Universal gravitation, discovered by Newton, is a force governing the entire universe. Understand that the gravity of Earth and the orbit of the Moon can be explained by the same equation.
Basic Knowledge
Law of universal gravitation:F = GMm/r²
(G = 6.67 × 10⁻¹¹ N·m²/kg²: gravitational constant, M, m: masses of the two objects [kg], r: distance between the objects [m])
Direction: Along the line connecting the two objects, in the direction in which they attract each other (action and reaction)
Relationship with gravity at Earth's surface:mg = GMm/R² (R: radius of Earth)
→ g = GM/R² (relationship among g, G, M, and R)
Gravitational acceleration at altitude h: g_h = GM/(R+h)²
Ratio to the surface value: g_h/g = R²/(R+h)²
Universal gravitation: F = GMm/r²
Gravitational constant: G = 6.67 × 10⁻¹¹ N·m²/kg²
Relationship between g and G: g = GM_E/R_E²
Inversely proportional to the square of r: When the distance doubles, the force becomes 1/4
Gravity at altitude h: g_h = GM/(R+h)²
Further Exploration (Background and Significance)
Newton is said to have derived universal gravitation from the idea that “the force causing an apple to fall and the force attracting the Moon to Earth might be the same kind of force.” The Moon's orbital radius is about 60 times Earth's radius. The Moon's centripetal acceleration, a_月 = g × (1/60)² ≈ 0.0027 m/s², agreed with its actual acceleration, confirming the idea.
Universal gravitation is a universal force in the cosmos that acts between all objects. It acts between people and Earth, people and the Sun, and people and stars in galaxies, etc., but it can be ignored in daily life because it is extremely small.
How to find Earth's mass: g = GM/R² → M = gR²/G = 9.8 × (6.4×10⁶)² / (6.67×10⁻¹¹) ≈ 6.0 × 10²⁴ kg.
Universal gravitation follows the inverse-square law (F ∝ 1/r²), which has the same form as Coulomb force (electrostatic force). Each is described in a unified way by field theory (gravitational fields and electric fields).
- F = GMm/r² (inversely proportional to the square of distance)
- Direction: The direction in which the objects attract each other
- g = GM/R² (relationship between surface g and the constants)
- Double the distance → one-fourth the gravitational force
- At altitude h, g decreases by the square of R/(R+h)
- The Moon's orbit can be explained by universal gravitation
- G = 6.67 × 10⁻¹¹ N·m²/kg²
Points of Caution (Common Confusions)
① r is the “distance between the objects.” For Earth and an object, it is the distance from Earth's center (at the surface, r = R). ② F ∝ 1/r² (inverse-square relationship), not simply the reciprocal of distance. ③ g is “gravitational acceleration” (m/s²), while G is the “gravitational constant” (N·m²/kg²). They are different quantities. ④ Understand the difference in units and meaning between mg and GMm/r² (mg is an approximate equation at Earth's surface).
Practice
- Given Earth's mass M, radius R, and gravitational constant G, find the gravitational acceleration g at Earth's surface.
- Express the gravitational acceleration at a height R above Earth's surface, where R is Earth's radius, using the surface value g.
- Express the gravitational force on an object of mass m located at distance r from Earth's surface (measured from Earth's center) in terms of G, M, m, and r.