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AP Physics 1 – Universal Gravitation: Force, Field & How g Changes

Newton’s Big Idea: The force that makes an apple fall is the same force that keeps the Moon in orbit. In this guide (Topics 3.4 & 3.5), we break down Newton’s Law of Universal Gravitation, the difference between $G$ and $g$ , and how to read the famous “Gravity Graph” that appears on the AP Exam.

1. Newton’s Law of Universal Gravitation

Every object in the universe attracts every other object. This force is proportional to their masses and inversely proportional to the square of the distance between them.

The “Universal” Equation

$F_g = G \dfrac{m_1 m_2}{r^2}$

$F_g$ : Gravitational Force (Newtons). Always attractive.
$m_1, m_2$ : The two masses (kg).
$r$ : Distance between the centers of the masses (meters).
$G$ : The Universal Constant ( $6.67 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2$ ).

⚠️ AP Trap: Do not confuse $G$ with $g$ .
$G$ is a universal constant. $g$ (9.8 m/s²) changes depending on the planet and your distance from it.

2. The Inverse Square Law

Gravity is an “inverse square” force. This means if you move twice as far away, the force drops by a factor of four.

Diagram illustrating Newton's Inverse Square Law showing gravitational force decreasing as 1/r^2 as distance increases. — **The Rule:** Double the distance ( $2R$ ) $\rightarrow$ Quarter the force ( $F/4$ ).

Double Distance ( $2r$ )

$F_{new} = \dfrac{1}{(2)^2} F = \dfrac{1}{4} F$

Triple Distance ( $3r$ )

$F_{new} = \dfrac{1}{(3)^2} F = \dfrac{1}{9} F$

Half Distance ( $\dfrac{1}{2}r$ )

$F_{new} = \dfrac{1}{(1/2)^2} F = 4 F$

3. Gravitational Field Strength ( $g$ )

We say $g = 9.8 \, \text{m/s}^2$ on Earth, but what about on Mars or in orbit? By setting $F_g = mg$ , we can solve for the field strength anywhere:

$g = \dfrac{GM}{r^2}$

This tells us:

Mass ( $M$ ): Heavier planets create stronger fields.
Distance ( $r$ ): The further you are, the weaker $g$ gets (inverse-square).

⚠️ Local g vs Global g:
g ≈ 9.8 m/s² is the local value near Earth’s surface.
Global formula: $g(r) = \dfrac{GM}{r^2}$ works for any planet/distance.

4. Graphing Gravity: Inside vs. Outside

This is a favorite AP Exam question (Topic 3.4). How does gravity change as you travel from the center of a planet out to deep space?

Graph of gravitational field strength g versus distance r, showing linear increase inside the planet and inverse-square decay outside. — **Inside vs. Outside:** Gravity grows linearly inside the planet, peaks at the surface, then fades away rapidly.

Inside ( $r < R$ ):
Gravity increases linearly. As you move out, there is “more planet” underneath you. ( $g \propto r$ )
The Surface ( $r = R$ ):
Gravity is at its maximum.
Outside ( $r > R$ ):
Gravity decreases as the inverse square. ( $g \propto 1/r^2$ )

5. Concept Check: Types of Mass

Topic 3.5: Inertial vs. Gravitational Mass

Inertial Mass: Resistance to acceleration ( $F=ma$ ).
Gravitational Mass: How strongly it interacts with gravity ( $F_g = mg$ ).

Key Idea: Experiments show these two are identical. This is why all objects fall at the same rate regardless of mass!

6. AP Practice Problems

Problem 1: Planet X

Planet X has twice the mass ( $2M$ ) and twice the radius ( $2R$ ) of Earth. What is the surface gravity ( $g$ ) on Planet X compared to Earth?