Physics 106 Lesson #6
Density, Buoyancy and
Archimedes’ Principle
Dr. Andrew Tomasch
2405 Randall Lab
atomasch@umich.edu
Last Time: Work and Energy
Energy ≡ the ability to do work
Work is a scalar
• Definition: Work is the action of a force through
a distance:
Example: You exert a horizontal
force F to push an object an object
through a horizontal displacement d
W Fd
F
d
Work equals parallel force times distance.
Work by Conservative Forces
•
•
The work a conservative force does on an object in
moving it from a point A to a point B is path
independent . It depends only on the path’s end
points.
For conservative
it is possible
Conservative Forces: forces
to define a potential
energy function
1. Gravity
D
2. Elastic (spring) Force
E
3. Electromagnetic Force
The work done by
gravity is the same
for path A-B-C-D-E
as it is for path A-E
B
A
C
Equivalently, the work done on an object traveling in a closed loop is zero.
Work Done by the Gravitational Force
Wgravity  mg (hinitial  hfinal )
 weight  vertical distance
PEgrav  mgh
hinitial
mg
hfinal
mg
PE=0
Wgravity  mghinitial  mghfinal  PEinitial  PEfinal
 difference in potential energy
The work done by gravity is just the difference in potential energy between where
you start and where you end in space, and only the difference in height matters
Energy Accounting
The Work-Energy Theorem is
energy conservation and turns
mechanics into accounting.
The Work-Energy
“Balance Sheet”
KEi  PEi  Wnc  KEf  PEf
The Work-Energy Balance Sheet
automatically accounts for the
work done by gravity as differences
In gravitational potential energy.
Wnc ≡ Work done by
nonconservative
forces other than
gravity → Friction
•KE ≡ Cash
•PE ≡ Deposits
•Wfriction ≡ Taxes!
A Bob-Sled Run
The bob-sled exchanges
gravitational potential energy for
kinetic energy as it slides
without friction down the hill.
Total energy, the sum of kinetic
and potential energy, remains
constant at every point along the
motion.
Density
• Density  (Greek letter rho):

Caution
Quiz
Ahead
mass
volume
The mass of uniform
object is its density
times its volume: m = V
water = 1000 kg/m3 (SI)
water = 1 g/cm3 (CGS)
The specific gravity is density expressed
in units of the density of water. The
specific gravity of x ≡ x /water
Concept Test #1
A ton of feathers and a ton of bricks
have the same mass. The feathers
make _____ pile than the bricks.
A. a bigger
B. a smaller
C. the same
m
   m  V
V
Buoyancy
T +
• Before: The upward
tension in the string equals
the downward weight of
the block →scale reads
true weight
W
• After: The upward buoyant
force helps to balance the FB
T +
downward weight,
reducing the tension in the
string, which is the
apparent weight of the
block → apparent weight is
W
less than the true weight
Read T
Demonstration
apparent weight  W  FB  true weight
Archimedes’ Principle
FB  (  fluidVobject ) g
Magnitude of
Buoyant Force
Weight of
Displaced Fluid
•The buoyant force acting on
a body is equal to the weight
of the fluid it displaces
Eureka! Eureka!
Caution
Quiz
Ahead
• Archimedes deduced the
equality between the
weight of his body and
the weight of the water
that had overflowed from
his bath tub
• He yelled triumphantly,
“Eureka! Eureka!” (“I
have found it” in Greek)
Don’t try this at home!
Concept Test #2
FBHg  mg (Floats)
FBH O  mg (Sinks)
2
A lead ball of mass m floats in liquid
mercury and sinks in water. Which
statement is true about the buoyant force
acting on the lead ball?
A. It is greater when the ball is floating in
the mercury.
B. It is greater when the ball is totally
submerged in the water.
C. It is the same in both cases.
In mercury the buoyant force is equal to the weight and the ball floats.
In water the buoyant force is less than the weight and the it sinks.
Archimedes’ Principle Applied
•
•
•
•
FB = ρfluid Vobject g
W = ρobject Vobject g
object sinks if ρobject > ρfluid
object floats if ρobject < ρfluid
FB
ρobject
Vobject
ρfluid
W
Completely immerse an
object of volume Vobject
with density ρobject in a
fluid of density ρfluid
Floating Objects
• An object floats with some portion of
its volume protruding from the liquid
• When Floating in Equilibrium:
FB  W  liquid gVin  object gVobject
object
Vin

Vobject
liquid
FB
Vin
W