Gravitational Force Equations

Why This Matters

Gravity isn't just about things falling down—it's the fundamental force that governs everything from why you stay on Earth to how satellites orbit and why planets follow predictable paths around the Sun. In Honors Physics, you're being tested on your ability to apply gravitational equations to real scenarios: calculating orbital speeds, comparing gravitational effects at different distances, and understanding how energy transforms in gravitational systems. These concepts connect directly to Newton's laws, circular motion, energy conservation, and inverse-square relationships.

The equations in this guide aren't isolated formulas to memorize—they're interconnected tools that describe the same phenomenon from different angles. When you see $G$, $M$, and $r$ appearing across multiple equations, that's not coincidence; it's showing you how force, field strength, energy, and velocity all stem from the same gravitational principles. Don't just memorize formulas—know which equation to reach for based on what the problem is actually asking.

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The Foundation: Newton's Universal Gravitation

Every gravitational calculation traces back to one core idea: masses attract each other with a force that depends on their masses and the distance between them.

Newton's Law of Universal Gravitation

• $F = G \frac{m_1 m_2}{r^2}$—this is the master equation; every other gravitational formula derives from it
• Inverse-square relationship means doubling the distance quarters the force—a concept that appears constantly on exams
• $G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$ is the gravitational constant; it's tiny, which is why gravity only matters for massive objects

Gravitational Force Between Two Masses

• Both masses matter equally—the force on mass A from mass B equals the force on mass B from mass A (Newton's Third Law in action)
• Distance $r$ is measured center-to-center, not surface-to-surface—a common exam trap
• This force provides the centripetal force for orbital motion, connecting gravity directly to circular motion problems

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Field Strength and Surface Gravity

Gravitational field strength tells you how strongly gravity pulls on each kilogram of mass at a given location—it's the "intensity" of the gravitational field.

Gravitational Field Strength (g)

• $g = \frac{GM}{r^2}$—this comes directly from Newton's Law divided by test mass; notice it depends only on the source mass, not the object being pulled
• On Earth's surface, $g \approx 9.81 \, \text{m/s}^2$, but this value decreases with altitude as $r$ increases
• Field strength is a vector pointing toward the center of the mass creating the field

Weight as a Function of Gravity

• $W = mg$ is the simplest gravitational equation—weight is just mass times local field strength
• Weight changes with location (Earth vs. Moon vs. Jupiter), while mass stays constant—exams love testing this distinction
• Weight is a force vector directed toward the gravitational center, not just "down"

Gravitational Acceleration on a Planet's Surface

• Surface gravity depends on both mass AND radius—a planet with twice Earth's mass but twice the radius has the same surface gravity
• $g = \frac{GM}{R^2}$ where $R$ is the planet's radius—this is identical to field strength at the surface
• Comparing planets requires analyzing how $M$ and $R$ change together, not just one variable

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Energy in Gravitational Fields

Gravitational potential energy describes how much work gravity can do on an object—or how much work you'd need to do against gravity.

Gravitational Potential Energy (Near Surface)

• $U = mgh$ works only for small height changes near a planet's surface where $g$ is approximately constant
• Reference point matters—you choose where $h = 0$, making $U$ a relative quantity in this form
• Energy is scalar, so no direction—just positive or negative depending on your reference

Gravitational Potential

• $V = -\frac{GM}{r}$—this is potential energy per unit mass, useful for describing the field itself
• The negative sign is crucial: potential is zero at infinity and becomes more negative as you approach the mass
• To escape a gravitational field, an object must gain enough kinetic energy to overcome this negative potential (total energy must reach zero or positive)

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Orbital Motion and Escape

When gravitational force provides centripetal acceleration, you get orbits. The balance between kinetic and potential energy determines whether an object stays bound or escapes.

Orbital Velocity

• $v_o = \sqrt{\frac{GM}{r}}$—derived by setting gravitational force equal to centripetal force ($\frac{mv^2}{r}$)
• Orbital speed decreases with distance—satellites farther from Earth move slower, which seems counterintuitive but follows from the math
• Low Earth orbit velocity ≈ 7.8 km/s—a useful benchmark for comparison problems

Escape Velocity

• $v_e = \sqrt{\frac{2GM}{r}}$—notice this is exactly $\sqrt{2}$ times orbital velocity at the same radius
• Derived from energy conservation: set kinetic energy equal to the magnitude of gravitational potential energy
• Earth's escape velocity ≈ 11.2 km/s—independent of the escaping object's mass (the $m$ cancels out)

Kepler's Third Law of Planetary Motion

• $T^2 = \frac{4\pi^2}{GM}r^3$—the full equation shows that orbital period depends on the central mass and orbital radius
• $T^2 \propto r^3$ means planets farther from the Sun have much longer years (Neptune's year is 165 Earth years)
• This law lets you calculate mass—if you know $T$ and $r$ for a satellite, you can find the mass of what it orbits

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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 if you double both masses while keeping distance constant?

2. Two planets have the same surface gravity. Planet A has twice the mass of Planet B. How do their radii compare? (Hint: solve $g = \frac{GM}{R^2}$ for both cases.)

3. Compare orbital velocity and escape velocity at the same distance from a planet. Why is escape velocity exactly $\sqrt{2}$ times orbital velocity? (Think about what energy condition each represents.)

4. A satellite moves to an orbit twice as far from Earth. Does its orbital velocity increase or decrease? By what factor? How does its orbital period change?

5. FRQ-style: An astronaut weighs 800 N on Earth's surface. Calculate their weight at an altitude equal to Earth's radius (i.e., at $r = 2R_E$). Then explain why gravitational potential energy is negative while this weight calculation gives a positive value.