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- Q 1- 2
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c. Students know how to solve problems involving conservation of energy in simple systems,
such as falling objects.
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- p. 159-160
- p. 161, 174-175
- p. 162-163,176-177
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Equations (4) and (5) can be used
to show that if the object dealt with in Standard 2.b is released
from rest and allowed to fall freely, it will strike the ground with
a speed v = (square root of ) 2gh , (eq. 16) and its
kinetic energy at the instant of impact will be
E = 1/2 mv2 = 1/2 m(2 gh) = mgh . (eq. 17)
The total energy T of the object is then defined as the sum of
kinetic plus potential energy
T = E + PE . (eq. 18)
This sum is conserved in a closed system
for such forces as gravity and electromagnetic
interactions and those produced by ideal springs. Thus,
DE +
DPE = 0 . (eq. 19)
Therefore, the change in kinetic energy
equals the negative of the change in potential energy. This principle
is a consequence of the law of the
conservation of energy. Energy can be converted from one form
to another, but in a closed system the total energy remains the same.
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- Q 2
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d. Students know how to calculate momentum as the product mv.
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- p, 179-180
- p. 181, 210-211
- p. 182, 211
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The momentum p of an object is defined as the product of its mass m and
its velocity v. Momentum
is thus a vector quantity,
having both a magnitude and a direction. The units of momentum are kg-m/s. The
magnitude of the momentum is mv, the product of the object’s
mass and its speed.
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- Q 2
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e. Students know momentum is a separately conserved quantity different from energy.
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- p. 187-189
- p. 190, 213
- p. 191-193, 213-214
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If no net force is acting on an
object or on a system of objects, the momentum remains constant.
That is, neither its magnitude nor its direction changes with time. Conservation of momentum is another fundamental
law of physics.
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- Q 2
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f. Students know an unbalanced
force on an object produces a change in its momentum.
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- p.
185
- p. 186-187, 212
- p. 212-213
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As
discussed in the section for Standard 1.c, if the net force on an
object is not zero, then its velocity and hence its momentum will
change. Motion resulting from a constant force F acting on
an object for a time Dt causes
a change in momentum of FDt. This
change in momentum is called an impulse. (Note that the units of impulse
are the same as those of momentum [i.e.,
Newton-second = kg-m/s].) Depending
on the direction of the force, the impulse can increase, decrease,
or change the direction of the momentum of an object.
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- Q 2
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g. Students know how to solve problems involving elastic and inelastic collisions in one dimension by using the principles of conservation
of momentum and energy.
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- p.
193-195,197-199
- p. 196, 199, 214
- p. 200-202, 215-216
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Momentum is always conserved in collisions. Collisions that also conserve
kinetic energy are called elastic collisions; that is, the kinetic energy before and after
the collision is the same. Billiard balls colliding on smooth pool
tables and gliders colliding on frictionless air tracks are approximate
examples. Collisions in which kinetic energy is not conserved are
called inelastic collisions. An
example is a golf ball colliding with a ball of putty and the two
balls sticking together. Some of the kinetic energy in inelastic
collisions is transformed into other types of energy, such as thermal
or potential energy. In all cases the total energy of the system
is conserved.
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NE- Q 2
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h.* Students know how to solve problems involving conservation of energy in simple systems with various sources
of potential energy, such
as capacitors and springs.
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- p.
163, 559
- p. 169, 174, 561
- p. 175-176, 562, 574-575
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An
ideal spring is an example of a conservative system. The force required either to stretch or
to compress a spring by a displacement x from
its equilibrium (unstretched) length is
F = kx , (eq. 20)
here k is the spring
constant that measures a spring’s stiffness. A graph of
the magnitude of this force as a function of the compression shows
that the force varies linearly from zero to kx as the spring
is compressed. The area under this graph is the work done in compressing
the spring and is equal to
½ (base)(height) = 1/2 kx 2. (eq. 21)
This is also the potential energy stored in the spring.
A capacitor stores charge. The charge Q that is stored depends
on the voltage V according to
Q = CV, (eq. 22)
where the constant C is called the capacitance. (Notice
that this equation and the equation for a spring [eq.
20] have the same form.) The energy stored in a capacitor is given
by the equation
E = 1/2 CV 2, (eq. 23)
which
also has the same form as the equation that gives the energy stored
by a spring.
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3rd Content
Standard The
concept of heat (thermal energy)
is related to all scientific disciplines. Energy transfer, molecular motion, temperature, pressure,
and thermal conductivity are integral parts of physics, chemistry,
biology, and earth science. Thermodynamics deals with exchanges of
energy between systems. If students in high school have not yet covered the
chemistry standards, the related topics from those standards should
be introduced. (See the following standards for chemistry 4.a through
4.h, “Gases and Their Properties," and 7.a through 7.d, “Chemical
Thermodynamics." Specific chemistry topics that are useful
or necessary for promoting a more complete understanding of Standard
Set 3 are specifically mentioned, when relevant, under the sections
with detailed descriptions. At
the atomic and molecular levels, all matter is continuously in motion.
For example, individual molecules of nitrogen, oxygen, and other
gases which makes up the air inside a balloon move at varying speeds
in random directions, vibrating and rotating. The collisions of these
molecules with the inner surface of the balloon create the pressure that
supports the balloon against atmospheric
pressure. Considerable
confusion exists in scientific literature about the definitions of
the terms heat and thermal energy. Some texts define heat strictly
as “transfer of energy." These science content standards use the
term heat interchangeably with thermal energy. However,
it is less confusing to reserve the term heat for thermodynamic
situations in which energy is transferred either because of differences
in temperature or through work done by or on a system. In this sense both heat and work have
meaning only as they describe energy exchanges into and out of the
system, adding or subtracting from a system’s store of internal energy. Students, just like scientists
of the eighteenth century, might easily fall prey to the misconception
that heat is a substance. Students should be cautioned that heat
is energy, not a material substance, and that heat flow refers
not to material flow but to the transfer of energy from one place
to another. Confusion is most apt to arise when dealing
with heat transfer by convection;
that is, when heat is transferred through actual motion of hot
and cold material along a thermal
gradient. Heating a material such as air causes it to expand
and leads to differences in density that
drive the movement of heated material. Students also often confuse temperature and heat. From a molecular viewpoint, temperature is a measure
of the average translational kinetic energy of a molecule, as shown
in equation (27). (See also Chemistry
Standard Set 7.) Studies of the temperature of materials as
they pass through phase transitions may also help students understand
the differences and relationships between heat and temperature. A
way to avoid confusion is to reserve the use of the word heat for
situations in which heat transfer is involved, as described in the
next section.
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Text Page Numbers
I- Introduced
P- Practiced
TM -Taught to Mastery
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Heat
and Thermodynamics (15% of CST: 9 items) - Q 2l - 3
[9 items] 3. Energy cannot be created or destroyed, although
in many processes energy is transferred to the environment as heat.
As a basis for understanding this concept:
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- Q 2l - 3
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a. Students know heat
flow and work are
two forms of energy transfer between systems
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- p. 404-408
- p. 408-410, 436
- p. 410-417, 437
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Heat transfer is energy flow from one system to another
because of differences in temperature or because of mechanical work.
The energy that flows into a pot of cold water put on a hot stove
is an example of heat transfer. This energy increases the kinetic energy of the random motion of the molecules of water and
therefore the temperature of the water rises. When the water reaches
100 oC,
a new phenomenon, a phase transition,
occurs: the water vaporizes, or boils. Although energy continues to flow into the water, the kinetic
energy of the water molecules does not increase; therefore the temperature
of the water remains constant. As the water changes from a liquid
to a gas, the energy goes instead into breaking the bonds that hold
one molecule of liquid water to another. The energy required (per
unit mass or mole of liquid) to change a particular liquid at its
boiling temperature into a gas is called the liquid’s latent heat of vaporization.
Mechanical work can change temperature too (e.g., when the forces of friction
heat objects or when a gas is compressed and so warms). Conversely,
changes in temperature can do mechanical work (e.g., warming a container
of gas that is sealed by a piston will cause the gas to expand and
the piston to move). Heat is energy that moves between a system and
its environment because of a temperature difference between them.
Every system has its internal energy, that is, the energy required
to assemble the system; and this energy is independent of any particular
path or means by which the system is assembled. The transfer of internal
energy from one system to another, because of a temperature difference,
is known as heat
flow. There
are three basic kinds of heat flow: conduction, convection, and radiation. Students should have first learned about these processes in the sixth grade.
As heat is transferred to a system (object), the temperature of the system
(object) may increase. Substances vary in the amount of heat necessary
to raise their temperatures by a given amount. More mass in the system
clearly requires more heat for a given temperature change. An expression
that illustrates the relationship between the amount of heat transferred
and the corresponding temperature change is shown in equation (24).
The change in temperature DT is proportional to the amount of heat
added. This relationship is specified by
Q = mCDT
, (eq.
24)
where Q is the internal energy added by heat transfer to the system from the surroundings, DT is the difference in temperature between
the final and initial states of the system, m is
the system’s mass, and C is
the specific heat of the
substance (joules/gram-oC or calories/gram-oC). Specific
heat is a characteristic property of a material.
The unit of specific heat is energy divided by mass and temperature
change (e.g., calories/gram-degree).
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Water, which serves as a standard against
which all other materials may be compared, has a specific heat of
one calorie/gram-degree. In other words, one calorie of heat is required to raise one gram of water one degree
Celsius. When a gram of water cools one degree, one calorie is liberated.
This value is large compared with those of other substances. Therefore,
it takes much more heat to warm water than it does to raise the temperature
of the same amount of most other substances. This fact has important
implications for weather and climate and is one reason the weather
is “tempered" in coastal areas (e.g., summers are cooler and winters
are warmer than they are in inland areas at a similar latitude).
Equation (24) makes the distinction between heat
and temperature quite clear. It specifies that heat can flow in or
out of a system because of temperature difference alone. There are,
however, other situations in which the addition or removal of heat is not accompanied by changes in temperature. These situations
occur when a substance undergoes a change
of phase, or state, such as when water evaporates or freezes.
During phase changes, the absorption or release of heat takes place
while the system remains at a constant temperature. For example,
when ice melts in a glass of water that is sufficiently well mixed,
the temperature of the water remains at the freezing point of water.
Additional heating of the water raises its temperature only after
the ice has melted.
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- Q 3
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d. Students know that most processes tend to decrease the order
of a system over time and that energy levels are eventually distributed
uniformly.
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- p. 418
- p. 419, 438
- p. 420
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Energy in the form of heat transfers from hot to cold, but not
from cold to hot, regardless of whether that energy transfers by radiation, conduction,
or convection. Why? Matter exists in discrete
energy states (or levels). For tiny objects, such as a single electron,
the difference in energy between one state and the next is big enough
to be detected and measured. For the larger objects of everyday experience,
such as a pebble, the difference is too small to detect; still, the
discrete states exist, and it makes sense to speak of the probability
with which any given system is to be found in any one of its possible
states.
A system of many components has many states of given total energy because
some components can have a larger fraction of that energy if others
have less. Such a system evolves so that all states with the same
total energy become equally probable. Heat flows from hot to cold because states in which components share
energy equally vastly outnumber states in which they do not. A copper
bar with one end hot and the other cold has many atoms with more
kinetic energy on the hot end and many atoms with less kinetic energy
on the cold end. Later, however, because the kinetic energy has been
transferred from the hot end of the bar to the cold end, all the
atoms will have nearly the same kinetic energy. The change can be
interpreted as heat flowing from hot to cold until the temperature
of the bar is uniform. Similarly, most physical processes disorder
a system because disordered states vastly outnumber ordered ones.
A drop of perfume evaporates because states in which molecules of
perfume are scattered throughout a large volume of air vastly outnumber
states in which the molecules are confined in the tiny volume of
a drop.
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- Q 2
|
e. Students know that entropy is
a quantity that measures the order or disorder of a system and that this quantity
is larger for a more disordered system.
|
- p. 419
- p. 420-421, 436
- p. 421-424, 442
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Students
know from Standard 3.d that energy transferred as heat leads to the
redistribution of energy among energy levels in the substances that
compose the system. This redistribution increases the disorder of
material substances. A quantity called entropy has been defined to track this process and to measure
the randomness, or disorder,
of a system. Entropy is larger for a disordered system than for an
ordered one. Thus, a positive change in entropy, in which the final
entropy is larger than the initial entropy, indicates decreasing
order, also considered as increasing disorder. The properties of
entropy fix the maximum efficiency with which energy stored as a
temperature difference can be converted into work.
For a system at constant temperature, such as during melting
or boiling, the change in entropy DS is given by
DS = Q/T , (eq. 28)
where Q is the heat (thermal energy) that flows into
or out of the system and T is the absolute temperature. The
units of entropy are joules/K. All processes that require energy,
for example, biochemical reactions that support life, occur only
because the entropy increases as a result of the process.
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4th Content
Standard
Students can be introduced to this standard set by learning
to distinguish between mechanical and electromagnetic waves. In general, a wave is
defined as the propagation of a disturbance. The nature of the disturbance
may be mechanical or electromagnetic. Mechanical waves, such as ocean waves, acoustic waves, seismic waves, and
the waves that ripple down a flag stretched taut by a wind, require
a medium for their propagation and gradually lose energy to that
medium as they travel. Electromagnetic waves can travel in
a vacuum and lose no energy even over great
distances. When electromagnetic waves travel through a medium, they
lose energy by absorption,
a phenomenon that explains why light signals sent through the most
transparent of optical fibers still need to be amplified and repeated. In contrast, light
emitted from distant galaxies has traveled great distances without
the aid of amplification, an indication that a relatively small amount
of material is in the light’s path.
Waves transfer energy from one place to another without net
circulation or displacement of matter. Light, sound, and heat energy
can be transmitted by waves across distances measured from fractions
of a centimeter to many millions
of kilometers. Exertion of
a direct mechanical force, such as a push or a pull, on a physical body is an example of energy
transfer by direct contact. However, for transfer to occur, objects
do not need to be in direct physical contact with a source of energy.
For instance, light transmits from a distant star, heat radiates
from a fire, and sound propagates from distant thunder. Energy may
be transferred by radiation, for example, from the Sun to Earth;
therefore, radiation is also an example of a noncontact energy transfer.
Both sight and hearing are senses that can perceive energy patterned
to convey information without direct contact between the source and
the sensing organ.
If students take physics before
they have studied other high school science courses, the teacher
may find it useful to cross-reference materials on pressure, heat,
and solar radiation from the following standards in: 4.a and 7.
of the chemistry standards and 4.a through 4.c of the earth sciences
standards. Algebra, geometry, and simple trigonometric skills are
required for some of the advanced topics in this standard set.
Students with a good foundation in algebra and geometry can be
taught the trigonometry necessary to solve problems in this standard
set.
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Text Page Numbers
I- Introduced
P- Practiced
TM -Taught to Mastery
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Waves (17% of CST: 10 items) -
Q 3
[10 items] 4. Waves have characteristic properties that
do not depend on the type of wave. As a basis for understanding this concept:
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- Q 3
|
a. Students know waves
carry energy from one place
to another.
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- p.
457
- p. 458, 474
- p. 465-467,475
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Waves may transport energy through a vacuum or through matter. Light waves, for example, transport energy
in both fashions, but sound waves and most other waves occur only
in matter. However, even waves propagating through matter transport
energy without any net movement of
the matter, thus differing from other means of energy transport,
such as convection, a waterfall,
or even a thrown object.
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- Q 3
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b. Students know how to identify transverse and longitudinal
waves in mechanical media,
such as springs and ropes,
and on the earth (seismic waves).
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- p.
459
- p. 474
- p. 460-464, 475
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Waves that propagate in mechanical media are either longitudinal or transverse waves. The disturbance in longitudinal
waves is parallel to the direction of propagation and causes compression and expansion (rarefaction) in the medium carrying the wave.
The disturbance in transverse waves is perpendicular to the direction of propagation
of the wave. Examples of longitudinal waves are sound waves and P-type earthquake waves. In transverse waves
a conducting medium, or a test particle inserted in the wave, moves
perpendicular to the direction in which the wave propagates. Examples
of transverse waves are S-type earthquake waves
and electromagnetic (or light) waves.
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|
- Q 3
|
c. Students know how to solve
problems involving wavelength, frequency, and wave speed.
|
- p.
458
- p. 459, 474
- p. 475
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All
waves have a velocity v (propagation
speed and direction), a property that represents the rate at which
the wave travels. Only periodic, sustained waves can be easily characterized
through the properties of wavelength and frequency. However, most real waves are composite, meaning they can be understood
as the sum of a few or of many waveforms, each with an amplitude, a wavelength, and a frequency.
Wavelength l is the distance
between any two repeating points on a periodic wave (e.g., between
two successive crests or troughs in
a transverse wave or between adjacent compressions
or expansions [rarefactions]
in a longitudinal wave). Wavelength is measured in units of length.
Frequency f is
the number of wavelengths that pass any point in space per second.
A wave will make any particle it encounters move in regular cycles,
and frequency is also the number of such cycles made per second and
is often abbreviated as cycles per second. The unit of frequency is the inverse second (s-1),
a unit also called the hertz (Hz). Periodic wave characteristics are related to each other.
For example,
v = f l . (eq.
31)
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- Q 3
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d. Students know sound is a longitudinal
wave whose speed depends on the properties of the medium
in which it propagates.
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- p.
478-479
- p. 480-484, 507
- p. 485-492, 508-509
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Sound waves, sometimes
called acoustic waves, are typically produced when a vibrating object is
in contact with an elastic medium, which may be a solid, a liquid,
or a gas. A sound wave is longitudinal, consisting of regions of
high and low pressure (and therefore of compression and rarefaction) that propagate away from the
source. (Note that sound cannot travel through a vacuum.) In perceiving sound, the human eardrum
vibrates in response to the pattern of high and low pressure. This
vibration is translated into a signal transmitted by the nervous
system to the brain and interpreted by the brain as the familiar
sensation of sound. Microphones similarly translate vibrations into
electrical current. Sound speakers reverse the process and change
electrical signals into vibrational motion, recreating sound waves.
An acoustic wave attenuates, or reduces in amplitude, with
distance because the energy in the wave is typically spread over
a spherical shell of ever-increasing area and because interparticle friction in the medium gradually
transforms the wave’s energy into heat. The speed of sound varies from one medium to
another, depending primarily on the density and elastic properties of the medium. The speed
of sound is typically greater in solid and liquid media than it is
in gases.
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|
- Q 3
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e. Students know radio
waves, light, and X-rays are different wavelength bands in
the spectrum of electromagnetic
waves whose speed in a vacuum is approximately 3 x 108 m/s (186,000 miles/second).
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Electromagnetic
waves consist of changing electric and magnetic
fields. Because these fields are always perpendicular to the direction
in which a wave moves, an electromagnetic wave is a transverse wave.
The electric and magnetic fields are also always perpendicular to
each other. Concepts of electric and magnetic fields are introduced
in Standard Set 5, “Electric and Magnetic Phenomena,". The
range of wavelengths for electromagnetic waves is very large, from
less than nanometers (nm)
for X-rays to more than kilometers for radio waves. The human eye
senses only the narrow range of the electromagnetic spectrum from
400 nm to 700nm. This range generates the sensation of the rainbow of colors from violet through the respective colors to red.
In a vacuum all electromagnetic waves travel at the same speed of
3x108 m/s
(or 186,000 miles per second). In a medium the speed of an electromagnetic
wave depends on the medium’s properties and on the frequency of the
wave. The ratio of the speed of a wave of a given frequency in a
vacuum to its speed in a medium is called that medium’s index of refraction. For visible light
in water, this number is approximately 1.33.
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|
- Q 3
|
f. Students know how to identify the characteristic
properties of waves: interference (beats), diffraction, refraction, Doppler effect, and polarization.
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- p.
494-497, 797-798
- p. 494, 498-500, 510, 811
- p. 799-802, 511-512
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A characteristic and unique property of waves is that two or
more can occupy the same region of space at the same time. At a particular
instant, the crest of one
wave can overlap the crest of another, giving a larger displacement
of the medium from its condition of equilibrium (constructive interference); or the crest
of one wave can overlap the trough of another, giving a smaller displacement (destructive
interference). The effect of two or more waves on a test
particle is that the net force on the particle is the algebraic sum
of the forces exerted by the various waves acting at that point.
If two overlapping waves traveling in opposite directions have
the same frequency, the result is a standing wave. There is a persistent pattern
of having no displacement in some places, called nulls or nodes, and large, oscillating displacements in others, called maxima or antinodes. If
two overlapping waves have nearly the same frequency, a node will
slowly change to a maximum and back to a node, and a maximum will slowly change to a node and back to a maximum.
For sound waves this periodic change leads to audible, periodic changes
from loud to soft, known as beats.
Diffraction describes
the constructive and destructive patterns of waves created at the
edges of objects. Diffraction can cause waves to bend around an obstacle
or to spread as they pass through an aperture. The nature of the diffraction patterns of a wave
interacting with an object depends on the ratio of the size of the
obstacle to the wavelength. If this ratio is large, the shadows are
nearly sharp; if it is small, the shadows may be fuzzy or not appear
at all. Therefore, a hand can block a ray of light, whose average
wavelength is about 500 nm, but cannot block an audible sound, whose
average wavelength is about 100 cm. The bending of water waves around
a post and the diffraction of light waves when passing through a
slit in a screen are examples of diffraction patterns.
Refraction describes
a change in the direction of a wave that occurs when the wave encounters
a boundary between one medium and
another provided that the media have either different wave velocities
or indexes of refraction and waves corresponding to one special direction
to pass through. Polarized sunglasses and
stretched cellophane wrap are examples of polarizing materials provided
that the wave arrives at some angle to the boundary other than perpendicular.
At a sharp boundary, the change in direction is abrupt; however,
if the transition from one medium to another is gradual, so that
the velocity of the wave
changes slowly, then the change in the wave’s direction is also gradual.
Therefore, a ray of light that passes obliquely from air to water
changes its direction at the water’s surface, but a ray that travels
through air that has
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a temperature gradient will follow a bent path.
A ray of light passing through a saturated solution of sugar (sucrose)
and water, which has an index of refraction of 1.49, will not change
direction appreciably on entering a colorless, transparent piece
of quartz submersed in the solution because the quartz has an almost
identical index of 1.51. The match in indexes makes the quartz nearly
invisible in the sugar-water solution.
Another interesting phenomenon, the Doppler effect, accounts for the shift
in the frequency of a wave when a wave source and an observer are
in motion relative to each other compared with when they are at relative
rest. This effect is most easily understood when the source is at
rest in some medium and the observer is approaching the source at
constant speed. The interval in time between each successive
wave crest is shorter than it would be if the observer were
at rest, and so the frequency observed is larger. The general rule,
for observers moving at velocities much less than the velocity of
the wave in its medium, is that the change in frequency depends only
on the velocity of the observer relative to the source. Therefore,
the shriek of an ambulance siren has a higher pitch when
the source approaches and a lower pitch when the source recedes.
For an observer following the ambulance at the same speed, the siren
would sound normal. Similar shifts are observed for visible light.
Polarization is a property of light and
of other transverse waves. Transverse
waves are those in which the displacement of a test particle is always perpendicular to
the direction in which the wave travels. When that displacement is
always parallel to a particular direction, the wave is said to be (linearly) polarized. A ray of light emitted
from a hot object, like a lamp filament or the sun, is unpolarized; such a ray consists of many component waves overlapped
so that there is no special direction perpendicular to the ray in
which a test particle is favored to move. The components of an unpolarized
ray can be sorted to select such a special direction and so make
one or more polarized rays. An unpolarized ray that is partly reflected and partly transmitted by an angled
sheet of glass is split into rays that are polarized; an unpolarized
ray can become polarized by going through a material that allows
only waves corresponding to one special direction to pass through.
Polarized sunglasses and stretched cellophane wrap are examples of
polarizing materials.
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