Gravitational Force Formulas

Why This Matters

Gravitational force is the glue holding the universe together—from the apple falling in your backyard to satellites orbiting Earth to galaxies spinning through space. In AP Physics 1, you're being tested on your ability to apply Newton's law of universal gravitation and connect it to concepts like weight, gravitational field strength, and the inverse-square relationship. The exam loves asking you to compare gravitational forces at different distances or explain why astronauts in orbit appear "weightless" even though gravity is still acting on them.

Don't just memorize these formulas—understand what each one reveals about how gravity works. The inverse-square law shows up repeatedly, linking gravitational force, field strength, and orbital motion. When you see a question about changing distance or mass, you should immediately think about proportional reasoning. Master the relationships between these formulas, and you'll be ready for both multiple-choice calculations and FRQ explanations that ask you to justify your reasoning.

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The Fundamental Force: Newton's Law of Universal Gravitation

Every gravitational calculation in AP Physics 1 traces back to one equation. This inverse-square law describes how the attractive force between any two masses depends on their masses and separation distance.

Newton's Law of Universal Gravitation

• $|\vec{F}_g| = G\frac{m_1 m_2}{r^2}$—the gravitational force is directly proportional to both masses and inversely proportional to the square of the distance between their centers
• Universal gravitational constant $G \approx 6.674 \times 10^{-11} \text{ N·m}^2/\text{kg}^2$—this tiny value explains why you need massive objects (like planets) to feel significant gravitational effects
• The force acts along the line connecting the centers of mass—both objects experience equal and opposite forces (Newton's third law), regardless of their relative sizes

Weight Near Earth's Surface

• $W = mg$—weight is simply the gravitational force acting on an object with mass $m$ in a gravitational field of strength $g$
• Weight is measured in newtons (N), not kilograms—mass stays constant everywhere, but weight changes depending on the local gravitational field
• This is the simplified form of universal gravitation—valid when you're close enough to Earth's surface that $g$ is approximately constant

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Gravitational Field Strength: Connecting Force to Acceleration

The gravitational field describes how strongly gravity pulls at any location in space. Field strength equals the gravitational force per unit mass, which is why it has the same numerical value as free-fall acceleration.

Gravitational Field Strength

• $g = \frac{GM}{r^2}$—field strength depends on the mass $M$ of the source object and your distance $r$ from its center
• Near Earth's surface, $g \approx 9.8 \text{ m/s}^2$ (often approximated as 10 m/s²)—this value decreases as you move farther from Earth's center
• Units are equivalent: N/kg = m/s²—this equivalence connects gravitational force concepts to kinematics and free-fall motion

Acceleration Due to Gravity

• $g \approx 9.8 \text{ m/s}^2$ is Earth's surface value—represents the acceleration any object experiences in free fall (ignoring air resistance)
• This value varies slightly with altitude and location—higher altitudes and equatorial regions have slightly lower $g$ values
• Essential for projectile motion and free-fall problems—whenever you see "dropped" or "thrown," you'll likely use this value

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Apparent Weight and Free Fall

Understanding the difference between actual gravitational force and what you feel is crucial for elevator problems and orbital mechanics. Apparent weight equals the normal force—what a scale would read—not the actual gravitational force.

Apparent Weight

• Apparent weight = Normal force $N$—this is what you feel pushing up on you and what a scale measures
• $N - mg = ma$ relates normal force to acceleration—when accelerating upward, apparent weight increases; when accelerating downward, it decreases
• In free fall, apparent weight is zero—this is why astronauts in orbit feel "weightless" even though gravity is still acting on them

Free Fall and Weightlessness

• Free fall occurs when gravity is the only force acting—no normal force means zero apparent weight
• Orbiting astronauts are in continuous free fall—they're falling around Earth, not floating in zero gravity
• The equivalence principle states that gravitational and inertial effects are indistinguishable—you can't tell if you're accelerating upward or in a stronger gravitational field just by feel

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Energy in Gravitational Systems

Gravitational potential energy describes the energy stored in the configuration of masses. The negative sign indicates that you must add energy to separate objects against gravity's attractive pull.

Gravitational Potential Energy

• $U = -G\frac{m_1 m_2}{r}$—potential energy between two masses separated by distance $r$
• The negative sign is crucial—it indicates that bound systems have negative total energy and that $U$ approaches zero as $r$ approaches infinity
• Work must be done against gravity to increase separation—as $r$ increases, $U$ becomes less negative (increases toward zero)

Escape Velocity

• $v_{\text{escape}} = \sqrt{\frac{2GM}{r}}$—minimum speed needed to escape a gravitational field without additional propulsion
• Derived from energy conservation—setting total mechanical energy equal to zero (just barely escaping)
• For Earth, $v_{\text{escape}} \approx 11.2 \text{ km/s}$—this doesn't depend on the escaping object's mass, only on Earth's mass and your starting distance

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Orbital Motion

Objects in stable orbits balance gravitational attraction with their tangential velocity. Setting gravitational force equal to centripetal force reveals the relationship between orbital speed, radius, and period.

Orbital Velocity

• $v_{\text{orbit}} = \sqrt{\frac{GM}{r}}$—speed required to maintain a circular orbit at radius $r$
• Derived from $F_g = F_c$—gravitational force provides the centripetal force needed for circular motion
• For low Earth orbit, $v_{\text{orbit}} \approx 7.9 \text{ km/s}$—faster than this at the same altitude means you'll spiral outward; slower means you'll fall inward

Kepler's Third Law

• $T^2 \propto r^3$ or more precisely $T^2 = \frac{4\pi^2}{GM}r^3$—orbital period squared is proportional to orbital radius cubed
• Applies to any orbiting object—planets, moons, satellites, or even binary stars
• Farther objects orbit more slowly—both because they travel a longer path and because they move at lower speeds

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Quick Reference Table

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Self-Check Questions

1. If you double the distance between two masses, what happens to the gravitational force between them? What happens to the gravitational potential energy?

2. An astronaut orbiting Earth feels weightless. Is the gravitational force on them zero? Explain using the concepts of actual weight vs. apparent weight.

3. Compare escape velocity and orbital velocity at the same distance from Earth's center. Which is larger, and by what factor? Why does this make physical sense?

4. A satellite moves to an orbit with twice the radius. Using Kepler's Third Law, by what factor does its orbital period change? Does it move faster or slower?

5. In an elevator accelerating upward at $2 \text{ m/s}^2$, what is your apparent weight compared to your actual weight? Write the equation you'd use and explain why the normal force changes.