In addition to the buoyant force, an object moving in a liquid experiences a linear drag force F_drag = (bv, direction opposite the motion), where b is a constant. For a sphere of radius R, the drag constant can be shown to be b = 6πηR, where η is the viscosity of the liquid. Consider a sphere of radius R and density ρ that is released from rest at the surface of a liquid with density ρ_f.
a. Find an expression in terms of R, η, g, and the densities for the sphere’s terminal speed v_term as it falls through the liquid.
b. Solve Newton’s second law to find an expression for v_y(t), the sphere’s vertical velocity as a function of time as it falls. Pay careful attention to signs!
c. Water at 20°C has viscosity η = 1.0 × 10^-3 Pa s. Aluminum has density 2700 kg/m³. If a 3.0-mm-diameter aluminum pellet is dropped into water, what is its terminal speed, and how long does it take to reach 90% of its terminal speed?