Gravitational Field Equations to Know for AP Physics 1 (2025)

What You Need to Know

Gravitational field equations let you connect mass distributions to forces, accelerations, and energies. On AP Physics 1, they show up in:

• Attraction between masses (Newton’s law of gravitation)
• Weight vs. mass and how $g$ changes with altitude
• Orbits (circular motion + gravity)
• Gravitational potential energy (near Earth and universal)

Core idea: a mass creates a gravitational field $\vec g$ that pulls other masses toward it. The field is defined by:

$\vec g \equiv \frac{\vec F_g}{m}$

So if you know $\vec g$ at a location, the gravitational force on a mass $m$ there is:

$\vec F_g = m\vec g$

AP Physics 1 focus: You do algebra-based gravitation: point masses / spherical bodies, inverse-square relationships, energy, and circular orbits. No calculus-based field integrals.

Step-by-Step Breakdown

Use this process whenever you see “gravitational force,” “field,” “weight at altitude,” or “orbit.”

1) Decide which model applies

1. Near Earth (constant field): Use $F_g = mg$ and $U_g = mgh$ when height changes are small relative to Earth’s radius.
2. Universal gravitation (inverse-square): Use $F_g = \frac{Gm_1m_2}{r^2}$ and $g = \frac{GM}{r^2}$ when distances are large (satellites/planets) or $g$ is changing.
3. Orbital motion: If something is in (circular) orbit, gravity provides centripetal force: $F_g = F_c$.

2) Draw the diagram and define $r$ carefully

• Mark centers of masses and the separation distance $r$.
• Direction: gravitational force points toward the attracting mass.

Decision point: If the problem says “altitude $h$ above Earth,” then $r = R_E + h$ (not just $h$).

3) Choose the right equation and solve algebraically

Common “moves”:

• Replace force with field: $g = \frac{F_g}{m}$.
• Replace field with source mass: $g = \frac{GM}{r^2}$.
• Set gravity equal to centripetal: $\frac{GMm}{r^2} = \frac{mv^2}{r}$.

4) Use superposition when multiple masses act

• Forces add as vectors: $\vec F_{net} = \sum \vec F_i$.
• Fields add as vectors: $\vec g_{net} = \sum \vec g_i$.

5) Check “reasonableness”

• Inverse-square check: doubling $r$ should make $F_g$ (or $g$) become $\frac{1}{4}$ as large.
• Units check: $G$ makes units work out so $F_g$ comes out in newtons.

Key Formulas, Rules & Facts

Constants and symbols you’re expected to know

• Universal gravitational constant: $G = 6.67\times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$
• Earth’s surface gravitational field: $g \approx 9.8\ \text{m/s}^2$ (often use $10\ \text{m/s}^2$ if told)

Force, field, and weight (most-tested relationships)

Potential energy (near Earth vs universal)

Important sign idea: With $U_g = -\frac{GMm}{r}$, decreasing $r$ makes $U_g$ more negative (decreases), meaning gravity releases energy.

Orbits (gravity + circular motion)

These are high-yield because AP loves connecting units.

Scaling relationships (fast comparisons)

• If $r$ doubles: $F_g \to \frac{1}{4}F_g$ and $g \to \frac{1}{4}g$.
• If one mass doubles: $F_g$ doubles.
• For circular orbit: if $r$ increases, $v \propto \frac{1}{\sqrt{r}}$ and $T \propto r^{3/2}$.

Superposition (multiple masses)

• Force from each source mass $M_i$ on a test mass $m$:

$\vec F_{net} = \sum_i \left(\frac{GmM_i}{r_i^2}\ \hat r_i\right)$

• Field at a point from source masses $M_i$:

$\vec g_{net} = \sum_i \left(\frac{GM_i}{r_i^2}\ \hat r_i\right)$

where $\hat r_i$ points toward each mass (direction of the field).

Examples & Applications

Example 1: Compare gravitational force at two distances

A satellite moves from $r$ to $2r$ from Earth’s center. What happens to gravitational force magnitude?

• Use inverse-square scaling:

$F_g \propto \frac{1}{r^2}$

So:

$\frac{F_{new}}{F_{old}} = \frac{1/(2r)^2}{1/r^2} = \frac{1}{4}$

Insight: You don’t need numbers—just the power law.

Example 2: Gravitational field (weight) at altitude

Find $g$ at altitude $h$ above Earth (assume Earth mass $M_E$ and radius $R_E$).

• Radius from Earth’s center:

$r = R_E + h$

• Field magnitude:

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

If they ask “weight,” then $F_g = mg(h)$.

Exam variation: They may ask for the ratio:

$\frac{g(h)}{g(0)} = \frac{GM_E/(R_E+h)^2}{GM_E/R_E^2} = \left(\frac{R_E}{R_E+h}\right)^2$

Example 3: Solve for orbital speed

A satellite of mass $m$ orbits Earth in a circle at radius $r$ from Earth’s center. Find $v$.

• Set gravity equal to centripetal:

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

• Cancel $m$ and solve:

$v^2 = \frac{GM_E}{r} \quad\Rightarrow\quad v = \sqrt{\frac{GM_E}{r}}$

Insight: Orbital speed does not depend on the satellite’s mass.

Example 4: Universal potential energy change (big-picture)

A probe moves from $r_1$ to $r_2$ from a planet of mass $M$. Find $\Delta U_g$.

• Use:

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

So:

$\Delta U_g = U_2 - U_1 = -\frac{GMm}{r_2} + \frac{GMm}{r_1} = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

Insight: If $r_2 > r_1$ (moving outward), then $\Delta U_g > 0$ (you must add energy).

Common Mistakes & Traps

1. Using $h$ instead of $r$ in inverse-square formulas

   • Wrong: $g = \frac{GM}{h^2}$ when altitude is $h$.
   • Right: $r = R_E + h$, so $g = \frac{GM}{(R_E+h)^2}$.

2. Forgetting gravity is always attractive (sign/direction errors)

   • Students sometimes point $\vec F_g$ away from the planet.
   • Fix: draw arrows toward the attracting mass; use a sign convention consistently.

3. Mixing up $g$ and $G$

   • $G$ is universal constant; $g$ depends on location.
   • Check: $g = \frac{GM}{r^2}$ includes $G$.

4. Treating $mg$ as universal

   • $F_g = mg$ is the near-Earth approximation (constant $g$).
   • For satellites/planets, use $F_g = \frac{GMm}{r^2}$ or $F_g = mg(r)$ with $g(r) = \frac{GM}{r^2}$.

5. Not canceling the orbiting mass $m$ in orbit equations

   • In $\frac{GMm}{r^2} = \frac{mv^2}{r}$, $m$ cancels.
   • If your final $v$ depends on satellite mass, you made an algebra mistake.

6. Confusing potential energy sign (especially with $U_g = -\frac{GMm}{r}$)

   • Trap: thinking “higher up means more negative.”
   • Reality: increasing $r$ makes $U_g$ less negative (increases toward $0$).

7. Assuming heavier objects fall faster because gravitational force is bigger

   • Yes, $F_g = mg$ increases with $m$, but $a = \frac{F}{m} = g$ stays the same (ignoring air resistance).

8. Forgetting superposition is vector-based

   • Two equal masses on opposite sides can cancel fields at a midpoint.
   • Add directions: $\vec g_{net} = \vec g_1 + \vec g_2$, not just magnitudes.

Memory Aids & Quick Tricks

Quick Review Checklist

• You can write and use Newton’s gravitation law: $F_g = \frac{Gm_1m_2}{r^2}$.
• You know the definition of field: $\vec g = \frac{\vec F_g}{m}$ and the field of a mass: $g = \frac{GM}{r^2}$.
• You correctly use $r = R_E + h$ for altitude problems.
• You distinguish near-Earth energy $\Delta U_g = mg\Delta h$ from universal $U_g = -\frac{GMm}{r}$.
• You can set up an orbit with $F_g = F_c$ and get $v = \sqrt{\frac{GM}{r}}$.
• You can get orbital period: $T = 2\pi\sqrt{\frac{r^3}{GM}}$.
• You remember superposition: $\vec g_{net} = \sum \vec g_i$ with directions.
• You avoid sign mistakes: gravity points inward; universal $U_g$ is negative.

You’ve got the toolkit—now practice picking the right model fast and keeping $r$ straight.