Question

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?

Answer #1

Very small objects, such as dust particles, experience a
linear drag force, D⃗ D→ = (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 gas.
Suppose a gust of wind has carried a 52-μm-diameter dust
particle to a height of 260 m. If the wind suddenly stops, how long
will it take the dust particle to settle...

93.
A spherical particle falling at a
terminal speed in a liquid must have the gravitational force
balanced by the drag force and the buoyant force. The buoyant force
is equal to the weight of the displaced fluid, while the drag force
is assumed to be given by Stokes Law,
?s=6????.Fs=6πrηv.
Show that the terminal speed is given by
?=2?2?9?(?s−?1)v=2R2g9η(ρs−ρ1)R
is the radius of the sphere,
?sρs
is its density, and
?1ρ1
is the density of the fluid, and
?η

calculate buoyant force acting on a solid sphere with radius
R, submerged into a liquid with density rho

A flower falls to the bottom of a lake. Once it is on the
bottom, it will release pollen (a little sphere) which begins to
float towards the surface The drag force on a sphere in fluid is
Fdrag=6πrηv,
where r=100 μm is the radius of the pollen,
η=10^−3 Ns/m^2 is the viscosity of water and v is the
velocity of the pollen. The density of pollen is
p=300kg/m^3 and the density of water is
ρw=1000kg/m3. The pollen floats due...

Large objects have inertia and tend to keep moving-Newton's
first law. Life is very different for small microorganisms that
swim through water. For them, drag forces are so large that they
instantly stop, without coasting, if they cease their swimming
motion. To swim at constant speed, they must exert a constant
propulsion force by rotating corkscrew-like flagella or beating
hair-like cilia. The quadratic model of drag given by the equation,
D⃗ = (12CρAv2, direction opposite the motion), fails for
very small...

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