Final answer:
The terminal velocity of a spherical particle in fluid relates to the balance between gravitational, buoyant, and viscous forces. The equation V = (2R²(g(Ps - P1)))/(9η) provides this relationship, showing that terminal velocity increases with lower fluid viscosity and with smaller, denser particles.
Step-by-step explanation:
The relationship between the terminal velocity of a particle and the viscosity of the fluid can be understood through the concepts of balance between gravitational force, drag force, and buoyant force. A spherical particle falling at a terminal velocity in a liquid must have the gravitational force counteracted by the drag force, as described by Stokes' Law (Fs = 6πrηv), and the buoyant force, which is equal to the weight of the displaced fluid. The terminal velocity V of a spherical particle in a fluid is given by the equation V = (2R²(g(Ps - P1)))/(9η), where R is the radius of the sphere, g is the acceleration due to gravity, Ps is the density of the particle, P1 is the density of the fluid, and η is the coefficient of viscosity.
For example, if we were to calculate the viscosity of motor oil with a given steel ball falling at terminal velocity, we'd rearrange the terminal velocity equation to solve for η and substitute the known values for the steel ball's radius, the terminal velocity, and the densities of the ball and oil into the equation. Terminal velocity is greatest for low-viscosity fluids and dense, small particles. Thus, in a low-viscosity fluid, a particle will reach a higher terminal velocity compared to a high-viscosity fluid.