1420
Chapter 14
14.75.
(a)
I
DENTIFY
:
Apply Newton°s 2nd law to the crown. The buoyancy force is given by Archimedes° principle.
The target variable is the ratio
c
w
/
ρ ρ
(c
crown,
=
w
water).
=
S
ET
U
P
:
The freebody diagram for the crown is given in Figure 14.75.
E
XECUTE
:
y
y
F
ma
=
∑
0
T
B
w
+
−
=
T
fw
=
w
c
,
B
V g
ρ
=
where
w
density
ρ
=
of water,
c
volume
V
=
of crown
Figure 14.75
Then
w
c
0.
fw
V g
w
ρ
+
−
=
w
c
(1
)
f w
V g
ρ
−
=
Use
c
c
,
w
V g
ρ
=
where
c
density
ρ
=
of crown.
c
c
w
c
(1
)
f
V g
V g
ρ
ρ
−
=
c
w
1
,
1
f
ρ
ρ
=
−
as was to be shown.
0
f
→
gives
c
w
/
1
ρ
ρ
=
and
0.
T
=
These values are consistent. If the density of the crown equals the density of
the water, the crown just floats, fully submerged, and the tension should be zero.
When
1,
f
→
c
w
ρ
ρ
>>
and
.
T
w
=
If
c
w
ρ
ρ
>>
then
B
is negligible relative to the weight
w
of the crown and
T
should equal
w
.
(b)
±apparent weight² equals
T
in rope when the crown is immersed in water.
,
T
fw
=
so need to compute
f
.
3
3
c
19.3
10
kg/m ;
ρ
=
×
3
3
w
1.00
10
kg/m
ρ
=
×
c
w
1
1
f
ρ
ρ
=
−
gives
3
3
3
3
19.3
10
kg/m
1
1.00
10
kg/m
1
f
×
=
×
−
19.3
1/(1
)
f
=
−
and
0.9482
f
=
Then
(0.9482)(12.9 N)
12.2 N.
T
fw
=
=
=
(c)
Now the density of the crown is very nearly the density of lead;
3
3
c
11.3
10
kg/m .
ρ
=
×
c
w
1
1
f
ρ
ρ
=
−
gives
3
3
3
3
11.3
10
kg/m
1
1.00
10
kg/m
1
f
×
=
×
−
11.3
1/(1
)
f
=
−
and
0.9115
f
=
Then
(0.9115)(12.9 N)
11.8 N.
T
fw
=
=
=
E
VALUATE
:
In part (c) the average density of the crown is less than in part (b), so the volume is greater.
B
is
greater and
T
is less. These measurements can be used to determine if the crown is solid gold, without damaging
the crown.
14.76.
I
DENTIFY
:
Problem 14.75 says
object
fluid
1
1
f
ρ
ρ
=
−
, where the apparent weight of the object when it is totally
immersed in the fluid is
fw
.
S
ET
U
P
:
For the object in water,
water
water
/
f
w
w
=
and for the object in the unknown fluid,
fluid
fluid
/
f
w
w
=
.
E
XECUTE
:
(a)
steel
fluid
fluid
,
ρ
w
ρ
w
w
=
−
steel
fluid
water
ρ
w
ρ
w
w
=
−
. Dividing the second of these by the first gives
fluid
fluid
water
water
.
ρ
w
w
ρ
w
w
−
=
−
(b)
When
fluid
w
is greater than
water,
w
the term on the right in the above expression is less than one, indicating that
the fluids is less dense than water, and this is consistent with the buoyant force when suspended in liquid being less
than that when suspended in water. If the density of the fluid is the same as that of water
fluid
w
=
water
w
, as
expected. Similarly, if
fluid
w
is less than
water
w
, the term on the right in the above expression is greater than one,
indicating that the fluid is denser than water.
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Fluid Mechanics
1421
(c)
Writing the result of part (a) as
fluid
fluid
water
water
1
,
1
ρ
f
ρ
f
−
=
−
and solving for
fluid
,
f
(
)
(
) (
)
fluid
fluid
water
water
1
1
1
1.220 0.128
0.844
84.4%.
ρ
f
f
ρ
=
−
−
=
−
=
=
E
VALUATE
:
Formic acid has density greater than the density of water. When the object is immersed in formic
acid the buoyant force is greater and the apparent weight is less than when the object is immersed in water.
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 Spring '06
 Buchler
 Physics, Buoyancy, Force, ρc, Vtot

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