Download Kinetic Energy Kinetic Energy Potential Energy

Kinetic Energy
Kinetic Energy
• Kinetic energy is the energy associated
with an object’s motion.
– Doing work on an object increases its kinetic
energy.
– Work done = change in kinetic energy
1
KE = mv 2
2
Potential Energy
• If work is done but no kinetic
energy is gained, we say that
the potential energy has
increased.
– For example, if a force is
applied to lift a crate, the
gravitational potential energy
of the crate has increased.
– The work done is equal to the
force (mg) times the distance
lifted (height).
– The gravitational potential
energy equals mgh.
• Negative work is the work done by a force
acting in a direction opposite to the object’s
motion.
– For example, a car skidding to a stop
– What force is acting to slow the car?
Work is done on a large crate to tilt the crate
so that it is balanced on one edge, rather than
sitting squarely on the floor as it was at first.
Has the potential energy of the crate
increased?
a) Yes
b) No
Yes. The weight of the crate
has been lifted slightly. If it is
released it will fall back and
convert the potential energy
into kinetic energy.
1
Potential Energy
• The term potential energy
implies storing energy to
use later for other
purposes.
Potential Energy
• An elastic force is a force that results from
stretching or compressing an object.
• Elastic potential energy is the energy gained
when work is done to stretch a spring.
– The spring constant, k, is a number describing the
stiffness of the spring.
– For example, the
gravitational potential
energy of the crate can be
converted to kinetic energy
and used for other purposes.
Potential Energy
• The increase in elastic potential energy is
equal to the work done by the average force
needed to stretch the spring.
• Conservative forces are forces for which the
energy can be completely recovered.
– Gravity and elastic forces are conservative.
– Friction is not conservative.
PE = work done = average force × distance
average force =
1
kx
2
1
PE = kx 2
2
2
Conservation of Energy
Energy Transformation for a Pendulum
• Conservation of energy
means the total energy
(the kinetic plus potential
energies) of a system
remain constant.
– Energy is conserved if
there are no forces doing
work on the system.
If W = 0, then TME = KE+PE
Total Mechanical Energy
A lever is used to lift a rock. Will the
work done by the person on the lever be
greater than, less than, or equal to the
work done by the lever on the rock?
a)
b)
c)
d)
Greater than
Less than
Equal to
Unable to tell
from this graph
– Work done in pulling a sled up a hill produces
an increase in potential energy of the sled and
rider.
– This initial energy is converted to kinetic
energy as they slide down the hill.
The work done by the person can never be less than the work
done by the lever on the rock. If there are no dissipative forces
they will be equal. This is a consequence of the conservation of
energy.
3
– Any work done by frictional forces is negative.
– That work removes mechanical energy from
the system.
A sled and rider with a total mass of 40 kg are perched at the
top of the hill shown. Suppose that 2000 J of work is done
against friction as the sled travels from the top (at 40 m) to
the second hump (at 30 m). Will the sled make it to the top of
the second hump if no kinetic energy is given to the sled at the
start of its motion?
a)
b)
c)
yes
no
It depends.
Yes.
The
difference
between the potential
energy at the first point
and the second point,
plus loss to friction is
less than the kinetic
energy given at the start
of the motion.
A sled and rider with a total mass of 40 kg are perched at the
top of the hill shown. Suppose that 2000 J of work is done
against friction as the sled travels from the top (at 40 m) to
the second hump (at 30 m). What is the maximum height that
the second hump could be in order for the sled to reach the
top?
a)
b)
c)
d)
e)
Energy Transformation on a Roller Coaster
10 m
35 m
32 m
40 m
30 m
35 m. This additional
height would allow the
body to reach this
point just as it has lost
all of its kinetic
energy.
If W = 0, then TME = KE+PE
4
DISSIPATIVE forces acting on
Downhill Skiing
If W = 0, then TME = KE+PE
But in this case W ≠ 0
+W
Internal
work
Work done
by external
forces
W
KE
+
PE
W
Internal
work
=
TME
-W
Work done
by external
forces
TME = PE + KE
Energy and Oscillations
+W
Work done
by external
forces
Internal
work
W
KE
+
PE
W
Internal
work
-W
Work done
by external
forces
Why does a
swinging
pendant
return to the
same point
after each
swing?
TME = KE + PE ± Wext
my system
5
Energy and Oscillations
The force
does work to
move the ball.
This increases
the ball’s
energy,
affecting its
motion.
Simple Machines, Work, and
Power
• A simple machine multiplies the effect of an
applied force.
– For example, a pulley :
A small tension applied to
one end delivers twice as
much tension to lift the box.
The small tension acting
through a large distance
moves the box a small
distance.
Simple Machines, Work, and
Power
• A simple machine multiplies the effect of an
applied force.
– For example, a lever :
A small force applied to one
end delivers a large force to
the rock.
The small force acting
through a large distance
moves the rock a small
distance.
• The mechanical advantage of a simple machine is
the ratio of the output force to the input force.
– For the pulley example, the mechanical advantage is 2.
• Work is equal to the force applied times the distance
moved.
– Work = Force x Distance:
– Work output = Work input
W=Fd
units: 1 joule (J) = 1 Nm
6
• Only forces parallel to the motion do work.
• Power is the rate of doing work
– Power = Work divided by Time:
P=W/t
units: 1 watt (W) = 1 J / s
A string is used to pull a wooden block
across the floor without accelerating the
block. The string makes an angle to the
horizontal. Does the force applied via the
string do work on the block?
a)
b)
c)
d)
Yes, the force F
does work.
No, the force F
does no work.
Only part of the
force F does work.
You can’t tell from
this diagram.
Only the part of the force that is parallel to the distance moved
does work on the block. This is the horizontal part of the force F.
If there is a frictional force opposing the
motion of the block, does this frictional
force do work on the block?
Does the normal force of the floor pushing
upward on the block do any work?
a)
a)
b)
c)
d)
Yes, the frictional
force does work.
No, the frictional
force does no
work.
Only part of the
frictional force
does work.
You can’t tell from
this diagram.
Since the frictional force is antiparallel to the distance moved,
it does negative work on the block.
b)
c)
d)
Yes, the normal
force does work.
No, the normal
force does no
work.
Only part of the
normal force does
work.
You can’t tell from
this diagram.
Since the normal force is perpendicular to the distance
moved, it does no work on the block.
7
A force of 50 N is used to drag a crate 4 m
across a floor. The force is directed at an
angle upward from the crate as shown. What
is the work done by the horizontal component
a)
120 J
of the force?
b) 160 J
c)
d)
e)
200 J
280 J
0J
The horizontal
component of force is
40 N and is in the
direction of motion:
W=F·d
= (40 N) · (4 m)
= 160 J.
What is the total work done by the 50-N
force?
a)
b)
c)
d)
e)
120 J
160 J
200 J
280 J
0J
Only the component of
force in the direction of
motion does work:
W=F·d
= (40 N) · (4 m)
= 160 J.
What is the work done by the vertical
component of the force?
a)
b)
c)
d)
e)
120 J
160 J
200 J
280 J
0J
The vertical component
of force is 30 N but isn’t
in the direction of
motion:
W=F·d
= (30 N) · (0 m)
= 0 J.
What are
temperature
and heat?
Are they the
same?
What causes
heat?
8
What Is Temperature?
How do we
measure
temperature?
What are we
actually
measuring?
• The first widely used
temperature scale was
devised by Gabriel Fahrenheit.
• Another widely used scale was
devised by Anders Celsius.
• The Celsius degree is larger
than the Fahrenheit degree:
the ratio of Fahrenheit degrees
to Celsius degrees is 180/100,
or 9/5.
5
TC = (TF − 32)
9
9
TF = TC + 32
5
Temperature and Its
Measurement
• If two objects are in contact with one another
long enough, the two objects have the same
temperature.
• This begins to define temperature, by
defining when two objects have the same
temperature.
– When the physical properties are no longer
changing, the objects are said to be in thermal
equilibrium.
– Two or more objects in thermal equilibrium
have the same temperature.
– This is the zeroth law of thermodynamics.
• The zero point on the
Fahrenheit scale was based
on the temperature of a
mixture of salt and ice in a
saturated salt solution.
• The zero point on the Celsius
scale is the freezing point of
water.
• Both scales go below zero.
• Is there such a thing as
absolute zero?
• They are both equal at -40°.
9
What is absolute zero?
• If the volume of a gas is
kept constant while the
temperature is increased,
the pressure will increase.
• This can be used as a
means of measuring
temperature.
• A constant-volume gas
thermometer allows the
pressure to change with
temperature while the
volume is held constant.
• The difference in height of
the two mercury columns
is proportional to the
pressure.
Can anything ever get colder than 0 K?
No.
Can absolute zero ever be reached?
No.
TK = TC + 273.2
• We can then plot the pressure of a gas as a function of the
temperature.
• The curves for different gases or amounts are all straight lines.
• When these lines are extended backward to zero pressure,
they all intersect at the same temperature, -273.2°°C.
• Since negative pressure has no meaning, this suggests that the
temperature can never get lower than -273.2°°C, or 0 K (kelvin).
TK = TC + 273.2
Heat and Specific Heat
Capacity
What happens when objects or
fluids at different temperatures
come in contact with one
another?
The colder object gets hotter, and
the hotter object gets colder, until
they both reach the same
temperature.
What is it that flows between the
objects to account for this?
• We use the term heat for this
quantity.
10
Heat and Specific Heat
Capacity
• Heat flow is a form of energy transfer between
objects.
• One-hundred grams of room-temperature water is
more effective than 100 grams of room-temperature
steel shot in cooling a hot cup of water.
Steel has a lower specific
heat capacity than water.
The
specific heat capacity
of a material is the relative
amount of heat needed to
raise its temperature.
• When two objects at different temperatures are
placed in contact, heat will flow from the object with
the higher temperature to the object with the lower
temperature.
• Heat added increases temperature, and heat
removed decreases temperature.
• Heat and temperature
are not the same.
• Temperature is a
quantity that tells us
which direction the
heat will flow.
• The specific heat capacity of a material is the
quantity of heat needed to change a unit mass of
the material by a unit amount in temperature.
– For example, to change 1 gram by 1 Celsius degree.
– It is a property of the material, determined by experiment.
– The specific heat capacity of water is 1 cal/g⋅C°: it takes
1 calorie of heat to raise the temperature of 1 gram of
water by 1°C.
• We can then calculate how much heat must be
absorbed by a material to change its temperature
by a given amount:
Q = mc∆T where Q = quantity of heat
m = mass
c = specific heat capacity
∆T = change in temperature
Phase Changes and Latent
Heat
• When an object goes through a change of phase
or state, heat is added or removed without
changing the temperature. Instead, the state of
matter changes: solid to liquid, for example.
• The amount of heat needed per unit mass to
produce a phase change is called the latent heat.
– The latent heat of fusion of water corresponds to the
amount of heat needed to melt one gram of ice.
– The latent heat of vaporization of water corresponds to
the amount of heat needed to turn one gram of water into
steam.
11
If the specific heat capacity of ice is
0.5 cal/g⋅C°, how much heat would have
to be added to 200 g of ice, initially at
a temperature of -10°C, to raise the ice
to the melting point?
a)
b)
c)
d)
1,000 cal
2,000 cal
4,000 cal
0 cal
m = 200 g
c = 0.5 cal/g⋅C°
T = -10°C
Q = mc∆T
= (200 g)(0.5 cal/g⋅C°)(10°C)
= 1,000 cal
(heat required to raise the temperature)
Joule’s Experiment
and the First Law of
Thermodynamics
• Rumford noticed that cannon
barrels became hot during
drilling.
• Joule performed a series of
experiments showing that
mechanical work could raise
the temperature of a system.
• In one such experiment, a
falling mass turns a paddle in
an insulated beaker of water,
producing an increase in
temperature.
If the specific heat capacity of ice is
0.5 cal/g⋅C°, how much heat would have
to be added to 200 g of ice, initially at
a temperature of -10°C, to completely
melt the ice?
Lf = 80 cal/g
a)
b)
c)
d)
1,000 cal
14,000 cal
16,000 cal
17,000 cal
Q = mLf
= (200 g)(80 cal/g)
= 16,000 cal
(heat required to melt the ice)
Total heat required to raise the ice to 0 °C
and then to melt the ice is:
1,000 cal + 16,000 cal = 17,000 cal = 17 kcal
Joule’s Experiment
and the First Law of
Thermodynamics
• Joule’s experiments led to Kelvin’s statement of the first law of
thermodynamics.
– Both work and heat represent transfers of energy into or out of a system.
– If energy is added to a system either as work or heat, the internal energy
of the system increases accordingly.
• The increase in the internal
energy of a system is equal
to the amount of heat added
to a system minus the
amount of work done by the
system. ∆U = Q - W
12
Joule’s Experiment
and the First Law of
Thermodynamics
• This introduced the concept of the internal energy of a
system.
– An increase in internal energy may show up as an increase in
temperature, or as a change in phase, or any other increase in the
kinetic and/or potential energy of the atoms or molecules making up
the system.
– Internal energy is a property of the system uniquely determined by the
state of the system.
• The internal energy of the system is the sum of the
kinetic and potential energies of the atoms and
molecules making up the system.
A hot plate is used to transfer 400
cal of heat to a beaker containing ice
and water; 500 J of work are also
done on the contents of the beaker
by stirring. How much ice melts in
this process?
a)
b)
c)
d)
0.037 g
0.154 g
6.5 g
27.25 g
Lf = 80 cal/g
= (80 cal/g)(4.19 J/cal)
= 335 J/g
∆U = mLf
m = ∆U / Lf
= (2180 J) / (335 J/g)
= 6.5 g
A hot plate is used to transfer 400
cal of heat to a beaker containing ice
and water; 500 J of work are also
done on the contents of the beaker
by stirring. What is the increase in
internal energy of the ice-water
mixture?
a)
b)
c)
d)
900 J
1180 J
1680 J
2180 J
W = -500 J
Q = 400 cal
= (400 cal)(4.19 J/cal)
= 1680 J
∆U = Q - W
= 1680 J - (-500 J)
= 2180 J
Gas Behavior and The First Law
Consider a gas in a cylinder with a movable piston.
If the piston is pushed inward by an external force, work
is done on the gas, adding energy to the system.
The force exerted on the piston
by the gas equals the pressure
of the gas times the area of the
piston: F = PA
The work done equals the force
exerted by the piston times the
distance the piston moves:
W = Fd = (PA)d = P∆V
13
• If the gas is being compressed, the change in volume
is negative, and the work done is negative.
– Work done on the system is negative.
– Negative work increases the energy of the system.
• If the gas is expanding, positive work is done by the
gas on its surroundings, and the internal energy of the
gas decreases.
• An ideal gas is a gas for which the forces between atoms are
small enough to be ignored.
– For an ideal gas, absolute
temperature is directly
related to the average
kinetic energy of the
molecules of the system.
– Most gases behave
approximately as
ideal gases.
• If the process is adiabatic, no heat flows into or out of the gas.
• Even though no heat is added, the temperature of a gas will
increase in an adiabatic compression, since the internal energy
increases.
• In an isothermal process, the temperature does not change.
– The internal energy must be constant.
– The change in internal energy, ∆U, is zero.
– If an amount of heat Q is added to the gas, an equal amount of work W
will be done by the gas on its surroundings, from ∆U = Q - W.
• In an isobaric process, the pressure of the gas remains
constant.
– The internal energy increases as the gas is heated, and so does the
temperature.
– The gas also expands, removing some of the internal energy.
– Experiments determined that the pressure, volume, and absolute
temperature of an ideal gas are related by the equation of state:
PV = NkT
where N is the number of molecules
and k is Boltzmann’s constant.
What process makes a hot-air balloon
rise?
• When gas is heated in a hotair balloon, the pressure, not
the temperature, remains
constant.
• The gas undergoes an
isobaric expansion.
• Since the gas has expanded,
the density has decreased.
• The balloon experiences a
buoyant force because the
gas inside the balloon is less
dense than the surrounding
atmosphere.
14
The Flow of Heat
• There are three basic processes for
heat flow:
– In conduction, heat flows through a material
when objects at different temperatures are placed
in contact with one another.
–Conduction
–Convection
–Radiation
(Conduction Contd.)
– The rate of heat flow
depends on the temperature
difference between the
objects.
– It also depends on the
thermal conductivity of the
materials, a measure of how
well the materials conduct
heat.
– For example, a metal block at
room temperature will feel
colder than a wood block of
the exact same temperature.
– The metal block is a better
thermal conductor, so heat
flows more readily from your
hand into the metal.
– Since contact with the metal
cools your hand more rapidly,
the metal feels colder.
– In convection, heat is transferred by the motion of a fluid
containing thermal energy.
• Convection is the main method of heating a house.
• It is also the main method heat is lost from buildings.
15
– In radiation, heat energy is
transferred by electromagnetic
waves.
• The electromagnetic waves involved
in the transfer of heat lie primarily in
the infrared portion of the spectrum.
• Unlike conduction and convection,
which both require a medium to
travel through, radiation can take
place across a vacuum.
• For example, the evacuated space
in a thermos bottle.
• The radiation is reduced to a
minimum by silvering the facing
walls of the evacuated space.
What heat-flow processes are involved in
solar collectors?
What process makes a car’s interior
heat up when parked in the sun?
Why are houses insulated with
material in the walls instead of just
empty space?
Why is this insulated material often
foil-backed?
Is a light-colored roof or a darkcolored roof more energy efficient?
What heat-flow processes are involved in
the greenhouse effect?
16
Pressure
explains...
Why does a small
woman wearing
high-heel shoes sink
into soft ground
more than a large
man wearing large
shoes?
floating
objects and
moving
fluids
Pressure
• The man weighs more, so he
exerts a larger force on the ground.
• The woman weighs less, but the
force she exerts on the ground is
spread over a much smaller area.
• Pressure takes into account both
force and the area over which the
force is applied.
– Pressure is the ratio of the force to the
area over which it is applied:
– Units: 1 N/m2 = 1 Pa (pascal)
– Pressure is the quantity that
determines whether the soil will yield.
P=
Pressure and Pascal’s Principle
Pascal’s Principle
• What happens inside a fluid
when pressure is exerted
on it?
• Does pressure have a
direction?
• Does it transmit a force to
the walls or bottom of a
container?
F
A
17
Pascal’s Principle
• Fluid pushes outward uniformly in all
directions when compressed.
• Any increase in pressure is transmitted
uniformly throughout the fluid.
• Pressure exerted on a piston extends
uniformly throughout the fluid, causing it
to push outward with equal force per
unit area on the walls and the bottom of
the cylinder.
• This is the basis of Pascal’s Principle:
– Any change in the pressure of a fluid
is transmitted uniformly in all
directions throughout the fluid.
A force of 10 N is
applied to a circular
piston with an area of 2
cm2 in a hydraulic jack.
The output piston for
the jack has an area of
100 cm2. What is the
pressure in the fluid?
a)
b)
c)
d)
0.002 Pa
5 Pa
10 Pa
50 kPa
How does a
hydraulic jack
work?
• A force applied to a piston with a small area can produce a
large increase in pressure in the fluid because of the small area
of the piston.
• This increase in pressure is transmitted through the fluid to the
piston with the larger area.
• The force exerted on the larger piston is proportional to the
area of the piston: F = PA.
• Applying the same pressure to the larger area of the second
piston results in a larger force on the second piston.
What is the force
exerted on the
output piston by the
fluid?
F1 = 10 N
A1 = 2 cm2 = 0.0002 m2
P = F1 / A1 = 10 N / 0.0002 m2
= 50,000 N/m2
= 50 kPa
a)
b)
c)
d)
50 N
500 N
5,000 N
50,000 N
P = 50 kPa
A2 = 100 cm2 = 0.01 m2
F2 = PA2 = (50,000 N/m2)(0.01 m2)
= 500 N
The mechanical advantage is
500 N / 10 N = 50.
E5, E6
18
Buoyant force
Buoyant force
FB = W − T
Archimedes’ Principle
• Archimedes’ Principle: The buoyant force acting
on an object fully or partially submerged in a
fluid is equal to the weight of the fluid displaced
by the object.
Pressure
FB = weight of water displaced
FB = mass of water displaced × g
FB = ρ water × V × g
19
Atmospheric Pressure and the
Behavior of Gases
• Living on the surface of the earth, we are at the
bottom of a sea of air.
• This sea of air is thinner at higher altitudes.
• It is also thinner during certain weather conditions.
• We describe this property by atmospheric
pressure: the pressure of the layer of air that
surrounds the earth.
– At sea level, the atmospheric pressure is 100 kPa, or
14.7 pounds per square inch, but it decreases with
altitude.
Pascal principle:
Application with
hydraulic jack
• Torricelli invented the barometer, a
device for measuring atmospheric
pressure, in an attempt to explain why
water pumps could pump water to a
height of only 32 feet.
• He filled a tube with mercury and
inverted it into an open container of
mercury.
• Mercury worked well because it is much
denser than water.
– Density is the mass of an object divided by
its volume.
• Otto von Guericke performed a famous experiment
to demonstrate the effects of air pressure.
• He designed two bronze hemispheres that could be
smoothly joined together at their rims.
• He pumped the air out of the sphere formed from
the two hemispheres.
• Two eight-horse
teams were unable
to pull the
hemispheres apart.
• Air pressure acting on the mercury in
the dish supported a column of
mercury, of height proportional to the
atmospheric pressure.
20
• In other experiments on variations in atmospheric
pressure, Pascal sent his brother-in-law to the top
of a mountain with a barometer and a partially
inflated balloon.
• The balloon expanded as the climbers gained
elevation.
• This was evidence of a
decrease in the external
atmospheric pressure.
Boyle’s Law
• Boyle discovered that the volume
of a gas is inversely proportional
to the pressure.
• Boyle’s Law: PV = constant
• If the pressure increases, the
volume decreases.
• The density of a column of air
decreases as altitude increases
because air expands as pressure
decreases.
• P1V1 = P2V2
In Boyle’s experiment, adding
mercury to the open side of the
bent tube caused a decrease in the
volume of the trapped air in the
closed side.
Boyle’s Law
• Variations in the volume
and density of a gas that
accompanies changes in
pressure were studied
by Boyle and Mariotte.
• The density of a column
of air decreases as
altitude increases
because air expands as
pressure decreases.
Boyle’s Law:
PV = constant
This law is only valid if the
gas is kept at constant
temperature while the
pressure and volume
change.
21
A fixed quantity of gas is held in a cylinder capped at one end
by a movable piston. The pressure of the gas is initially 1
atmosphere (101 kPa) and the volume is initially 0.3 m3. What
is the final volume of the gas if the pressure is increased to 3
atmospheres at constant temperature?
a)
0.1 m3
b)
0.3 m3
c)
d)
1 m3
3 m3
P1 = 1 atm
V1 = 0.3 m3
P2 = 3 atm
V2 = ?
V2 = P1V1 / P2
= (1 atm)(0.3 m3) / 3 atm
= 0.1 m3
E9
Archimedes’ Principle
Why do some things float while others do not?
Is floating determined by the weight of the object?
Archimedes’ Principle
The key is: DENSITY.
Objects that are denser than the fluid they are
immersed in will sink – those less dense will float.
What is DENSITY? It is MASS divided by VOLUME.
Not a matter of the total weight of the object.
What is the key then?
MASS (m) is measured in kg
(SI)
VOLUME (V) is measured in m3
(SI)
DENSITY (ρ) is measured in kg/m3 in the international
system of units, but very often it is also given in g/cm3
22
This is the definition of density:
ρ=
Density of interstellar space
10-20 kg/m3
Density of air
1.21 kg/m3
Density of styrofoam
m
V
This is the definition of density:
ρ=
m
V
100 kg/m3
Density of water
1000 kg/m3
Density of iron
7900 kg/m3
What if, in a problem, I have the density and
the volume, but don’t know the mass.
13600 kg/m3
Density of mercury
Density of the Earth (average)
5500 kg/m3
Density of the Sun (average)
1400 kg/m3
(core)
160 000 kg/m3
Archimedes: a little bit of history
How do I calculate the mass?
m = ρ ×V
What have you learnt so far from the story?
Archimedes was given the task of determining whether King Hiero's goldsmith was
embezzling gold during the manufacture of a wreath dedicated to the Gods and
replacing it with another, cheaper alloy.
Archimedes knew that the irregular shaped wreath could be smashed into a cube or
sphere, where the volume could be calculated more easily when compared with the
weight; the king did not approve of this.
Baffled, Archimedes went to take a bath and observed from the rise of the water upon
entering that he could calculate the volume of the crown through the displacement of
the water.
Allegedly, upon this discovery Archimedes went running though the streets in the nude
shouting, "Eureka! Eureka!" (Greek for "I have found it!"). As a result, the term "eureka"
entered common parlance and is used today to indicate a moment of enlightenment.
This story first appeared in written form in Vitruvius' books of architecture, two
centuries after it supposedly took place. Some scholars have doubted the accuracy of
this tale, saying among other things that the method would have required precise
measurements that would have been difficult to make at the time.
To find the volume of an object, you can
measure the volume of water displaced,
and that will be the volume of the object
you are searching for. This is particularly
useful for irregular objects (like the wreath).
Another way of determining volume is
by measuring the dimensions of the
object and calculating the volume
mathematically.
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Archimedes’ Principle
Archimedes’ Principle
Is it hard to submerge in water a large block of wood
or a rubber inner tube filled with air?
Legend has it that Archimedes was sitting in the public
baths observing floating objects when he realized what
determines the strength of the BUOYANT FORCE.
They keep popping back to the surface, don’t they?
When an object is submerged, its volume takes up
space occupied by water: it DISPLACES the water, in
other words.
The upward force that pushes such objects back
toward the surface is called the BUOYANT FORCE
(FB).
The more water displaced by the object as you push
downward, the greater the upward buoyant force.
final level of water
Initial level of water
Block of wood
water
Archimedes’ Principle
• Archimedes’ Principle: The buoyant force acting
on an object fully or partially submerged in a
fluid is equal to the weight of the fluid displaced
by the object.
FB
W
Aluminium (Al)
Density (near r.t.) 2.70 g·cm−3
Copper (Cu)
Density (near r.t.) 8.96 g·cm−3
Tin
Density (near r.t.)(white) 7.265 g·cm−3
FB = weight of water displaced
Exercises
E10, 11, 12
FB = mass of water displaced × g
FB = ρ water × V × g
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• The source of the buoyant force is the increase in
pressure that occurs with increasing depth in a fluid.
– Atmospheric pressure is greater near the surface of the
earth than at higher altitudes.
– The pressure near the bottom of a pool is larger than near
the surface.
– The weight of the fluid above contributes to the pressure
that we experience.
• For example, consider a block submerged in water,
suspended from a string.
– The pressure of the water pushes on the block from all sides.
– Because the pressure increases with depth, the pressure at the bottom
of the block is greater than at the top.
– There is a larger force (F = PA) pushing up at the bottom than there is
pushing down at the top.
– The difference between these two forces is the buoyant force.
Water emerging from a hole near
the bottom of a can filled with
water has a larger horizontal
velocity than water emerging
from a hole near the top.
Weight = mg = Vdg
Volume = Ah
W dgAh
Excess Pressure ∆P =
=
= dgh
A
A
The buoyant force is proportional
to both the height and the crosssectional area of the block, and
thus to its volume.
The volume of the fluid displaced is
directly related to the weight of
the fluid displaced.
• What forces act on a floating object?
– For the block shown, the weight W is balanced by the string’s tension T
and the buoyant force.
– If there are no strings attached or other forces pushing or pulling on the
object, only the weight of the object and the buoyant force determine
what happens.
– The weight is proportional to the density and volume of the object, and
the buoyant force depends on the density of the fluid and the volume of
the fluid displaced by the object.
What happens if the density of the
object is:
greater than that of the fluid?
less than that of the fluid?
the same as that of the fluid?
How does an
airplane wing
work?
• The shape and tilt of the wing cause the air to move
faster across the top than across the bottom.
• This causes a lower pressure on the top of the wing.
• The pressure difference produces a net upward
force, or lift, acting on the wing.
• When the lift balances the airplane’s weight, the
airplane will fly.
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How can a
ball be
suspended in
mid-air?
A ball is suspended in an upwardmoving column of air produced by a
hair dryer. The air pressure is
smallest in the center of the column,
where the air is moving the fastest.
Why does a
curveball
curve?
The whirlpool of air created by the
spin of the ball causes the air to move
more rapidly on one side than the
other. The difference in pressure
produces a force toward the lowerpressure, higher-airspeed side.
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