Physics 110
Spring 2006
Fluid Mechanics – Their Solutions
1. Pressure in a fluid varies linearly with depth.
a.
What is the absolute pressure at a depth of 1000m in the ocean if the
density of seawater is 1024 kg/m3?
b.
At this depth what force must the frame around a circular submarine
porthole having a diameter of 30cm exert to counterbalance the force
exerted by the water?
a.
Pd = Pair + ρgh = 1.013 ×105
b.
F = PA = 1.0137 ×107
N
m2
(
N
m2
(
)(
)
+ 1024 mkg3 9.8 sm2 (1000m ) = 1.0137 ×107
)
N
m2
× π (0.30m ) = 2.87 ×106 N
2
2. A light balloon can be filled with either 400m3 of helium (ρHe = 0.179 kg/m3) or
400m3 of hydrogen (ρH = 0.0899 kg/m3) and floated to carry a payload.
a.
If the balloon is nearly in equilibrium (meaning all accelerations are
zero), what mass of a payload could a balloon filled with helium
support at 0oC?
b.
If the balloon is nearly in equilibrium (meaning all accelerations are
zero), what mass of a payload could a balloon filled with hydrogen
support at 0oC?
∑ Fy : Fb,air − mgas g − mhanging g = ma y = 0
Fb ,air − m gas g
a.
= ρ airVair − ρ gasV gas = (ρ air − ρ gas )Vgas
g
for He, we have mhanging = 444.4kg
b.
for H, we have mhanging = 480.0kg
→ mhanging =
where ρ =
M
and volume of air displaced by gas is the same as the volume of gas.
V
3. The U.S. Navy has the largest warships in
the world, aircraft carriers of the Nimitz
class (for example, the USS Ronald Regan
shown on the right.) Suppose that 50,
29,000kg airplanes take off from the flight
deck and the ship bobs up to float 11cm
higher in the water, in an area where g =
9.78m/s2. What is the horizontal area
enclosed by the waterline of the ship?
Compare this to the deck of an aircraft
which has an area 18,000 m2.
www.aerospaceweb.org/ question/history/q0226.shtml
M total , planes = 50 × 29, oookg = 1.45 × 10 6 kg
The weight of the planes is equal to the weight of a slice of water
11cm thick with cross sectional area bounded by the waterline of the ship.
(
)
M water = ρV = ρ water dA → 1.45 × 10 6 kg = 1030 mkg3 (0.11m )A → A = 1.28 ×10 4 m 2
∴ ratio =
Awaterline
= 0.71
A flightdeck
4. A Boeing 777 has a mass of 2.43x105 kg and each wing has an area of 189m2.
During level flight, the pressure on the lower wing surface is 700x104 Pa.
a.
What is the pressure on one of the upper wings?
b.
What is the upward acceleration of the aircraft if the
pressure on the lower surface were to increase to
702x104 Pa? (This increase in pressure is due to the
aircraft increasing its forward velocity and assumes
that Pupper remains constant.)
www.boeing.com
a.
If the plane is not accelerating up or down then the difference in pressures
above and below the wing gives the lifting force (when multiplied by the wing area.)
Plower − Pupper =
b.
(P
lower
Flift
A
=
m plane g
Awing
→ Pupper = Plower −
m plane g
Awing
= 6.987 × 10 6
N
m2
− Pupper )Awing = Funbalanced → Funbalanced = 6.73 × 10 6 N
∴ Funbalanced − m plane g = m plane a → a = 15.8 sm2 .
5. A helium filled balloon is tied to a 2.0m long string where the mass of the string
is 0.05kg. The balloon is spherical with radius of 0.40m. When released the
balloon lifts a length of string h and then remains in equilibrium. What length
of the string is lifted if the balloon has a mass of 0.250kg?
From a free body diagram we find
Fair − Fg ,balloon − Fg , He − Fg , string = ma y = 0 →
0 = ρ air gV − mballoon g − ρ He gV − mstring g
∴h =
(ρ air − ρ He )V − mballoon L = 1.91m
h
L
mstring
6. A uniform disk with a mass of 10kg and a radius of 0.250m spins at 300 rpm on
a low-friction axle. The disk must be brought to rest in 1.0min by a brake pad
that makes contact with the disk at an average distance of 0.220m from the axis.
The coefficient of friction between the brake pad and the disk is 0.5. A piston
in a cylinder with a diameter of 5.0cm presses the brake pad against the disk.
What is the pressure of the brake fluid in the cylinder?
rev
ωi = 300 min
= 31.42 rads → ω f = ωi + αt → α = −0.524 rad
s2
(
)
2
1
Iα
2 MR α
τ = rF friction = Iα → F friction =
=
= −0.744 N
r
r
F friction
= −1.49 N
F friction = µ K Fapplied → Fapplied =
µK
∴P =
Fapplied
A
=
1.49 N
= 189.5 mN2
2
π (0.05m )
7. Suppose that the pressure in the cerebrospinal fluid
(CSF) is measured as shown below, using a spinal tap
with the patient sitting erect. The pressure due to the
weight of the CSF (given that its density is 1050
kg/m3) in the spinal column increases the pressure?
a. What is the pressure measured in Pascal and
in centimeters of water if the pressure around
the brain is 10mm Hg and the tap is at a point
60cm lower than the brain?
b. What is the pressure measured in Pascal and in centimeters of water if the
person is lying down?
a.
P = Pbrain + ρgh = 10mmHg ×
133.3 mN2
1mmHg
(
height ofwater : P = ρg water hwater → hwater =
b.
)
+ 1050 mkg3 × 9.8 sm2 × 0.6m = 7507 mN2 = 56.3mmHg
P
= 0.766m = 76.6cm
ρg water
Plying down = 1333 mN2 = 10mmHg
hwater =
P
= 0.136m = 13.6cm
ρg water
8. The total flow rate of the adult circulatory system is about 10-4 m3/s. This is the
flow that passes through either the right or left side of the heart. The aorta, which
carries the blood from the left side of the heart, has a diameter of 2.5cm. What is
the flow velocity of the aorta?
1×10−4 ms
flow rate
=
= 0.204 ms = 20.4 cms
flow rate = Av → v =
A
π (0.0125m )2
3
9. Suppose that the aorta has a radius of about 1.25cm and that the typical blood
velocity is around 30 cm/s and that it has an average density of 1060 kg/m3.
a. What is the average blood velocity in the major arteries if the total cross
sectional area of the major arteries is 20cm2?
b. What is the total flow rate?
c. On the assumption that all the blood in the circulatory system goes
through the capillaries, what is the total cross sectional area of the
capillaries if the average velocity of the blood in the capillaries is
0.03cm/s?
d. If a typical capillary has a cross sectional area of 3x10-11 m2, about how
many capillaries are there in the human body?
e. What are the kinetic energy per unit volume for blood in the aorta, the
major arteries, and the capillaries?
f. If a capillary has an average length of 0.75mm what is the average time
that a red blood cell remains in a capillary?
flow rate 1.47 × 10 −4 ms
=
=
= 0.0735 ms = 7.35 cms
2
0.002m
Aarteries
3
a.
varteries
b.
flow rate = 1.47 × 10 − 4
m3
s
1.47 × 10 − 4 ms
=
= 0.49m 2
−4 m
3 × 10 s
3
c.
flow rate = Acap vcap → Acap
d.
# capillaries =
e.
KEaorta
0.49m 2
= 1.6 × 1010 = 16billion
3 × 10 −11 m 2
2
= 12 ρ blood vaorta
= 47.7 mJ3
2
KEarteries = 12 ρ blood varteries
= 2.9 mJ3
2
−5
KEcapillaries = 12 ρ blood vcapillaire
s = 4.8 × 10
f.
v=
J
m3
d
d 0.75 × 10 −3 m
→t = =
= 2 .5 s
t
v 3.0 × 10 − 4 ms
10. Estimate the amount by which the blood pressure P2 in the anterior tibial artery at
the top of the foot exceeds the blood pressure P1 in the aorta at the heart when the
body is reclining horizontally and when standing erect. Assume that the distance
between the aorta and the artery in the top of the foot is 1.35m and express your
answer in Pascal and in mm Hg, were 1 mmHg = 133.3 Pa.
a.
reclining → ∆h = 0;
from #9 vaorta = 0.30 ms and vartery = 0.0735 ms
(
)
2
2
2
2
Paorta + 12 ρ blood vaorta
= Partery + 12 ρ blood vartery
→ Paorta − Partery = 12 ρ blood vartery
− vaorta
=
−44.7 mN2 = −0.34mmHg ~ 0mmHg
b.
standing → ∆h = 1.35m;
from #9 vaorta = 0.30 ms and vartery = 0.0735 ms
2
2
Paorta + 12 ρ blood vaorta
+ ρ blood ghaorta = Partery + 12 ρ blood vartery
+ ρ blood ghartery →
(
)
2
2
Paorta − Partery = 12 ρ blood vartery
− vaorta
+ ρ blood g∆h = 10468.5 mN2 = 78.5mmHg