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Sect. 14.4: Buoyant Forces
Archimedes’ Principle
• Experimental facts:
1. Objects submerged (or partially submerged)
in a fluid APPEAR to “weigh” less than
in air.
2. When placed in a fluid, many objects float!
Both are examples of BUOYANCY
• See figure. An object submerged in a static fluid experiences
an upward BUOYANT FORCE B as well as the downward
gravitational force Fg
• This Buoyant Force B is exerted by a fluid on an immersed
object. Since the fluid is static, the object doesn’t move.
• So, the buoyant force balances the downward gravitational
force, so that Newton’s 2nd Law in the vertical direction
∑Fy = 0, is satisfied.
• By Newton’s 2nd Law, ∑Fy = 0,
the magnitude of the buoyant B
force must be equal (in magnitude)
to the downward gravitational
force. Fg
• B is the resultant force due to all
forces applied by the fluid
surrounding the object.
• The person who first realized how to
calculate B was an ancient Greek
mathematician named Archimedes
Archimedes
• C. 287 – 212 BC
• Greek mathematician,
physicist and engineer
• First to calculate the ratio of
a circle’s circumference to
it’s diameter (π)
• Calculated volumes of
various shapes
• Discovered the nature of the
buoyant force
• Inventor of many practical
devices:
– Catapults, levers, screws, etc.
• Buoyant Force: Occurs because the
pressure in a fluid increases with depth!
P = ρg h (fluid at REST!!)
• Experiments find that the magnitude of
the buoyant force B is always equal to the
weight of the fluid displaced by the object
– This is called Archimedes’ Principle
• Archimedes’s Principle does not refer to
the makeup of the object experiencing the
buoyant force
– The object’s composition is not important
since the buoyant force is exerted by
the fluid
• To understand Archimedes’ Principle
in more detail, consider the simple case of
a cubic shaped object (height h, cross
sectional area A) floating submerged in a
fluid of density ρfluid as in the figure.
• Use the definition of pressure in terms of area
& force: The pressure Ptop at the top of
•
•
•
•
the cube causes a downward force
Fdown = PtopA on the upper face.
The pressure Pbot at the bottom of the
cube causes an upward force Fup =
PbotA on the lower face.
Total force the fluid exerts on the cube: B = (Pbot – Ptop)A
(1)
Use the dependence of pressure on depth: (Pbot – Ptop) = ρfluidgh (2)
Combining (1) & (2) gives B = ρfluidghA = ρfluidgV = Mg.
Mg is the weight of the FLUID displaced by the object!
Archimedes Principle
 The total (upward) buoyant force B on an object of
volume V completely or partially submerged in a fluid
with density ρfluid:
B = ρfluidVg
(1)
ρfluidV  M  Mass of fluid which would take up same
volume as object, if object were not there. (Mass of fluid
that used to be where object is!)
 Upward buoyant force
B = Mg
(2)
B = weight of fluid displaced by the object!
(1) or (2)  Archimedes Principle
Proved for cylinder. Can show valid for any shape
Archimedes Principle
B
B
B
B
B
• Archimedes Principle says that
if an object of volume Vobj is
totally submerged in a fluid of
density ρfluid
• The upward buoyant force is
B = ρfluidgVobj
• Of course, the downward gravitational force is
Fg = Mg = ρobjgVobj
• If the object is floating submerged & not moving, by
Newton’s 2nd Law in the vertical direction,
∑Fy = 0, the net vertical force must be zero:
B - Fg = (ρfluid – ρobj)gVobj = 0
• Of course, objects don’t always float
motionless when submerged! From the
previous slide, for this to happen, we
must have: (ρfluid – ρobj)gVobj = 0
or, ρfluid = ρobj. That is, the object &
the fluid must have the same density!
• If the object’s density is less than the
fluid density, (ρfluid > ρobj), by
Newton’s 2nd Law, ∑Fy = Ma, the
object accelerates upward, as in Fig. a
• If the object’s density is greater than the fluid density, (ρfluid < ρobj),
by Newton’s 2nd Law, ∑Fy = -Ma, the object accelerates
downward (it sinks!), as in Fig. b.
• So, Archimedes’ Principle tells us that:
The direction of the motion of an object in a fluid is
determined only by the densities of the fluid and the object
Archimedes Principle: Floating Objects
• Consider an object of volume Vobj &
Free body diagram 
density ρobj floating (partially
submerged) at top of a fluid of
density ρfluid.
See figure:
• Object is not moving so Newton’s 2nd Law in the vertical
direction, ∑Fy = 0, tells us that the upward buoyant force B is
balanced by the downward force of gravity Fg. This gives:
B - Fg = ρfluidgVfluid - ρobjgVobj = 0 or ρfluidVfluid = ρobjVobj
• Here, the volume of the fluid displaced by the object, Vfluid
isn’t the object volume but is the portion of
the volume of the object beneath the fluid
Vfluid
level. The last relation can be rewritten as: 
Vobj
obj

fluid
• Archimedes Principle: Floating objects in
equilibrium:  ∑Fy = B - Fg = 0
Gives: ρfluidVfluid = ρobjVobj
or
Vfluid obj

f = Vobj fluid
f  Fraction of the volume of floating object which is
submerged.
 Archimedes’ Principle: The fraction of a floating
object which is submerged is equal to the ratio of the
density of the object to the density of the fluid.