Gravitation and Fluid Dynamics
Core concepts
Gravitation covers Newton's inverse-square law, gravitational potential energy (–GMm/r), orbital velocity, escape velocity, and Kepler's three laws. Fluid dynamics adds pressure, Pascal's principle, Archimedes' buoyancy, continuity (A₁v₁ = A₂v₂), and Bernoulli's equation (P + ½ρv² + ρgh = constant). Viscosity and Stokes' law appear in terminal-velocity problems.
Traps
- Forgetting the negative sign of gravitational potential energy, or confusing it with the potential (per unit mass).
- Using g as constant over large heights when the inverse-square variation matters.
- Applying Bernoulli's equation across a viscous flow or across a pump (energy is added).
- Mixing gauge and absolute pressure, or forgetting atmospheric pressure in barometer/manometer readings.
- Assuming orbital speed depends on satellite mass - it does not.
Rule of thumb
Escape velocity is √2 times orbital velocity at the same radius. For a floating body, the fraction submerged equals the density ratio ρ_body/ρ_fluid.
Worked example
Find the orbital speed of a satellite at 400 km altitude. r = 6.4×10⁶ + 4×10⁵ = 6.8×10⁶ m. v = √(GM/r) = √(6.67×10⁻¹¹ × 5.97×10²⁴ / 6.8×10⁶) ≈ √(5.86×10⁷) ≈ 7.65×10³ m/s, about 7.7 km/s. The satellite's own mass cancels.