A water droplet evaporates before they reach the ground. Figure 1: Water droplets [source] In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rois the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is r(t) = (-)t+ro- %3D (Hint: m = pV,V =tr³, S = 4ar2). %3D of the water dronlet is
A water droplet evaporates before they reach the ground. Figure 1: Water droplets [source] In this situation, a few assumptions are made: a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical while evaporating. b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S. c) There is no air-resistance and downward direction is the positive direction. To describe this problem, given that p is the mass density of water, rois the radius of water before it drops, m is the water mass, V is the water volume and k is the constant of proportionality. QUESTION: (1) From assumption (b), show that the radius of the water droplet at time t is r(t) = (-)t+ro- %3D (Hint: m = pV,V =tr³, S = 4ar2). %3D of the water dronlet is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question no. 1
![A water droplet evaporates before they reach the ground.
ond orde
E
Figure 1: Water droplets [source]
In this situation, a few assumptions are made:
a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical
while evaporating.
b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S.
c) There is no air-resistance and downward direction is the positive direction.
quati
To describe this problem, given that p is the mass density of water, rois the radius of water before it
drops, m is the water mass, V is the water volume and k is the constant of proportionality.
QUESTION:
(1) From assumption (b), show that the radius of the water droplet at time t is
on
er
r(t) = () t + ro-
%3D
(or
(Hint: m = pV,V =nr³, S = 4r2).
4
%3D
3
%3D
(2) From assumption (c), the velocity (v) at time (t) of the water droplet is
ou
3
v' +
t+ro
If the water drops from stationary, solve for v(t).
(3) Determine the time when the water droplet has evaporated entirely, given that
ro = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8fbbe91e-b101-441b-b5fc-9477065585b8%2F65db3bcb-78f7-4546-ba5b-88fb3e7b2697%2Fsu2acrg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A water droplet evaporates before they reach the ground.
ond orde
E
Figure 1: Water droplets [source]
In this situation, a few assumptions are made:
a) At initial point, a typical water droplet is in sphere shape with radius r and remain spherical
while evaporating.
b) The rate of evaporation (when it loses mass (m)) is proportional to the surface area, S.
c) There is no air-resistance and downward direction is the positive direction.
quati
To describe this problem, given that p is the mass density of water, rois the radius of water before it
drops, m is the water mass, V is the water volume and k is the constant of proportionality.
QUESTION:
(1) From assumption (b), show that the radius of the water droplet at time t is
on
er
r(t) = () t + ro-
%3D
(or
(Hint: m = pV,V =nr³, S = 4r2).
4
%3D
3
%3D
(2) From assumption (c), the velocity (v) at time (t) of the water droplet is
ou
3
v' +
t+ro
If the water drops from stationary, solve for v(t).
(3) Determine the time when the water droplet has evaporated entirely, given that
ro = 3mm. Then, 10 seconds after the water drops, the radius r = 2mm.
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