Gravitation — NDA Physics

Formulas (3)

• Newton's law of gravitation — the inverse-square law · Newton's law of gravitation
  $$F = \dfrac{G\,m_1 m_2}{r^2}$$
• Scaling the force — changing masses and distance together · Force scaling factor
  $$\dfrac{F'}{F} = \dfrac{a \cdot b}{c^2}$$
• Gravitational force is action-reaction — equal and opposite · Action-reaction pair
  $$\vec{F}_{12} = -\,\vec{F}_{21}, \qquad |\vec{F}_{12}| = |\vec{F}_{21}| = \dfrac{G m_1 m_2}{r^2}$$

Reference tables (1)

Watch out for (5)

• Distance enters as a square, masses do not
  → Newton's law of gravitation — the inverse-square law
• G is universal, but the FORCE is not
  → The universal gravitational constant G
• Don't confuse G with g
  → The universal gravitational constant G
• Square only the distance factor
  → Scaling the force — changing masses and distance together
• The bigger mass does NOT exert the bigger force
  → Gravitational force is action-reaction — equal and opposite

Formulas (6)

• Surface gravity — g = GM/R² · Surface gravity
  $$g = \dfrac{GM}{R^2}$$
• Surface gravity from density — g = (4/3)πGρR · Surface gravity from density
  $$g = \dfrac{4}{3}\pi G \rho R$$
• Average density of a composite body · Average density of a composite body
  $$\bar{\rho} = \dfrac{m_1 + m_2}{V_{\text{total}}} = \dfrac{\rho_1 V_1 + \rho_2 V_2}{V_1 + V_2}$$
• Gravitational field versus potential · Work and equal potential
  $$W = -\,m\,(V_B - V_A); \qquad V_A = V_B \Rightarrow W = 0$$
• Weightlessness in orbit — zero normal reaction · Apparent weight = normal reaction
  $$W_{\text{apparent}} = N = 0 \quad (\text{free fall}); \qquad F_{\text{gravity}} \neq 0$$
• g is the same for all bodies — free fall and weight · Weight and spring extension scale with g
  $$W = mg, \qquad x = \dfrac{mg}{k} \propto g$$

Watch out for (7)

• Radius enters as a square in g, just like in F
  → Surface gravity — g = GM/R²
• Same density does NOT mean same gravity
  → Surface gravity from density — g = (4/3)πGρR
• Average density is volume-weighted, not the mean of densities
  → Average density of a composite body
• Equal potential, not equal field, decides the work
  → Gravitational field versus potential
• Weightless does NOT mean gravity-free
  → Weightlessness in orbit — zero normal reaction
• In vacuum, mass and shape don't change the fall time
  → g is the same for all bodies — free fall and weight
• Spring extension follows g, not just the mass
  → g is the same for all bodies — free fall and weight

Formulas (4)

• Kepler's third law — T² ∝ a³ · Kepler's third law
  $$T^2 \propto a^3 \qquad\Longleftrightarrow\qquad \dfrac{T_1}{T_2} = \left(\dfrac{a_1}{a_2}\right)^{3/2}$$
• Orbital velocity — v_o = √(GM/R) · Orbital velocity
  $$v_o = \sqrt{\dfrac{GM}{R}}$$
• Escape velocity — v_e = √(2GM/R) and how it scales · Escape velocity and its density scaling
  $$v_e = \sqrt{\dfrac{2GM}{R}} = \sqrt{2gR}; \qquad v_e \propto R\sqrt{\rho}$$
• What keeps a satellite up — no fuel required · Orbit is sustained by gravity alone
  $$F_{\text{gravity}} = \dfrac{GMm}{R^2} = \dfrac{mv_o^2}{R} \;\Rightarrow\; \text{no propulsion needed}$$

Watch out for (5)

• It's T² ∝ a³, not T ∝ a
  → Kepler's third law — T² ∝ a³
• Orbital speed is set by the orbit, not the satellite
  → Orbital velocity — v_o = √(GM/R)
• Halving R while quadrupling ρ leaves v_e UNCHANGED
  → Escape velocity — v_e = √(2GM/R) and how it scales
• Escape velocity is independent of the projectile's mass and launch angle
  → Escape velocity — v_e = √(2GM/R) and how it scales
• Orbiting needs no fuel — gravity does the work
  → What keeps a satellite up — no fuel required