 \approx 9.8$ m/s²

At altitude $h$ above the surface ( $r = R_E + h$ ):

$g(h) = \frac{GM_E}{(R_E + h)^2} = g_0 \left(\frac{R_E}{R_E + h}\right)^2$

Surface Gravity on Other Bodies

Body	$g$ (m/s²)	Compared to Earth
Moon	1.6	0.16 $g$
Mars	3.7	0.38 $g$
Jupiter	24.8	2.53 $g$
Sun

Gravitational Field Concepts 🎯

Gravitational Field Calculations 🧮

Use $G = 6.67 \times 10^{-11}$ N·m²/kg², $g_E = 10$ m/s², $R_E = 6.4 \times 10^6$ m.

What is the gravitational field strength at twice Earth's radius from Earth's center (in m/s²)?
Mars has mass $6.42 \times 10^{23}$ kg and radius $3.39 \times 10^6$ m. What is its surface gravity (in m/s², round to 3 significant figures)?
At what distance from Earth's center (in units of , round to 3 significant figures) is m/s²?

Field Concepts 🔍

Exit Quiz — Gravitational Field ✅

Why orbital speed is independent of satellite mass

The relationship between orbital radius and speed

Gravity = Centripetal Force

For a satellite of mass $m$ orbiting a planet of mass $M$ at radius $r$ :

$F_{gravity} = F_{centripetal}$

$\frac{GMm}{r^2} = \frac{mv^2}{r}$

Solving for Orbital Velocity

Cancel $m$ (satellite mass doesn't matter!):

$\frac{GM}{r} = v^2$

$v = \sqrt{\frac{GM}{r}}$

Key Insights

Mass of satellite doesn't matter: A feather and a bowling ball orbit at the same speed at the same radius!
Higher orbit = slower speed: $v \propto 1/\sqrt{r}$
- Doubling orbital radius → speed decreases by factor $\sqrt{2}$

Orbital Period

$T = \frac{2\pi r}{v} = \frac{2\pi r}{\sqrt{GM/r}} = 2\pi\sqrt{\frac{r^3}{GM}}$

Energy in Orbit

$KE = \frac{1}{2}mv^2 = \frac{GMm}{2r}$

$PE = -\frac{GMm}{r}$

$E_{total} = KE + PE = -\frac{GMm}{2r}$

Total energy is negative (bound orbit) and equals half the PE.

Orbital Motion Concepts 🎯

Orbital Calculations 🧮

Use $G = 6.67 \times 10^{-11}$ N·m²/kg², $M_E = 5.97 \times 10^{24}$ kg, $R_E = 6.37 \times 10^6$ m.

What is the orbital speed of the ISS at altitude 400 km above Earth's surface (in m/s, round to nearest 100)? Hint: $r = R_E + h$ .
What is the orbital period of the ISS (in minutes, round to nearest whole number)?
What orbital radius gives a period of exactly 24 hours (geosynchronous orbit)? Express as a multiple of $R_E$ (round to 3 significant figures).

Orbital Relationships 🔍

Exit Quiz — Orbital Motion ✅

How to derive it from Newton's Law of Gravitation

Using Kepler's Law to compare orbits

Calculating unknown orbital properties

Deriving Kepler's Third Law

Starting from gravity = centripetal force:

$\frac{GMm}{r^2} = \frac{mv^2}{r} = \frac{m(2\pi r/T)^2}{r} = \frac{4\pi^2 mr}{T^2}$

Cancel $m$ and rearrange:

$T^2 = \frac{4\pi^2}{GM} r^3$

The Key Relationship

$T^2 = \left(\frac{4\pi^2}{GM}\right) r^3$

This is a linear relationship between $T^2$ and $r^3$ .

Comparing Two Orbits (Same Central Body)

$\frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3}$

This ratio form is incredibly useful — you don't even need to know $G$ or $M$ !

Kepler's Three Laws Summary

Law	Statement
1st (Ellipses)	Planets orbit in ellipses with the Sun at one focus
2nd (Equal Areas)	A line from the Sun sweeps equal areas in equal times
3rd (Harmonics)	$T^2 \propto r^3$ for all objects orbiting the same body

Important: Same Central Body Only!

$T^2/r^3 = 4\pi^2/(GM)$ depends on $M$ — the mass of the body being orbited. You can only compare orbits around the central body.

Kepler's Third Law Concepts 🎯

Kepler's Third Law Calculations 🧮

Earth orbits the Sun at 1 AU with $T = 1$ year. What is the orbital period of an asteroid at 2.5 AU (in years, round to 3 significant figures)?
A moon orbits a planet with period 10 days at radius $r$ . Another moon orbits the same planet at radius $2r$ . What is its period (in days, round to 3 significant figures)?
Jupiter's moon Io orbits at $r_1 = 4.22 \times 10^8$ m with $T_1 = 1.77$ days. Europa orbits at $r_2 = 6.71 \times 10^8$ m. What is Europa's period (in days, round to 3 significant figures)?

Kepler's Law Applications 🔍

Exit Quiz — Kepler's Third Law ✅

Energy required to launch satellites

Transfer orbits and orbit changes

Common Orbit Types

Orbit	Altitude	Period	Speed	Use
LEO (ISS)	~400 km	~92 min	~7.7 km/s	Space station, imaging
MEO (GPS)	~20,200 km	~12 h	~3.9 km/s	Navigation
GEO	~35,800 km	24 h	~3.1 km/s	Communications, weather
Lunar	~384,400 km	~27.3 days	~1.0 km/s	Moon's orbit

Key Pattern

Higher orbit → slower speed, longer period

Escape Velocity

The minimum speed needed to escape a gravitational field (reach infinite distance with zero final speed):

$\frac{1}{2}mv_{esc}^2 = \frac{GMm}{r}$

$v_{esc} = \sqrt{\frac{2GM}{r}} = \sqrt{2} \cdot v_{orbit}$

Escape Velocity from Earth's Surface

$v_{esc} = \sqrt{2gR_E} = \sqrt{2 \times 10 \times 6.37 \times 10^6} \approx 11{,}200 \text{ m/s} \approx 11.2 \text{ km/s}$

Key Insight

Escape velocity = $\sqrt{2}$ × orbital velocity at the same radius. A satellite in orbit needs to increase its speed by only about 41% to escape!

Satellite Concepts 🎯

Satellite Calculations 🧮

Use $g = 10$ m/s², $R_E = 6.4 \times 10^6$ m, $M_E = 6.0 \times 10^{24}$ kg, $G = 6.67 \times 10^{-11}$ .

What is the orbital velocity for a satellite at altitude $h = 200$ km above Earth (in m/s, round to nearest 100)?
What is the escape velocity from Earth's surface (in km/s, round to 3 significant figures)?
The Moon orbits at $r = 3.84 \times 10^8$ m. What is the Moon's orbital speed (in m/s, round to nearest 10)?

Energy in Satellite Problems

Total Energy of a Satellite

$E = -\frac{GMm}{2r}$

Negative: the satellite is gravitationally bound
Less negative at higher orbits: higher orbit = more total energy
To move a satellite to a higher orbit: must ADD energy (fire rockets forward, then forward again)

Energy to Launch a Satellite

From Earth's surface to orbit at radius $r$ :

$\Delta E = E_{orbit} - E_{surface} = -\frac{GMm}{2r} - \left(-\frac{GMm}{R_E}\right)$

$\Delta E = GMm\left(\frac{1}{R_E} - \frac{1}{2r}\right)$

Binding Energy

The energy needed to completely remove a satellite from orbit to infinity:

$E_{bind} = |E| = \frac{GMm}{2r} = \frac{1}{2}mv^2$

The binding energy equals the kinetic energy! You need to add exactly one more KE worth of energy to escape.

Satellite Energy Concepts 🔍

Exit Quiz — Satellite Problems ✅

Problem-Solving Steps

Identify: What are you given? What do you need to find?
Choose: Which formula connects your known and unknown quantities?
Check units: Make sure everything is in SI units (m, kg, s)
Solve: Algebra first, numbers last
Verify: Does the answer make physical sense?

Warm-Up Problems 🧮

Use $G = 6.67 \times 10^{-11}$ , $M_E = 6.0 \times 10^{24}$ kg, $R_E = 6.4 \times 10^6$ m.

A satellite orbits Earth at twice Earth's radius from Earth's center. What is the gravitational field strength at that location (in m/s², round to 3 significant figures)?
What is the orbital speed at that altitude (in m/s, round to nearest 100)?
What is the orbital period (in hours, round to 3 significant figures)?

Multi-Step Problems 🎯

Challenge Problems 🧮

Use $G = 6.67 \times 10^{-11}$ .

A planet has surface gravity $g = 20$ m/s² and radius $R = 8 \times 10^6$ m. What is the planet's mass (in kg, express as $X \times 10^{25}$ — give $X$ to 3 significant figures)?
Using the planet from problem 1, what is the escape velocity from its surface (in km/s, round to 3 significant figures)?
A satellite orbits this planet at altitude $h = R$ (i.e., at $r = 2R$ from center). What is its orbital period (in hours, round to 3 significant figures)?

Problem-Solving Strategy 🔍

Exit Quiz — Problem-Solving Workshop ✅

The Power of Proportional Reasoning

On the AP exam, most gravitation questions test proportional reasoning — not plug-and-chug. Know these:

If you change...	Force changes by...	$g$ changes by...	$v_{orbit}$ changes by...	$T$ changes by...
$r \rightarrow 2r$	$1/4$	$1/4$	$1/\sqrt{2}$	$2\sqrt{2}$
$M \rightarrow 2M$	$2$	$2$	$\sqrt{2}$
$r \rightarrow 3r$	$1/9$	$1/9$	$1/\sqrt{3}$

Common AP Mistakes

Mistake 1: Using altitude instead of orbital radius

$r$ is measured from the center of the planet
$r = R_{planet} + h_{altitude}$
ALWAYS add the planet's radius!

Mistake 2: Confusing $g$ and $G$

$G = 6.67 \times 10^{-11}$ — universal constant, same everywhere
$g$ — local gravitational field strength, varies with location

Mistake 3: Thinking astronauts are "outside gravity"

The ISS at 400 km altitude: $g \approx 8.7$ m/s² (only 11% less than surface!)
Astronauts float because they're in free fall, not because gravity is absent

Mistake 4: Wrong proportional reasoning with $v$ and $T$

$v \propto 1/\sqrt{r}$ (NOT $1/r$ )
$T$ (NOT or )

Mistake 5: Drawing centripetal force separately from gravity

For orbiting objects, gravity IS the centripetal force
Don't draw both on a free body diagram

AP-Style Multiple Choice 🎯

FRQ Practice

Sample FRQ: "Weighing Jupiter"

Astronomers observe that Jupiter's moon Io orbits at radius $r$ with period $T$ .

(a) Derive an expression for Jupiter's mass in terms of $r$ , $T$ , and fundamental constants.

Gravity provides centripetal force: $\frac{GM_J m}{r^2} = \frac{4\pi^2 m r}{T^2}$

Cancel $m$ : $\frac{GM_J}{r^2} = \frac{4\pi^2 r}{T^2}$

$M_J = \frac{4\pi^2 r^3}{GT^2}$

(b) If Europa orbits at $r_E = 1.59r_{Io}$ , find the ratio $T_E/T_{Io}$ .

$\frac{T_E^2}{T_{Io}^2} = \frac{r_E^3}{r_{Io}^3} = (1.59)^3 = 4.02$

$T_E/T_{Io} = \sqrt{4.02} = 2.00$

(c) Does the mass of the moon matter? Justify.

No. The moon's mass cancels in the derivation (Step 2 above). Orbital properties depend only on the central body's mass and the orbital radius, not the orbiting body's mass. This is analogous to all objects falling with the same acceleration $g$ near Earth's surface.

AP-Style Calculations 🧮

Saturn's moon Titan orbits at $r = 1.22 \times 10^9$ m with $T = 15.95$ days. Calculate Saturn's mass (in kg, express as $X \times 10^{26}$ — give $X$ to 3 significant figures). Use $G = 6.67 \times 10^{-11}$ .
A planet has mass $5M_E$ and radius $2R_E$ . What is the escape velocity from this planet's surface as a multiple of Earth's escape velocity? (round to 3 significant figures)
Two satellites orbit Earth. Satellite A has period 6 hours and Satellite B has period 24 hours. What is the ratio $r_B/r_A$ (round to 3 significant figures)?

Conceptual Review 🔍

Final Exit Quiz — Universal Gravitation ✅

Only one speed works for each radius: You can't orbit at any speed you want — the speed is determined by $r$ and $M$

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation - Complete Interactive Lesson

Part 1: Newton\'s Law of Gravitation

🌍 Newton's Law of Universal Gravitation

The Law of Universal Gravitation

Part 2: Gravitational Field Strength

🌐 Gravitational Field: g=GM/r2g = GM/r^2g=GM/r2

Part 3: Orbits & Satellite Motion

🛰️ Orbital Motion: Gravity as Centripetal Force

Part 4: Kepler\'s Laws

🪐 Kepler's Third Law: T2∝r3T^2 \propto r^3T2∝r3

Part 5: Gravitational PE (Universal)

🚀 Satellite Problems & Orbital Velocity

Part 6: Problem-Solving Workshop

🔧 Problem-Solving Workshop

Formula Decision Guide

Part 7: Synthesis & AP Review

🎓 Synthesis & AP Review

Complete Gravitation Toolkit

Essential Formulas

Key Features

Why Don't We Feel Gravity Between Everyday Objects?

The Gravitational Field

Definition

Weight and Gravitational Field

How ggg Varies with Distance

Surface Gravity on Other Bodies

Gravity = Centripetal Force

Solving for Orbital Velocity

Key Insights

Orbital Period

Energy in Orbit

Deriving Kepler's Third Law

The Key Relationship

Comparing Two Orbits (Same Central Body)

Kepler's Three Laws Summary

Important: Same Central Body Only!

Common Orbit Types

Key Pattern

Escape Velocity

Escape Velocity from Earth's Surface

Key Insight

Energy in Satellite Problems

Total Energy of a Satellite

Energy to Launch a Satellite

Binding Energy

Problem-Solving Steps

The Power of Proportional Reasoning

Common AP Mistakes

Mistake 1: Using altitude instead of orbital radius

Mistake 2: Confusing ggg and GGG

Mistake 3: Thinking astronauts are "outside gravity"

Mistake 4: Wrong proportional reasoning with vvv and TTT

Mistake 5: Drawing centripetal force separately from gravity

FRQ Practice

Sample FRQ: "Weighing Jupiter"

🌐 Gravitational Field: $g = GM/r^2$

🪐 Kepler's Third Law: $T^2 \propto r^3$

How $g$ Varies with Distance

Mistake 2: Confusing $g$ and $G$

Mistake 4: Wrong proportional reasoning with $v$ and $T$