Fallingsphere

Fallingsphere is a term used in physics and engineering to describe a spherical object descending through a fluid under the influence of gravity. The concept is used as a simple model to study the balance of forces on a particle, including gravity, buoyancy, and aerodynamic or hydrodynamic drag. In a typical setting, a sphere of radius R and density ρp falls through a fluid of density ρf and dynamic viscosity μ.

The motion is governed by Newton's second law: m dv/dt = m g - ρf V g - F_drag, where m is the sphere mass, V its volume, and F_drag is the resistive force. The buoyancy force is ρf V g, and the weight is ρp V g. When the downward forces balance upward forces, the sphere reaches terminal velocity vt. In the laminar regime (low Reynolds number), Stokes' law gives F_drag = 6 π μ R v, and vt = (2/9) (ρp - ρf) g R^2 / μ. At higher Reynolds numbers, the drag force is often written as F_drag = (1/2) Cd ρf A v^2 with drag coefficient Cd depending on Re.

Applications of the fallingsphere model include measuring fluid properties, calibrating instrumentation, studying sedimentation and settling velocities, and illustrating drag and buoyancy in education and demonstrations. Variants consider non-Newtonian fluids, milling operations, or microgravity conditions to isolate specific forces. Related concepts include terminal velocity, Stokes' law, drag coefficient, and sedimentation physics.