Question

# In addition to the buoyant force, an object moving in a liquid experiences a linear drag...

In addition to the buoyant force, an object moving in a liquid experiences a linear drag force Fdrag = (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 ρf.

a. Find an expression in terms of R, η, g, and the densities for the sphere’s terminal speed vterm as it falls through the liquid.

b. Solve Newton’s second law to find an expression for vy(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?