Transcript Specific Gravity – The REST of the story…

Specific Gravity – The REST of
the story…
AKA
“Owl” Archimedes saved his life
by taking a bath!
Hiero of Syracuse (really!)…
challenged Archimedes to determine if his
new crown was made of pure gold.
Archimedes realized (eventually)
that the crown would weigh LESS
when weighed in water!
What is going on?
FT
Fg
Fg = FT
Has the gravity stopped
pulling as hard on the crown?
Fg
No! So…
Fg
FT
Fg > FT
Therefore…
F ? FT
Fg
Fg = FT + F?
This new force is called…
F ? FT
Fg
Fg = FT + F?
the BUOYANT force!
F ? FT
Fg
Fg = FT + F?
the BUOYANT force!
FB FT
Fg
Fg = FT + FB
What causes the BUOYANT
force?
FB FT
Fg
Fg = FT + FB
To keep this simple, let’s look
at a cylinder in water.
Does the fluid exert any force
on the cylinder?
Because h2 > h1, the force up is
greater than the force down.
FB = |Fup| - |Fdown|
= PbotAbot - PtopAtop
= ρgh2Abot – ρgh1Atop
{ Abot = Atop }
h2
h1
= ρgA(h)
= ρgV
h = h 2 - h1
Since the fluid was at rest…
|FB| = Fgfluid
= ρfluidgV
= mfluidg
h2
h1
Remember that:
{ ρ = m/V }
h = h 2 - h1
{ m = ρV }
So | FB |= | Fgfluid |
h2
h1
h = h 2 - h1
So | FB |= | Fgfluid |
h2
h1
h = h 2 - h1
Let’s consider a still fluid:
The fluid was there already, and
if we encase it in a plastic bag
that has the same density as the
fluid, it will continue to be there.
| FB |= | Fgfluid |
So the buoyant force on an
object equals the weight of the
fluid displaced, regardless of the
shape of the object.
Now let’s look at some situations
(and assume that down is negative):
In all cases, Fnet = Fgobject + FB
If Fnet is positive, then the object is
positively buoyant and it will rise
upward.
Fnet =
FB
Fg
Fgobject
+
FB
If Fnet is negative, then the object is
negatively buoyant and it will sink
downward.
Fnet =
FB
Fg
Fg
object
+
FB
If Fnet is zero, then the object is
neutrally buoyant and it will neither
rise nor sink while it is under the
surface.
Fg
Fnet = 0 N
Fnet =
FB
Fg
object
+
FB
Please note, FB was ALWAYS
______________
FB
FB
FgFB Fg
Fg
Please note also, if Fnet were positive,
the object would simply __________ if
it were not held below the surface.
(i.e. The object would displace enough
fluid such that Fnet = ________.
What does all of this have to
do with SG?
FB FT
Fg
Fg = FT + FB
Remember that SG = ρo/ρf
FB FT
Fg
Fg = FT + FB
So…
SG = ρo/ρf
SG = mo/Vo/mf/Vf
SG = mo/mf
SG = mog/mfg
SG = Fgo/Fgf
SG = Fgo/|FB|
SG = Fgo/ΔFgo
So…
Archimedes realized that an
object’s specific gravity (relative to the
fluid used) equaled the weight of the
object out of the water divided by the
change of weight that occurred when
the object was submerged in the fluid!
SG = Fg/ΔFg
Note: SG = Fgo/ΔFgo
SG = Fgo/|Fgo – Fgo’|
SG = ρoVo/ρfVf
SG = ρo/ρf
If the fluid involved is water at 4ºC,
then this ratio is the specific gravity
for the object, not just a relative
SG.
For floating objects, objects
sink enough that:
ρ =
o
|FB| = |Fg|
0.2 g/mL
ρf.Vdisplaced.g=ρo.Vo.g
Vdisplaced/Vo = ρo/ρf
So 80% of the ball is above
the water.
Hydrometers follow:
ρ1/ρ2 = V2/V1
ρ1/ρ2 = A2h2/A1h1
ρ1/ρ2 = h2/h1
Ex. .8/1 = .8 /1
If placed into a denser fluid: .8/2 =
x /1
Hydrometers follow:
ρ1/ρ2 = V2/V1
ρ1/ρ2 = A2h2/A1h1
ρ1/ρ2 = h2/h1
Ex. .8/1 = .8 /1
.8/2 = x /1
x=
0.4