Small, slowly moving spherical particles experience a drag force given by Stokes' law: Fd = 6πηrv where r is the radius of the particle, v is its speed, and η is the coefficient of viscosity of the fluid medium.
(a) Estimate the terminal speed of a spherical pollution particle of radius 9.00 10-6 m and density of 2166 kg/m3.
cm/s
(b) Assuming that the air is still and that η is 1.80 10-5 N · s/m2, estimate the time it takes for such a particle to fall from a height of 100 m.
h
mass = 2166 * (4/3)pi r^3
weight = 9.81 (2166)(4/3)pi r^3
balance weight with Stokes force
weight = 6 pi η r v
so
6piη r v=9.81 (2166)(4/3)pi r^3
9 η v=9.81 (2166)(2)(9*10^-6)^2
9 η v=9.81 (2166)(2)(9*10^-6)^2
v = 382472 *10^-12/η
have to know η to estimate v, see part b
v =382472 *10^-12 /1.8*10^-5
= 212485 *10^-7
= 0.0212485 m/s
v is constant so time= 100/.021
To estimate the terminal speed (v_terminal) of the particle, we can equate the drag force (Fd) to the gravitational force acting on the particle when it reaches terminal velocity.
(a) Terminal Speed (v_terminal):
The gravitational force acting on the particle is given by Fg = m * g, where m is the mass of the particle and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The mass of the particle can be calculated using its density (ρ) and volume (V):
m = ρ * V
The volume of a sphere is given by V = (4/3) * π * r^3, where r is the radius of the particle.
Now, let's calculate the mass of the particle:
ρ = 2166 kg/m^3
r = 9.00 * 10^-6 m
V = (4/3) * π * (9.00 * 10^-6)^3
V ≈ 3.05 * 10^-16 m^3
m = ρ * V
m ≈ 2166 kg/m^3 * 3.05 * 10^-16 m^3
m ≈ 6.61 * 10^-13 kg
The drag force can be expressed as Fd = 6πηrv_terminal, where η is the coefficient of viscosity, and v_terminal is the terminal velocity of the particle.
We can set up an equation equating the drag force (Fd) to the gravitational force (Fg) at terminal velocity:
Fd = Fg
6πηrv_terminal = m * g
Solving for v_terminal:
v_terminal = (m * g) / (6πηr)
Now, let's calculate the terminal speed:
g ≈ 9.8 m/s^2
η = 1.80 * 10^-5 N · s/m^2
r = 9.00 * 10^-6 m
v_terminal = (6.61 * 10^-13 kg * 9.8 m/s^2) / (6π * 1.80 * 10^-5 N · s/m^2 * 9.00 * 10^-6 m)
v_terminal ≈ 0.019 m/s
Converting to cm/s:
v_terminal ≈ 1.9 cm/s
Therefore, the estimated terminal speed of the spherical pollution particle is approximately 1.9 cm/s.
(b) Time taken to fall from a height (h):
The time taken for the particle to fall from a height (h) can be calculated using the equation for free fall:
h = (1/2) * g * t^2
Solving for time (t):
t^2 = (2 * h) / g
t = √((2 * h) / g)
Now, let's calculate the time taken for the particle to fall from a height of 100 m:
h = 100 m
t = √((2 * 100 m) / 9.8 m/s^2)
t ≈ 4.52 s
Therefore, the estimated time taken for the pollution particle to fall from a height of 100 m is approximately 4.52 seconds.
To estimate the terminal speed of the spherical pollution particle, we can use the equation for the drag force given by Stokes' law:
Fd = 6πηrv
Where Fd is the drag force, η is the coefficient of viscosity of the fluid medium (in this case air), r is the radius of the particle, and v is its speed.
(a) To find the terminal speed, we need to find the speed (v) at which the drag force (Fd) equals the weight of the particle (mg), where m is the mass of the particle and g is the acceleration due to gravity.
In this case, we know the radius of the particle (r = 9.00 × 10^(-6) m) and the density of the particle (ρ = 2166 kg/m^3).
The mass of the particle (m) can be calculated using the formula:
m = (4/3)πr^3ρ
Substituting the given values:
m = (4/3)π(9.00 × 10^(-6))^3(2166)
Now, we can equate the drag force with the weight of the particle:
6πηrv = mg
Simplifying:
6πηrv = (4/3)πr^3ρg
Cancelling out the common terms:
6ηv = (4/3)r^2ρg
Now, we can solve for v:
v = [(4/3)r^2ρg] / (6η)
Substituting the given values, with g = 9.8 m/s^2:
v = [(4/3)(9.00 × 10^(-6))^2(2166)(9.8)] / (6 × 1.80 × 10^(-5))
Evaluating this using a calculator, we get:
v ≈ 1.81 cm/s
Therefore, the estimated terminal speed of the spherical pollution particle is approximately 1.81 cm/s.
(b) To estimate the time it takes for the particle to fall from a height of 100 m, we can use the following equation of motion:
s = ut + (1/2)gt^2
where s is the distance fallen, u is the initial velocity (which is 0 in this case), g is the acceleration due to gravity, and t is the time.
Rearranging the equation:
t = sqrt((2s)/g)
Substituting the given values, with s = 100 m and g = 9.8 m/s^2:
t = sqrt((2 × 100) / 9.8)
Evaluating this using a calculator, we get:
t ≈ 4.52 s
Therefore, the estimated time it takes for the particle to fall from a height of 100 m is approximately 4.52 seconds.