Final answer:
The terminal speed of a spherical particle in a fluid is derived by balancing the gravitational force against the drag and buoyant forces, where the gravitational force is volumetric and the buoyant force is equivalent to the weight of the displaced fluid. Stokes' Law describes the drag force, leading to the derived formula for terminal speed, which is dependent on the sphere's radius, the densities of the sphere and fluid, and the fluid's viscosity.
Step-by-step explanation:
A student asked how to show that the terminal speed of a spherical particle falling in a liquid is given by the equation v = 2R²g/(9η(ρs - ρl)) where R is the sphere's radius, ρs is its density, ρl is the liquid's density, and η is the viscosity coefficient. At terminal speed, the gravitational force acting downward on the particle is balanced by the upward buoyant force and drag force. The gravitational force is the product of the mass of the sphere (volume × density) and acceleration due to gravity, while the buoyant force is the weight of the displaced fluid, and the drag force follows Stokes' Law, Fs = 6πrηv.
Using the balance of forces at terminal velocity (gravitational = buoyant + drag force), we can express the volume of the sphere as (4/3)πR³, allowing us to calculate the mass. Cancelling out common terms and rearranging the equation, we find the formula for terminal speed v, accounting for the density difference between the sphere and the fluid, and the viscosity of the fluid, which is key for understanding sedimentation and particle motion in fluids.