🌍Newton's Law of Universal Gravitation
Every Mass Attracts Every Other Mass
Newton realized that the force pulling an apple down is the same force keeping the Moon in orbit. He formulated:
F_g = G × m₁ × m₂ / r²
Where: - G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant) - m₁, m₂ = masses of the two objects - r = distance between their centers
Key features: - Force is attractive (always pulls together, never pushes apart) - Follows inverse square law: double the distance → force drops to 1/4 - Acts through empty space (no contact needed)
Think About It
If the Moon were twice as far from Earth as it currently is, how would the gravitational force change?
Gravitational Field
The gravitational field strength g at any point is the gravitational force per unit mass:
g = F_g/m = GM/r²
Near Earth's surface, g ≈ 9.8 m/s². This is both: 1. The acceleration due to gravity (how fast objects fall) 2. The gravitational field strength (N/kg)
g varies with altitude: farther from Earth → smaller g.
✏️ Worked Example
Problem: Find the gravitational force between Earth (M = 6×10²⁴ kg) and a 70 kg person standing on the surface (r = 6.4×10⁶ m).
📐 Key Equations
Newton's Law of Gravitation
F_g = (Gm_1 m_2)/(r²)g = (GM)/(r²)G = 6.674 × 10^-11 N·m²/kg²⚠️ Common Mistakes
Misconception: Using the altitude above Earth's surface as r instead of the distance from Earth's center.
✓ Correct thinking: r in the gravitational force formula is always measured from the center of each object, not from the surface.
Why: For an object at altitude h, r = R_Earth + h. Using h alone gives a much smaller r and an enormously inflated force.
Misconception: Thinking the gravitational force is zero in space (e.g., on the ISS).
✓ Correct thinking: Gravity extends infinitely (it weakens with distance but never reaches zero). The ISS at 400 km still experiences about 88% of surface gravity.
Why: Astronauts feel "weightless" because they are in free-fall, not because gravity is absent. This is a critical conceptual distinction.
Misconception: Confusing gravitational field strength g (N/kg) with gravitational acceleration (m/s²) and thinking they are different quantities.
✓ Correct thinking: They are the same quantity numerically. g = GM/r² has units of N/kg = m/s² — both express the same physical reality.
Why: F = mg works with either interpretation: force per unit mass (field view) or mass times acceleration (Newton's second law view).
📝 Practice Problems
Try these problems. Check your answer when ready.
Two 5 kg masses are placed 0.5 m apart. What is the gravitational force between them? (G = 6.67×10⁻¹¹ N·m²/kg²)
F_g = (Gm_1 m_2)/(r²)If the distance between two masses is tripled, by what factor does the gravitational force change?
Calculate the gravitational field strength g at an altitude of 3×R_Earth above Earth's surface. (g_surface = 9.8 m/s², R_Earth = 6.4×10⁶ m)
g = (GM)/(r²) ∝ (1)/(r²)Planet X has twice the mass of Earth and the same radius. What is g on its surface compared to Earth's?
The Moon has mass 7.3×10²² kg and radius 1.74×10⁶ m. Find g on the Moon's surface.
g = (GM)/(R²)A planet has the same density as Earth but twice the radius. What is g on its surface compared to Earth? (Hint: M = ρV = ρ × (4/3)πR³)
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