strong (ams4684) – HW14 – ditmire – (58216)
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001
(part 1 of 2) 10.0 points
In 1657, Otto von Guericke, inventor of the
air pump, evacuated a sphere made of two
brass hemispheres. Two teams of eight horses
each could pull the hemispheres apart only on
some trials, and then only “with the greatest
of di
ffi
culty.”
F
F
P
R
P
a
If
P
a
is the atmospheric pressure,
P is
the pressure inside the hemispheres and R
is the radius of the hemispheres, then what is
the force F required to pull the hemispheres
apart?
1.
F
= 4
π
R
2
(
P
a

P
)
2.
F
=
π
R
2
(
P
a

P
)
correct
3.
F
=
4
π
R
3
3
(
P
a

P
)
4.
F
=
π
R
2
2
(
P
a

P
)
5.
F
= 2
π
R
2
(
P
a

P
)
6.
F
=
2
π
R
3
3
(
P
a

P
)
Explanation:
Force
=
(Di
ff
erence
in
pressure
on
2
sides)
×
(Area). We must choose the area care
fully. Each team of horses is pulling in the
z
direction with a force
F
.
The hemispheres
will come apart only when
F
is
≥
the
z
com
ponent of the net force on each hemisphere
due to the pressure di
ff
erence (see figure). We
must therefore pick the e
ff
ective area which is
perpendicular to the
z
direction. If you stand
far away on the
z
axis and look at the hemi
sphere, you see a circle of area
π
R
2
. Hence,
F
=
π
R
2
(
P
a

P
)
.
θ
φ
x
y
dA
F
z
Hemisphere of radius R
dF
If we wanted to justify this more rigorously,
we would have to examine the amount of force
dF
caused by the pressure di
ff
erence acting on
a small area of the hemisphere
dA
(see figure).
We would then take the
z
component of this
and integrate this amount over the surface
of the hemisphere (a double integral).
The
result is the same as our “intuitive” argument
above.
002
(part 2 of 2) 10.0 points
Determine
the
force
if
P
=
0
.
0977
atm
and
R
= 0
.
23 m.
Atmosphesic pressure is
1
.
013
×
10
5
Pa.
Correct answer: 15
.
1903 kN.
Explanation:
Let :
P
= 0
.
0977 atm = 9897
.
01 Pa
,
R
= 0
.
23 m
,
and
P
a
= 1
.
013
×
10
5
Pa
.
F
=
π
R
2
(
P
a

P
)
=
π
(0
.
23 m)
2
×
(1
.
013
×
10
5
Pa

9897
.
01 Pa)
×
1 kN
1000 N
=
15
.
1903 kN
.
003
(part 1 of 2) 10.0 points
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strong (ams4684) – HW14 – ditmire – (58216)
2
In a car lift, compressed air exerts a force on a
piston with a radius of 1
.
88 cm. This pressure
is transmitted to a second piston with a radius
of 14
.
8 cm.
How large a force must the compressed air
exert to lift 10700 N car?
Correct answer: 172
.
654 N.
Explanation:
Let :
r
1
= 1
.
88 cm
,
r
2
= 14
.
8 cm
,
and
F
2
= 10700 N
.
P
1
=
P
2
F
1
=
F
2
A
2
·
A
1
=
F
2
π
r
2
2
·
(
π
r
2
1
)
=
F
2
·
r
2
1
r
2
2
= 10700 N
·
(1
.
88 cm)
2
(14
.
8 cm)
2
=
172
.
654 N
.
004
(part 2 of 2) 10.0 points
What pressure produces this force?
Neglect
the weight of the pistons.
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