Rucete ✏ AP Physics 1 In a Nutshell
5. Gravitation
This chapter introduces Newton’s law of universal gravitation, gravitational energy, and orbital motion. It also explores how gravity varies with distance, including escape velocity and Kepler’s third law of planetary motion.
Newton’s Law of Universal Gravitation
• All masses attract each other with a force: F = G·(m₁·m₂)/r²
• G = 6.67 × 10⁻¹¹ N·m²/kg² (universal gravitational constant).
• r is the distance between centers of mass; F is directed inward toward center.
• The law is an inverse-square relationship: doubling r → force becomes one-fourth.
• Equal and opposite forces act on both masses (Newton’s third law).
Acceleration Due to Gravity
• g = G·(ME)/(rE)² = 9.8 m/s² at Earth’s surface.
• On other planets: g = G·M / r², where M is the planet's mass and r its radius.
• g decreases with altitude: use r = rE + h.
• At 400 km above Earth, g ≈ 8.67 m/s² → ~11.5% weight reduction.
Gravitational Potential Energy
• For large distances: U = –G·(m₁·m₂)/r
• Potential energy is negative; zero at infinite distance.
• More negative U → stronger gravitational binding.
• Near Earth’s surface: U = mgh is an approximation (valid for small h).
Escape Velocity
• Minimum speed to escape a planet’s gravity without additional propulsion.
• Derived by setting total energy = 0: ½mv² = G·Mm / r → vescape = √(2GM / r)
• Independent of object’s mass.
• Earth’s escape velocity ≈ 11.2 km/s.
Orbital Motion
• For circular orbits: gravitational force provides centripetal acceleration.
• Set Fgravity = Fcentripetal: G·Mm / r² = mv² / r → v = √(GM / r)
• Orbital period: T = 2πr / v = 2π√(r³ / GM)
• Satellites stay in orbit by falling around Earth with just enough tangential velocity.
Kepler’s Third Law (for Planets)
• T² ∝ r³ → T² / r³ = constant (same for all planets orbiting same star)
• Useful for comparing orbits: (T₁ / T₂)² = (r₁ / r₂)³
• Only applies to objects in orbit around same central mass.
In a Nutshell
Gravitational forces govern motion on all scales, from falling apples to orbiting planets. Newton’s law quantifies the attractive force between any two masses, while gravitational potential energy and escape velocity describe energy in gravitational systems. Orbital motion relies on the balance between gravity and inertia, and Kepler’s laws provide insight into planetary dynamics.