Grade 11 Physics, The Abelard School
Review: Historical Survey and Chapters 1-23
Please note: this is a summary outline only. You cannot learn the
material from scratch by reading this outline. To do well in the course,
you must listen in class, take good notes, and read the textbook. However,
given that you do all these things, this outline can be a helpful study
guide. Particular attention is paid to certain material that is
not in the textbook.
I. Historical Survey: know the basic importance
of these historical figures.
Ancient Greeks:
Thales: everything is made of water, compressed or expanded
Anaximander: everything is made of the indefinite/infinite, or apeiron.
The cosmos is formed out of the apeiron through a vortex motion.
The apeiron is preferred over something common like water, because
water as the ultimate substance tries to explain something in terms of
that which is to be explained. (Also thought that humans evolved from fish-like
creations in the ocean.)
Pythagoras: thought the universe was made of numbers.
Parmenides and Zeno: neither motion through time nor objects separated
by space should be possible, since this would mean there was something
that could clearly be conceived of that was not (did not exist). This requires
us to make sense of the idea of nonexistence, or nothing, which cannot
be rationally conceived of. For instance, if object A is "here" and B is
"there", then that precludes A being "there" and B being "here". But one
cannot rationally say what it means for one perfectly imaginable situation
to exist while another does not. Therefore, all space and time are completely
filled up with all possible objects and movements. Nothing is left out.
Every imaginable thing is included. But this leaves a completely undifferentiated
universe, with no such thing as one thing "here" and another "there". No
plurality or motion. This argument shocked the ancient Greeks, and no physical
theory since has been able to avoid attempting to answer Parmenides, at
least to some degree.
Empedocles: attempted to answer Parmenides. Everything is made of combinations
of Earth, Air, Fire and Water (the four elements). (Also said that humans
evolved through some kind of process of combining and recombining parts,
through a process of natural selection, or survival of the fittest.)
Democritus: another attempt to answer Parmenides. Everything is made of
tiny atoms (each one by itself having, internally, the undifferentiated
character of Parmenides' entire universe). These atoms have differing external
properties, like shape and size, and so can combine in a huge variety of
different ways to form compounds, which give us the substances we experience
in everyday life. The atoms move in empty space, or void.
Anaxagoras: another attempt to answer Parmenides. There is a little bit
of everything in everything. So every bit of bread has a tiny bit of blood
and every bit of blood has a tiny bit of bread. So the actual composition
of things does not change, just the concentrations of the various parts.
Plato: attempted to answer Parmenides with the eternal "forms", which were
something like mathematical structures, and determined the structure of
the world. For Plato, it was the forms that had the objective reality of
Parmenides' undifferentiated universe, and the forms were actually,
for Plato, more real than the real world.
Aristotle: thought that all this rationalism (instigated by Parmenides
and culminating in Plato), trying to account rationally for the structure
of the world, was hogwash. He was an empiricist, not a rationalist, and
did not care that the world was not completely 100% intelligible. He thought
we should just start with what we observe with our common, everyday senses,
and work from that. He (mistakenly) thought that objects in motion tended
to come to rest unless kept in motion by some force. He divided the forces
in the world into four different types of "cause": (1) material cause:
the matter from which a thing was built, (2) formal cause: the structure
of the thing, (3) efficient cause: cause and effect acting through
time, essentially what we think of as physical force, and (4) final
cause: whatever state the system tends to move towards over time.
Dark Ages and Medieval Times:
When Greek/Roman civilization crumbled, science stopped progressing for
a long time. Not much of importance happened during this time. The church
kept some of the ancient teachings alive, but there was little innovation,
and most of ancient Greek literature was lost. There had been copies of
most of the books in the great Library of Alexandria, part of an immense
university complex called the Museum. The library, alas, was burnt during
the early years of the Dark Ages. Hypatia, one of the last of the great
scientist-philosophers of the ancient age and curator of the Museum during
its last great years, was lynched by a church mob, ripped to pieces and
burnt. Science was pagan and thus scientists were the enemy. The Academy
in Athens, a university founded by Plato that had been in operation for
hundreds of years, was shut down by the church, branded as pagan. Things
settled down during medieval times, civilization began to rebuild, and
science began to recover. But by this time innovation and creative thinking
had been largely snuffed out. Towards the end of this era, Aristotle (physics
blunders and all) was practically taken as scripture, and few people dared
question him.
Modern Times: (from the Renaissance to today)
Descartes: broke away from Aristotle's teachings. Rediscovered rationalism
and thinking for oneself. Based his entire system on the starting point
of his own thinking ("I think, therefore I am"), instead of, like Aristotle,
on common sense objects in the world around him. Established the modern
practise of describing the world mathematically, as a mechanism (although
he kept the medieval notion of a nonmechanical soul).
Galileo: founded the modern scientific method by doing systematic experiments
to verify his theories, something the ancients never really mastered. He
studied motion by rolling balls down an incline and taking measurements.
He taught us that objects in motion tend to stay in motion (contrary to
Aristotle), and objects at rest tend to stay at rest (this is the principle
of inertia). He discovered the principle of relativity, which states that
when an object moves, it moves only relative to some other object. There
is no experiment that will determine absolutely which object moves and
which stands still.
Newton: Newton took Galileo's ideas about motion and came up with a mathematically
rigorous theory, which stood the test of time until the twentieth century.
He said that objects at rest stay at rest (or objects in motion stay in
motion) unless acted on by a force, which causes the object to accelerate
(or decelerate). He said that F = ma (force equals mass times acceleration).
He said that the total energy in a system is always conserved (the law
of conservation of energy). His laws of motion explained how objects reacted
to forces. He said that for every action, there is an equal and opposite
reaction. This means that for any force acting on an object, an equal force
acts in the opposite direction. These laws are symmetrical in time, which
means that they do not account for the "arrow of time". For example, if
the motion of every molecule in a broken egg were reversed, the egg would
reassemble itself (if Newton's theories were right, which we now know they
aren't quite). There is nothing in Newton's laws which says that
eggs are more likely to go from whole to broken than the other way around.
An explanation for that would have to wait for thermodynamics (see below).
Newton also explained much about the behaviour of light. He showed that
white light consisted of a combination of many colours (the colour spectrum).
He believed that light consisted of tiny particles. He also derived the
law of gravity, which stated that every mass in the universe attracts every
other mass by a force proportional to the inverse of the square of the
distance between them.
Kant: enormously influential, was believed in his time to have resolved
the conflicts between rationalism and empiricism. He taught that physical
objects are in some sense dependent on the observer for their physicality,
their objecthood. This idea is called idealism and was later rejected in
the early twentieth century in favour of material realism (which states
that physical objects are completely independent of, and external to, the
observer). Idealism is now being taken seriously once again, due to the
evidence of quantum mechanics (see below).
Maxwell: came up with a rigorous mathematical description of light, which
convinced most scientists that light was a wave, and not a particle as
Newton had thought. His equation also unified light, electricity and magnetism
as all one and the same phenomenon. Light is an electromagnetic wave, with
an electrical and magnetic component, neither of which can exist without
the other. It travels at a known speed through a vaccum (3.00 x 108
m/s).
Boltzmann (and others): founded thermodynamics. Explained things like breaking
eggs in terms of probability theory. Going from a whole to a broken egg
is much more likely than going from a broken one to a whole one (i.e.,
eggs break more often than they spontaneously put themselves back together).
But even though breaking is far more likely to occur than putting-back-together,
both are equally allowed by Newton's laws. The more likely things (like
broken eggs) are said to have more disorder, or entropy, than the less
likely things (like whole eggs). The second law of thermodynamics says
that the universe as a whole is going towards disorder, or increasing entropy
(perhaps one day it will be totally disordered; this is sometimes called
the heat death of the universe). Of course, we can sometimes put things
into greater order, but this takes work, and work will always produce enough
disordered waste heat to more than offset the order it creates, so the
universe as a whole is still less ordered after the work is complete. Energy
transfer can thus be divided into two kinds: work and heat (the former
ordered, the latter disordered). Energy is always conserved, but work and
heat are not (since work energy will always tend to eventually end up as
disordered heat energy). What is or is not disorder depends on subjective
human judgement, and so is not absolute.
Bohr (and others): developed a model of the atom based on negatively charged
electrons orbiting a positively charged nucleus.
Einstein: showed that not only is motion through space relative (as Galileo
thought), but time is also relative, being just another (fourth) dimension
of space. This is only possible, however, if the speed of light is not
relative
to anything, but absolutely the same to anyone observing from any frame
of reference whatsoever. He showed that it is impossible to ever speed
up to the speed of light, although as one gets closer to it, objects appear
to contract and time for the object appears (to others) to slow down. He
also showed that gravity was due to the warping of space and time by a
body with mass.
Heisenberg, Schrödinger: developed quantum mechanics. Showed that
light is both a wave (like Maxwell thought) and a particle (like
Newton thought). The same is true, in fact, for electrons and all other
forms of matter or energy as well. These quantum waves show the probability
(not the certainty) that a particle will be here or there, moving or still.
The full quantum theory has the particle trying to do every possible thing
that it can (harking back to the ideas of Parmenides). When we observe
the particle, one of these possibilities is realized.What exactly this
means is highly controversial, and there remains no clear consensus.
Wheeler, Hawking (and others): developing the still young field of quantum
cosmology. This requires an as-yet unsuccessful unification of Einstein's
relativity and quantum mechanics. Some, like Hawking, are even more like
Parmenides than basic quantum theory, arguing that all motion and matter
is a "local" phenomena, while the total mass and energy of the universe
is always zero (any positive matter or energy is ultimately cancelled out
by some negative mass or energy somewhere else).
II. 1-D Motion (Displacement,
Velocity, Acceleration): (Ch. 1-2)
kinematics: study of motion (ignoring
what causes it), Ch. 1-3
dynamics: study of motion and
the forces that cause it, Ch. 4-6
Quantities: (sometimes have "units",
sometimes not)
scalar: magnitude
vector: magnitude plus direction (for
1-D or straight-line motion, there are two directions, generally positive
and negative).
indicated by a bold-face
symbol
(as in d) or by placing a small arrow above the symbol.
Change in a quantity: represented by
a D:
Dd
= d2 - d1
= change in displacement, d1
= initial position or displacement from zero, d2
= final position or displacement from zero,
Dt
= t2 - t1
= change in time
Uniform velocity: v = Dd/Dt,
the slope of a straight-line d-t graph. Acceleration, or a,
is of course zero.
Uniform acceleration: a = Dv/Dt,
the slope of a straight-line v-t graph.
Finding slopes we can go from
dÞvÞa.
Instantaneous slopes (whether v
or a) not covering a period of time can be found from the slope
of the tangent line to the curve.
To find an average slope (whether v
or a), just draw a straight line from the starting to the ending
point and find the slope of that.
To go in the opposite direction, from
aÞ
v Þd,
we accumlate all the area under the curve up to the point in time we are
interested in (unlike slope, this accumulates over time, rather than being
instantaneous via the tangent). As long as the curve consists of straight
line segments, the area can easily be calculated as the sum of a series
of areas of rectangles and triangles. Recall that Arect
= hw, and Atriangle = (1/2) hb. Even if
the curve is smooth, you can find an approximate answer by roughly dividing
it up into a series of straight lines (a flat line segment gives a rectangle,
while a non-flat but straight line segment gives a triangle).
Remember that area under the zero line
is negative area, and must be subtracted from the accumulative total, unless
you
are calculating a scalar instead of a vector quantity, in which case you
just find the total area, ignoring the sign (as in finding distance instead
of displacement, or speed instead of velocity).
When going from a Þ
v Þd,
we can only draw the correct shape; we cannot know where the starting y-value
(v or d) should be. The entire graph could be shifted up or down (this
makes sense, since a starting velocity of 8 m/s tells us how much to change
displacement over time, but says nothing about the starting displacement,
or position). It is common practise to assume a starting y-value of 0 if
none is given, but if you do this, you must state it as an assumption.
Finding the slope can be done more generally
for smooth curves using calculus (differentiation). This is not covered
in this course.
Finding the area can be done more generally
for smooth curves using calculus (integration). This is not covered in
this course.
Some useful equations (note: no need
to memorize these, since they are just constructed for some common cases
from the principles above):
See pages 42-43.
It is much more important to understand how to solve things from the graphs
than to memorize these formulas.
However, these formulas
will allow you to solve some problems more quickly.
III. 2-D Motion (Displacement,
Velocity, Acceleration): (Ch. 3)
Drawing vectors. Positive and negative
integers are 1-D vectors (they have direction, but only in 1 dimension).
Vectors can have any number of dimensions. When they have 2 dimensions,
they are 2-D vectors, and can be drawn on paper as an arrow. The length
of the arrow is the magnitude of the vector, and the angle of the arrow
is the direction.
Multiply a vector by a scalar: multiply
the length of the arrow, but leave the direction alone.
Adding two vectors: head-to-tail
rule -- place the tail of one vector on the head of the other. Draw
the resulting vector as an arrow from the tail of the first vector to the
head of the second (see diagram, pg. 61).
The d-t, v-t and a-t graphs from previous
chapters on 1-D motion are harder to draw for 2-D motion, since adding
the t axis for time means we need a 3-D graph. So we will stick to drawing
2-D
motion without time, so the x and y axes are both
displacement,
or both velocity or both acceleration. This means the time information
must be added to the graph by labelling specific points. Be careful, and
do not forget that graphs like the ones on page 60 are
displacement
only,
with no time axis. You can view them like maps with the path of motion
drawn on them (but with no information as to how long the trip took). When
time is factored in by dividing the displacement vector by the change in
time, we get a velocity vector, like those on page 65. These are still
drawn
on what is essentially a map of 2-D space, except the lengths of the arrows
show speed, not distance. There is no separate axis for time.
Just as total displacement for a 1-D
trip can be found by adding up all the separate displacements, the total
displacement for a 2-D trip is also found by adding the individual displacements,
which are now vectors (use the head-to-tail rule).
Velocity is never absolute for an object,
but always relative to some frame of reference. However, that frame
of reference may itself be an object moving relative to some other frame
of reference. To find the velocity of the first object relative to this
last frame of reference, we must add the velocity of the second objects
(its frame of reference) to the first object's velocity. This continues
for arbitrary levels of frame of reference. We can formalize this as a
chain
rule, where ovfis
the velocity of object o relative to frame of reference f.
ovf = ova
+
avb
+
bvf
E.g., the velocity of a passenger (p)
on a train (t) moving across the Earth (e), which is moving around the
Sun (s) can be expressed as:
pvs = pvt
+
tve+
evs
When using the chain rule to solve 2-D
velocity problems, you can add the vectors with the head-to-tail rule,
and use the Pythagorean theorem to find the length of the resulting arrow,
so long as the original arrows were at right angles to each other. This
theorem says that the sum of the squares of the lengths of the two right
angle arrows is equal to the square of the length of the resulting arrow
(which is, as you can see by examining a diagram such as that on page 67,
the hypotenuse of a triangle).
The angle of the resulting arrow can
be found using trigonometry, but we will be satisfied with drawing a diagram
and measuring the angle with a protractor.
It is possible for an object to be accelerating
with a constant speed. When a force constantly pulls a moving object towards
a fixed point, and the object is not moving directly towards or away from
the point, this can cause the object to move in a circle around the fixed
point. This is called angular acceleration, and can be nonzero
even when the speed is constant, since the object can be moving at a constant
speed around the fixed point, while the acceleration causes only its direction
to change. The velocity vector at any point on the 2-D displacement graph
(which looks like a circle) will be an arrow pointing out on a tangent
to the circular path. The acceleration vector will be an arrow pointing
inwards, towards the centre of the circle.
IV. Newton's Laws of
Motion: (Ch. 5-6)
Newton's Three Laws of Motion
are part of dynamics (up until now, we have studied kinematics).
I. Inertia. An object at rest
(or in uniform motion) will remain so until acted on by a force. An object
at rest (or in uniform motion) experiences no forces, and hence no stress.
II: F = ma. A force acts on an
object to accelerate it. This force is a vector quantity, and can be thought
of as the cause of the acceleration. The vector has the same direction
as the acceleration vector, and differs from it only in magnitude (think
of force as acceleration with mass factored in, since it takes more force
to accelerate a more massive object by the same amount as a less massive
object). When an object accelerates, there is always a force on it, and
the object feels stress. Unlike uniform motion, which is not a property
of the object but only of the object relative to some frame of reference,
acceleration is a property of the object itself, independent of
our choice of a frame of reference. Since F=ma, the units of force are
kg-m/s2, which we also call a Newton, abbreviated
N.
III. Interaction. For every action,
there is an equal and opposite reaction. In a closed system (with no energy
entering or escaping), forces always balance out to zero, since every force
is counteracted by an equal and opposite force (push on the table with
5 N of force, and it pushes back with 5 N of force).
Unbalanced forces: We normally
speak of an object accelerating because of an "unbalanced force" ... there
is more force pushing the object one way than there is pushing it the other
way. But how is this possible if for every action there is an equal and
opposite reaction? The answer is that the forces are only unbalanced when
we ignore part of the system (usually labelling it as part of the "environment"
instead of the "object" we are studying). For instance, we say that a rocket
accelerates forward when its engines are fired because there is more force
pushing the rocket forward than backward. That is true, but only because
we artificially separated the rocket from the fuel, considering the fuel
as not part of the object we were interested in. But if we consider the
whole rocket + fuel system as a single entity, then that entity
in fact does not move. The rocket's forward action is exactly balanced
by the exhaust's backward reaction.
Force problems: Usually the force
vector problems you are asked to solve assume some such division of the
system into the object of interest and the environment. Draw your force
vectors only for the object of interest. Adding all vectors (using the
same mathematical tools you used to solve vector problems in kinematics)
gives you the final resultant vector, which is (if nonzero) the
unbalanced force on the object that causes it to accelerate. See sample
problems on page 120-121 and 124-125.
Coefficient of friction: The
force of friction, or Ff, pulling in the direction opposite
of the force accelerating the object, is proportional to the "normal force",
FN, the force pressing the surfaces perpendicular to the direction
of motion together. So a book's force due to gravity on a table is opposed
by an equal and opposite force from the table, pushing the book and table
together, and creating friction. The proportionality constant is called
the "coefficient of friction", called m,
and is different for different types of material. See practise problems
on pages 100-101.
Ff = mFN
V. Gravity: (Ch. 4-5)
Free-fall and the Principle of Equivalence:
although
Newton said that a force was acting on an object to accelerate it in a
gravitational field, it turns out that this kind of force acts on the space
and time around the object (as Einstein taught us), rather than the
object itself. So the object itself is in a state of rest when it is falling,
and is accelerating when on the surface of the Earth, with the Earth pushing
up on it to oppose the force due to gravity. This is the reverse of what
one might expect. When a body is falling, we say it is in "free-fall".
When a body's free-fall is halted and opposed by an equal and opposite
force, like the surface of the Earth, we thus have a kind of acceleration,
even though the object is now standing still! This is why Einstein declared
that gravity and acceleration were equivalent (at least locally equivalent,
since the gravitational force at your head is a little bit lower
than that at your feet, being further from the centre of the Earth, causing
a very slight "tidal" force difference that does not exist in an
accelerating elevator, for instance). This principle equating acceleration
and gravity is called Einstein's Principle of Equivalence.
Orbits: When a cannon shoots
a projectile out horizontally, it has a horizontal velocity and a vertical
acceleration down. Thus, its horizontal velocity is eventually halted when
it hits the ground, due to the vertical acceleration due to gravity. If
the Earth was flat, that would be that. But, since the Earth is curved,
there is a certain special speed (escape velocity) where the curve of the
projectile falling down towards the Earth exactly equals the curvature
of the Earth, so the Earth falls away from the object at the same rate
that it accelerates down. Thus, it just continually circles, or orbits,
the Earth. An object in orbit is in free-fall, and feels no force, and
is thus "weightless" (however, the Earth's gravity is still pulling quite
strongly; the object is after all falling.
Centripetal Force and Angular Acceleration: a object in orbit is
being pulled in towards the center of the Earth, and so the situation is
exactly analagous to a weight being swung around one's head on the end
of a string. In the latter case, you are pulling in on the string. In the
case of the orbit, the Earth (or other body) is pulling in on the object
with gravity. The difference is that the object on the string feels stress,
since a force is being applied, but the object in orbit feels no stress,
since the force is applied to the space and time around it, not the object
itself. The force felt by the object on the string is called "centripetal
force". Such an object moves at constant speed, and yet is still accelerating,
since its direction is being constantly changed by a force (and hence its
velocity is changing even though its acceleration is not). For someone
on the inside face of the object being swung around, this force is locally
indistinguishable
from gravity, since it is due to an angular acceleration, and acceleration
and gravity are locally equivalent (Einstein's Principle of Equivalence
again). Note that this force is towards the centre of rotation, and not
as is commonly thought directly away in the opposite direction. One could,
for instance, rotate a space station to create a centripetal force, or
artificial gravity. But the floor would be pushing up on your feet, rather
than a force of gravity pulling down on you, which is exactly the case
when you stand on the Earth. You are standing still, yet accelerating!
Gravity d-v-a problems: gravity
acceleration/velocity/displacement problems are just a special case of
the earlier problems types we studied, but with a=9.8 m/s2 [down]near
the Earth's surface. Always assume a gravity question is on Earth unless
it is stated otherwise. See sample problems on page 78-79.
Newton's Law of Gravity: Newton
said that any two objects in the universe will attract each other with
a force, Fg, proportional to the inverse square of the distance,
d, between the centres of mass of the two objects (where the m's
are the masses of the two objects, and G is the universal gravitational
constant). See the sample problems on page 98.
Fg = Gm1m2 / d2
Pendulum Motion: the swing of a pendulum takes just as long, even
as the pendulum slows down. The equation describing its motion is below
(where T is the period in seconds, L is the length of the pendulum, and
g is the acceleration due to gravity). See practise problems on page 80.
T = 2p(L/g)1/2
VI. Thermodynamics:
(Ch. 7-8)
All energy transfer is of one
of two types:
Work: transfer of energy in an ordered
(less probable) fashion, as in moving a book across the table or roasting
a turkey. The simplest example is moving an object over a distance through
the application of a force: (see problems pp. 150-152)
W = F • Dd
(units:
N-m = J, or
kW-h)
Heat: transfer of energy in a disordered
(more probable, random) fashion.
Energy can thus be said to be
the ability to do work. The work done is at maximum equal to the
energy transferred. Normally, at least some energy is lost in an interaction
as heat. (see problems pp. 155-156)
Wmax = DE
DEtotal
= W + EH
Efficiency measures the percentage
of energy transferred as work, not heat: (see problems pp. 170-171)
efficiency = (W / DEtotal)
x 100%
Heat engine: sometimes heat can be made to do work, but only if
there is an existing differential in temperature between a hot and cold
object. Such a heat differential is a kind of order, and the flow of energy
from hot to cold is highly probable, resulting in an increase in total
disorder or entropy of the system (2nd law of thermodynamics). So
long as we start out with this ordered heat differential, then some of
the energy transferred can be converted into work. However, this can never
be
done with 100% efficiency (3rd law of thermodynamics). This means also
that it is impossible to pump thermal energy from a cold place to a hot
place without doing work.
EHin = W + EHout
Power is the amount of work done
per unit of time: (see problems pp. 152-153)
P = W / Dt
(units:
J/s = W)
Kinetic energy is the energy
of motion. All energy can be viewed as kinetic from some point of view,
although it can be convenient to sometimes think of it as stored up all
in one place, as potential energy. For example, gravitational
potential energy gets stored when we raise an object a distance from
the surface of the Earth. When we let go, the object falls, and the potential
energy gets released as kinetic energy. The total mechanical energy
in
a system is conserved, so long as we view all the energy as kinetic and
potential. This is due to the principle of conservation of energy (first
law of thermodynamics). As one kind of energy increases the other decreases.
(Use your usual kinematics equations here to find
Dh at a particular point in time,
if necessary.) We can derive both of the equations for kinetic and gravitational
potential energy directly from the equation for work; see derivations on
pp. 159, 162. (See problems pp. 159-163, 165-168).
DEg
= mgDh
Ek = ½ mv2 (note:
v here is the speed, not velocity, since it is squared; energy is a scalar
like work, not a vector like force)
Etotal = Ekbefore
+ Egbefore = Ekafter + Egafter
Energy transferred as heat is called
thermal
energy and is due to random motion of molecules. It is usually thought
of as kinetic energy of the molecules as they move around at random and
(in solids) vibrate back and forth. However, it also includes the potential
energy created when such motion pulls atoms apart, causing an electrostatic
potential energy due to the distance between the molecules, exactly analagous
to gravitational potential energy, but due to electrostatic forces attracting
the molecules to each other, rather than gravity. (In a solid, the molecules
are close enough together that they start to repel each other, and the
attraction and repulsion cancel out, leaving the molecules at a fixed distance
from each other.)
Temperature measures the average
kinetic energy of the particles (usually molecules or atoms) in a
system. In other words, it measures the kinetic part of the thermal energy
(leaving out the potential energy). This leaves out ordered kinetic energy
(nonthermal), such as that due to the entire object moving because it was
put in the trunk of a car. Temperature thus tells us how hot an object
is. It is not quite the same as thermal energy because (a) it excludes
the disordered potential energy in the system and (b) it is an average
rather than a total for the particles in the system, and thus is not dependent
on the size of the object in question.
Scientists always use degrees Kelvin
rather than degrees Celsius to measure temperature. A degree is
the same in both units, but Kelvin starts at absolute zero (not
molecular motion at all), or -273 ºC. Always convert to K from ºC
if necessary to solve a problem:
K = ºC + 273
Thermal expansion: objects expand when heated, since the molecules
have greater kinetic energy and thus move away from each other. The change
in length of an object is proportional to the change in temperature. Likewise,
liquids increase in volume when heated, also in proportion to the change
in temperature: (see problems pp. 201-202; use table on page 201 for the
constants of proportionality or expansion coefficients)
DL = aL0DT
(a: coefficient of linear expansion; see table
)
DV = bV0DT
(b: coefficient of volume expansion)
Heat: defined earlier as random energy transfer. Since total random
energy is thermal energy, this is the thermal energy transferred due to
random motion of particles, always from a hotter to a colder object.
Specific heat capacity, c: the amount of heat energy required to
raise the temperature of 1 kg of a substance by 1K (units are thus J/kg).
This varies depending on the type of material. When heat is transferred,
the total heat lost by the hotter object is equal to the heat gained by
the cooler object, due to conservation of energy, so long as none is transferred
as work or lost to the surroundings. (See problems pp. 206-208; use specific
heat capacity chart on pg. 205)
EH = mcDT
Latent heat is the heat that goes into potential thermal energy
instead of kinetic (and thus temperature). I.e., the heat that does into
pulling particles apart against their attractive electrostatic forces.
This heat is minimal compared to kinetic energy except during a phase change,
when most of the heat goes towards ripping the molecules apart rather than
making them move faster. (See problems pg. 213-214; use latent heat chart
on pg. 212.)
EL = ml
The Zeroth Law of Thermodynamics:
was
added later to the original three when it was realized that the distinction
between thermal energy and thus temperature and other more ordered kinds
of energy was completely arbitrary. Who was to say, for instance, that
the kinetic energy of molecules does not count towards the temperature
just because they are all moving in the same direction (if the object is
in the trunk of our car)? The 0th Law was invented to define temperature
in a reasonable everyday sense, using the notion of thermal equilibrium
(the state when hot and cold are completely mixed and at even temperature).
This law, however, can be changed to define what is ordered/disordered
differently, and reasonable (although perhaps less useful) results still
follow.
The Three Laws of Thermodynamics sum up what we have learned in
this chapter.
0th Law: let temperature be arbitrarily defined as the average random
kinetic
energy of an object's particles. (Note: the full version of this law,
which you need not know, states that if A is in thermal equilibrium with
B and B with C, then A is in thermal equilibrium with C. This yields an
intuitive, common-sensical result for what is ordered and what is random
thermal energy).
1st Law: total energy is always conserved in any transfer of energy.
2nd Law: heat always flows from a hotter to a colder object, resulting
in an increase in entropy (disorder). Any transfer of energy thus results
in a net increase of entropy in the universe.
3rd Law: heat cannot be converted completely into work (heat cannot
flow from cold to hot without work being done on it).
VII. Fluids: (Ch. 9)
density: r
= m / V
relative density: r
/ rH2O
pressure: F / A
Pascal's Principle: when a fluid
is redistributed, the pressure is everywhere undiminished. E.g.: when a
tube of water at a certain pressure is split into ten smaller tubes, the
pressure in each small tube is the same as it was in the large tube
(not 1/10 as much). This principle allows hydraulic systems (like
brakes) to use fluid to redistribute applied pressure very efficiently.
Fluids do this well because they are relatively incompressible.
Buoyant force: When a container
of fluid has a certain depth, the pressure from the mass of fluid above
that point being pulled on by gravity gets distributed evenly on all sides
of an object (Pascal's Principle). Thus an object will tend to stay in
one place at a particular depth, and considering only the fluid pressure.
However, in practise, there is also a force of gravity acting down. As
well, the depth of fluid is less near the top of the object than the bottom,
and so there is more pressure pushing up on the object than down, which
creates a "buoyant force" on the object. If this force is greater than
that of gravity, the object will float; if less, it will sink. The difference
in pressure between the top and bottom of the object that gives this buoyant
force is the pressure due to the downward force of a volume of water equal
in volume to the object (i.e., the force of gravity acting on the volume
of water displaced by the object). Since the buoyant force is due
to the force of gravity on the volume of displaced fluid, and this pressure
must offset the force of gravity acting down on the object if the object
is to float, we see that whether an object floats or sinks depends on whether,
when submerged, the volume of fluid displaced weighs more or less than
the object itself (since F=mg). If the fluid displaced weighs more (for
a less dense object), then the buoyant force will be greater than the force
of gravity, and the object floats. If the fluid displaced weighs less (for
a dense object), then the buyoant force will be less than the force of
gravity, and the object sinks. These ideas are illustrated in more detail
below.
Archimedes' Principle:
The buoyant force (FB) on an object is equal to the weight (ma
= mdisplacedg) of the fluid displaced by the object. The object
displaces a volume of fluid equal to the volume of the object that is submerged.
FB = mdisplacedg
= rdisplacedVsubmergedg
The net force (Fnet)
acting on an object in a fluid is the gravitational force pulling down
(assume this is in the negative direction) plus the buoyant force pushing
up (in the positive direction): Fnet = Fg
+ FB
= mdisplacedg - mobjectg
Flotation: for a floating object,
the mass of fluid displaced is equal to the mass of the object
From Archimedes' Principle, it can be
seen that an object of the same density as the fluid will be on the borderline
of floating and sinking, since its net buoyant force will be zero, since
the mass of fluid displaced will be equal to the mass of the object, and
the buoyant force will be equal to the force due to gravity, giving a net
force of zero on the object (since the mass of fluid displaced is here
equal to the mass of the object, if their densities are equal):
rfluid
= robject,
Vdisplaced = Vobject, mdisplaced = mobject
Fnet = Fg + FB
= mdisplacedg - mobjectg
Fnet = 0
From Archimedes' Principle, it can also
be seen that an object less dense than the fluid will float, since when
the object is completely submerged, there is a buoyant force greater than
that due to gravity, since the density of the fluid is greater than the
density of the object:
rfluid
>robject,
Vdisplaced = Vobject, mdisplaced > mobject
Fnet = Fg + FB
= mdisplacedg - mobjectg
Fnet = positive (the object
accelerates upwards)
If such an object is allowed to continue
to rise, it will stop only when the forces balance (the magnitude of the
buoyant and gravitational forces is equal). This will occur, according
to Archimedes' Principle, only when the mass of the fluid displaced is
equal to the mass of the object. This is why an object floating on the
surface of the water does not continue to rise. Once on the surface, less
than its full volume is submerged, since some of the object is sticking
up out of the water. So the volume of fluid (water) displaced is less,
and at a certain point as the object rises out of the water, the mass of
the water displaced (lower volume, higher density) will equal the mass
of the entire object (higher volume, lower density). At this point, the
object is at rest, since the net force is once again zero:
rfluid
>robject,
Vdisplaced < Vobject, mdisplaced =
mobject
Fnet = Fg + FB
= mdisplacedg - mobjectg
Fnet = 0
From Archimedes' Principle, it can also
be seen that an object denser than the fluid will sink, since such an object
cannot displace more than its own volume in fluid, and this volume of displaced
fluid will thus always weigh less than the object itself, and so the buoyant
force will always be less than the gravitational force, and the object
will sink:
rfluid
<robject,
Vdisplaced = Vobject, mdisplaced <
mobject
Fnet = Fg + FB
= mdisplacedg - mobjectg
Fnet = negative (the object
accelerates downwards)
Bernoulli's Principle: a higher
speed of fluid flow creates a lower pressure on the walls of a container
than a lower speed. This explains, for instance, why airplane wings create
lift.
VIII. Waves: (Ch. 10-12)
Wave: periodic motion through
a medium. The high point is called the peak, the low point the trough
(although
it is sometimes simply a matter of perspective which extreme is high and
which is low). Examples: sound, light, a pendulum, a spinning wheel. Anything
that repeats can be thought of as a wave behaviour.
Wavelength (l):
the length of one cycle.
Period (T): the time for one
cycle.
Frequency (f): the number
of cycles per unit time (usually measure in Hz, meaning "per second").
Amplitude (A): the intensity
or strength of the wave at its peak.
f = 1/T
T = 1/f (see
sample exercises on pg. 275)
Two kinds of physical waves: (see
the pictures of the slinky as an example, pg. 278, 280.
Longitudinal (e.g., sound): the
motion creating the wave is in the same direction as the motion of the
wave.
intensity of sound is measured in decibels,
dB
(not a linear scale, see table, pg. 320).
Transverse (e.g., light, water):
the motion creating the wave is perpendicular to the direction of motion
of the wave.
Speed of a wave is measured as
for any speed, by Dd/Dt,
which can be translated into wave terms:
v = Dd/Dt,
Dd
= l,
Dt
= T
v = l/T
v = fl
(see sample exercises on pg. 282, Ch. 11: 317, 319, 328, Ch. 13:
373)
Interference: When two waves meet, their amplitudes combines to
form a new wave. If the amplitudes add up to form a stronger wave, this
is constructive interference. If the amplitudes are opposite in
sign and cancel out to give zero amplitude, this is destructive interference.
When there is a combination of both, you can get interesting interference
patterns (as when two ripples on a pond meet). If the peak of one wave
adds directly to the trough of another, and vice versa, you get total destructive
interference, and the wave disappears (flat-lines).
Standing Waves: If two identical wave moving in opposite directions
meet, they form a standing wave pattern, with the zero amplitude points
fixed in position (called nodes), and the rest of the wave (a series
of loops in between the stationary nodes) oscillating up and down
(see diagrams on pp. 291-292).
Two-dimensional Interference: Waves can be of any dimension. Pg.
294 shows the characteristic interference pattern of two transverse (as
in water or light) waves.
IX. Quantum Mechanics -- Microscopic Behaviour of Light
and Other Particles:
(Note: this material is not in your book, so is described here in as
much detail as you need to know. You do not need to be able to solve any
mathematical problems at the quantum level. Those interested in more details
can look at the books I've recommended.)
Wave-particle Duality: Newton thought light consisted of particles.
By the 1800's, it was pretty much agreed that light was a wave. Electrons,
however, were thought to be particles. Quantum mechanics, however, has
shown us that photons and electrons are both waves (in a
sense) and particles (in another sense). The wave describes all
the different possible paths the particle could take. In fact, it turns
out that the particle, in a way, actually does take all the possible
paths, but when we observe the particle, it "collapses" into one
particular path (this is called the collapse of the wavefunction).
Double-slit Experiment: Young's double-slit experiment involves
either a light or an electron beam (it doesn't matter which) aimed at a
barrier with two small slits in it. The wave passes through just the two
slits, creating two new point sources of light on the other side of the
barrier, which then interfere with each other. If a piece of film is placed
on the other side of the barrier, it will show a complex interference pattern,
resembling the one on page 294.
Single-Particle Double-slit Experiment: if Young's experiment is
then refined, so that only one photon (or electron) is fired at the
barrier at a time, then, one might ask, do we get an interference pattern
or not? If light were a wave, we should get such a pattern even for the
tiniest amount of light possible. If light were made of particles, which
formed waves in large numbers (as in water waves), then we should only
get the pattern for large amounts of light, and not for a single photon,
which should just show a dot on the screen. But what actually happens is
far more mysterious than either of these possibilities: single photons
do indeed make small little dots on the screen, with no interference pattern.
However, if we do the experiment over and over again, with the same
screen, but still firing only one photon at a time, the little dots gradually
build up into an overall interference pattern! But this cannot be like
water particles adding up into a wave. If we send water molecules into
a double-slitted barrier one at a time, we would not get any such
interference! Somehow, it seems each photon (or electron) is interfering
with itself!
Higher dimensions: One possible explanation is that photons travel
as waves, but collapse down to particles when they are absorbed. But it
cannot be quite as simple as that. It turns out that if we have two
photons
interfering with each other, the resulting wave actually requires
four
dimensions
of space instead of three! Add yet another particle, and we need five
space
dimension (six dimensions in total if you include time). So, to describe
the entire universe as a wave would, it seems, require a stupendous number
of dimensions, on the order of the number of particles in the universe!
This is because a quantum wave is a wave of all the possible things
the particle could do. When we observe the particle, we only ever observe
one
of
these possibilities, yet to describe the physical world, we seem to need
to take into account all
of them. If this seems bizarre, you are
in good company. Physicists are not yet agreed on exactly how all this
should be interpreted and thought of. So don't worry if you can't figure
out how the world can be that way (or perhaps I should say go ahead and
worry about it, but remember that some of the best minds in physics are
just as confused as you are!).
Heisenberg's Uncertainty Principle: named after the originator of
quantum mechanics. Since the wave describes all the different paths the
particle could be taking, and the particle exists as a wave until we observe
it, this means that when unobserved the particle is not even on any particular
path (it is taking a whole bunch of different path at the same time). Hence,
it is impossible to measure both the position and momentum of a particle,
not because measuring one ruins your chances of looking at the other one,
but simply because the particle has no precise position or momentum
(or velocity, etc.) until you look at it. When you look at the particle,
you find yourself suddenly looking at one of the possible paths.
It turns out, there are numerous different kinds of experiments you can
do that measure a particle. Some measure the position, and some measure
the momentum. Exactly how the wave collapses depends on how you measure
it. If you do a position-only measurement, you will see a clear position
(actually, you will never be able to see a completely accurate position,
but you can come close), but you will notice little about the momentum.
If you do a momentum-measuring experiment, you will measure a momentum,
but see virtually nothing about where the particle was (because,
in fact, the particle still is actually in all the different possible
positions--not all of the different properties of the particle collapse
into only one possibility, only the ones your experiment is set up to measure).
Nonlocality: Since the particle is taking all possible paths, and
all but one path (or more correctly, all but one path with respect to the
properties being measured) collapses out when you look at the particle,
this means that the entire path collapses out, right from the start to
the finish, instantaneously. The wave could occupy a space the size of
a galaxy (not unusual for a particle travelling in interstellar space),
and once you observe it, that huge wave collapses everywhere all at the
same time. This seems to be a nonlocal effect where something happening
at one spot can influence something light-years away faster than the speed
of light--violating Einstein's principle that nothing can travel faster
than light, perhaps? Although this aspect of quantum mechanics worried
many people for years, it turns out that since the wave describes all the
possible things the particle could do, collapsing down to one possibility
still just gives one path from point A to point B, and the particle does
not travel along this path faster than light. It is the collapse that is
faster than light, not the travelling of the particle, and the collapse
is just the selection of one possible path over a bunch of others, each
path of which obeys Einstein perfectly. Indeed, it turns out that this
so-called "nonlocal" effect can never be used to send someone a message
faster than the speed of light, so Einstein's theory is quite safe!
Probabilities: Since only one possibility is realized in an observation,
and many, many possibilities are there in the wave, no one can predict
which possibility will really happen. We can, however, look at the wave,
at any point, and take its amplitude, square it, and it turns out that
this is the probability of finding the particle at that point:
P = A2
Schrödinger's Cat and the Interpretation of Quantum Mechanics:
one
of the first thought experiments that really showed the paradox of quantum
theory. Imagine a cat in a black box that is completely cut off from the
outside (we cannot make a black box that is 100% isolated, but this is
just a thought experiment, so go with it!). There is a bit of radioactive
material in the box, and we have calculated that there is a 50-50 chance
of its decaying within a certain time, according to the quantum equations.
If it does decay, a Geiger counter device is set up to detect it and smash
a phial of poison gas, killing the cat. Otherwise the cat lives. Schrödinger
pointed out that if quantum mechanics is taken literally, the cat will
be in a higher-dimensional state (superposition of states), being
both dead and alive. Since we never see half-dead, half-alive cats,
Schrödinger concluded that there was something wrong with quantum
theory. He was thus advocating what is called a hidden variables
view, in which quantum mechanics is thought to be an incomplete picture,
and in some way actually incorrect. Another view called Copenhagenism
is
that the wave function actually "collapses" -- so that the cat literally
collapses into one state and not the other when we open the box and look
inside (other views of wavefunction collapse might place the collapse
earlier, and posit different sources for the collapse, but they all agree
with the Copenhagenists that at some point there is an actual collapse,
at the very latest when we look inside the box). A third point of view
is the many-worlds interpretation, which claims that there is
not
collapse at all -- when we open the box, the universe in a sense "splits"
into two copies, one in which a version of us sees a dead cat, and the
other in which we see a living one -- we are now in a superposition,
along with the cat. Then there is still another group that just
ignores the whole issue, quietly declaring that they only care that the
theory makes correct predictions, and what it means about the world is
of no interest, or even meaningless.
X. Optics -- Macroscopic Behaviour
of Light (Ch. 13-17): On the level of macroscopic light (many,
many photons), light just seems to be a wave. Even though what each photon
does can only be described with probabilities, not deterministically, one
the large scale of huge numbers of photons, they seems to act together
as a single deterministic wave. This is the classical pre-quantum wave
described by the equations of Maxwell.
Note, however, that even
though electrons are also quantum waves, on the macroscopic scale, they
look more like solid object, due to the property of not wanted to jam together
into the same space (this keeps them separated into different atoms, and
prevents us from noticing their wave behaviour on the macroscopic level).
Rectilinear Propagation: Even though photons can go through many
different paths to get from A to B, on the large scale, light travels in
straight lines. This is because the waves describing the paths close to
a straight line are closer to being in phase with each other, and they
constructively
interfere with each other (they add up). The paths farther from the
straight line path differ in length one from the other more than the ones
close to the straight line, and this makes their wave descriptions more
out of phase with each other, resulting in destructive interference, and
so the waves tend to cancel out more. Thus, we have lots of constructive
interference near the straight line paths, and destructive interference
farther away. The most probable path for the photon is near the straight
line. Although an individual photon could quite easily deviate from this,
on the large scale, we will see light travel (usually) in straight lines.
There are, however, some exceptions, were interference effects can be seen
on the large scale (light scatters, instead of travelling straight, for
instance, when sent through a narrow passageway).
The speed of light (c): Although photons at the very small
scale can travel at any speed, over long distances, all but the speed of
light cancel out. The speed of light is 3 x 108 m/s (you
will be given this on tests and exams).
Reflection: the angle of incidence (angle at which the light is
coming at the object it reflects off of) is equal to the angle of reflection
(the angle at which it exits the object). This principle is a result of
rectilinear propagation.
Refraction (Ch. 15): Here, we have both an angle of incidence (i),
of reflection (r) and an angle of refraction (R). See the diagram
on page 420. Light is refracted when it goes from air to glass because
it travels slower through glass, and it angle thus decreases (imagine you
are swimming away from the shore at 30°, and suddenly you hit an oil
spill; your speed would suddenly slow down as you plow through the thick
oil--presuming you didn't turn around a swim back!--so your angle swimming
away from the shore is now much less than 30°).
Snell's Law: the factor by which the light slows down is the index of refraction
(n), and is the ratio of the speed of light through the initial medium
to the new speed (vi/vR). The index of refraction
of any one particular substance is measured with respect to the speed of
light, with vi=c:
n = c / v
inR = vi/vR ...
but since n = c / v,
inR = nR/ni
The index of refraction is also the ratio of the component of the angle
i parallel to the barrier to the component of the angle R parallel to the
barrier (i.e., the ratio of the sines of the angles--available on
your calculator): (see practise problems pg. 422-426)
inR = nR/ni = sin i
/ sin R
n1 sin q1 = n2 sin
q2
(memorize only the vi/vR = nR/ni
version;
the sine version will be given to you)
Magnification: When light is sent through a pinhole, it can form
an image on a screen. Through the principle of similar triangles,
it can be shown that the ratio of the height of the image to the height
of the object is the same as the ratio of the distance of the image from
the pinhole to the distance of the object from the pinhole. If the rays
of light end up inverted (as they do fo a pinhole), the magnification is
sometimes said to be negative (the image appears upside down). See diagram
on pg. 375. This results in the magnification equation: (see exercises,
pg. 375-376).
M = di / do = hi
/ ho
Lenses and Mirrors (Ch. 14,16): There are two kinds of both lens
and curved mirror: convex (bulging outward) and concave (dipping inward--as
in "cave"). Convex lenses and concave mirrors are the most
interesting, because they focus light onto a focal point (see
diagrams on pg. 395 for mirrors and 445 for lenses). The distance from
the lens/mirror to the focal point is called the focal length, or
f. A concave lens and a convex mirror can be said to have a (nonexistent
in a way) focal point on the opposite side of the lens/mirror, but
this virtual focal point is just imaginary--its where the light
rays would go if you extended them from the other side. The magnification
equation above applies to both concave mirrors and convex lenses.
Apparent magnification: if we ignore the actual distance to the
real object, and the height of the real object, and just take the object
as it appears before the light is reflected/refracted (its apparent
size), we get the traditional "magnification" used when taking about magnifying
glasses, telescopes and microscopes (written as 2X for a magnification
of 2). Here, dobject is the focal length f, and dimage
is the distance from the focal point to the image; hobject is
the apparent height of the image before magnification, and himage
is the apparent height after magnification (of course, magnification can
be less than one, and the image can actually look smaller).
Telescopes and Microscopes: a telescope focuses light either by
a refracting convex lens (a "refracting telescope") or reflecting concave
mirror (a "Newtonian telescope", named after Newton who invented them).
At some point after the focal point (depending on how much magnification
you want), a convex lens (the eyepiece) is placed to take the resulting
image and magnify it (there can be even more complex arrangements of greater
numbers of lenses). A microscope works exactly the same way, but is designed
to look at small object instead of large far away objects. The main mirror/lens
is the objective mirror or len, and the bigger it is, the more light
it will gather and focus, and the sharper the image will be. The actual
magnification depends on the eyepiece, however. Without the eyepiece, a
refractor becomes a simple magnifying glass.
XI. Electromagnetism (Ch. 18-23):
Electricity flows when, on a large macroscopic scale, there is an area
of excess charge near and area with a deficit of the same kind of charge,
as in the case of an area with too many electrons that is near an area
with too few electrons (more protons than electrons). This will generally
cause the electrons from the negatively charged area to flow into the positively
charged area, since opposite charges attract (whereas like charges repel).
Usually, the charge flows from negative to positive, although it can in
principle happen the other way. However, in most materials, the positive
charge is relatively fixed in the atomic nuclei, whereas the electrons
are in orbit around the atom and can relatively easily move from one atom
to another.
In conductors (such as many metals), the electrons are normally moving
around from atom to atom anyway, so they can easily move in unison in one
direction when a difference in charge is introduced. In insulators, each
electrons is relatively fixed to one particular atom, and so electrical
flow is less likely. (In insulators with high resistence, the flow occurs,
but is difficult, and generates a lot of heat.)
Electrical fields: the "lines of force" created by electrically charged
particles. See the examples on page 500. The force lines require arrowheads
to indicate the direction of the force. Each force line represents the
path that a small positive test particel would take if placed at that position
in the field.
If one of the charges is allowed to move (accelerated by the electrical
field force), the field will change around it as it moves, and the effect
will ripple away through the electromagnetic field, at the speed of light.
In fact, the ripple of change through an electromagnetic field caused by
an acclerated charged particle is precisely the definition of a light wave.
The wave is transverse, and has two components, an electrical and a magnetic
a right angles to it (that is why we can induce a magnetic field perpendicular
to an electrical current--an electromagnet--and we can induce a electrical
flow perpendicular to a moving magnetic field--a generator). Both of these
components, electrical and magnetic, are at right angles to the direction
of the accelerating particle, and together form the electromagnetic wave
(i.e., light or radiation). Since magnetism is a complimentary force to
electricity, a magnetic field diagram can be drawn in a very similar fashion
(see pg. 585-587).
Electrical charge, at the quantum level, measures the probability (actually,
the square root of the probability--the amplitude) of the charged particle
to emit/absorb a photon of light.
On the higher classical (nonquantum) level, the charge is the property
of the particle that determines the amount of force between it and other
charged particles. This force is described exactly like the gravitational
force, except that there is a different constant of proportionality and
we use charge instead of mass. Since the charges can be oppositely charged,
the resulting force can be negative (attractive) if the charges are opposite,
or positive (repulsive) if the charges are alike:
Fe = kq1q2 / d2
Charge is usually measured in coulombs, or C. At the quantum level, however,
it can be measured in elementary units equal to the charge of a single
electron: e = 1.60 x 10-19 C. (Thus, an object with an
excess of 1000 electrons will have a charge of (1000e) Coulombs.
(See
sample problems on page 503.)
Current electricity: if excess charge flows from one object to another
in one quick burst, it is called static electricity. But if there is a
steady, relatively uninterupted flow, it is called current electricity.
To have a steady current, there must be a circular loop of conducting material
to allow the electrons to flow around in a neverending circle, called a
"closed circuit".
The amount of current is measured in terms of the amount of charge that
flows past a given point per unit of time: (see sample problems, pg.
517)
I = q / t (measured in Amperes or amps, A =
1 C/s)
The electric potential is the potential energy caused by pulling two oppositely
charged particle apart (or two similarly charged particles together), and
it works much like gravitational potential energy. Potential is usually
measured in terms of the work that would be done by a small positive test
charge, were it to be placed at that point in the electric field. It is
thus measured in work (ordered energy) per unit positive charge: (see sample
problems, pg. 521)
V = W / q (measured in volts, V = 1 J/C)
The more resistance is experienced by the flowing electrons, the less current
will flow for a given electric potential, and the higher will be the ratio
of voltage (or potential) to amps (or current). This ratio thus measures
resistance: (see sample problems on pg. 541)
R = V / I (measured in Ohms, W
=
1 V/A)
There are two ways to hooks conductors up into a circuit. A simple loop
is called a "series" (see diagram on pg. 543). A loop with numerous short-cuts
being taken throughout the circuits (see diagram on pg. 545) is called
a "parallel circuit".
XII. Special Theory of Relativity: (not
in textbook)
By the early twentieth century, Maxwell's equations were the accepted description
of light, which was thought to be a wave (its particle nature was still
unknown, as quantum mechanics did not arrive until 1925). It was thought
that any wave needed a medium. But light seemed to travel through a vacuum.
So vacuum was thought of as a tangible medium, called the ether. Imagine
you are in a closed automobile on the highway and you speak to the person
next to you. The sound waves from your voice will travel in the still air
inside the car. But imagine you stick your head out the window. Now, the
speed of your sound wave depends on whether you shout against the oncoming
wind created by the car's motion through the outside air, or with the
wind. So if you shout against the wind (straight ahead), your voice travels
slower than if you shout with the wind (straight behind). Likewise, it
was believed that the speed of light should measure as greater or smaller
depending on whether the light was directed with or against the "ether
wind" created by the motion of the Earth through the ether. However, the
experiments of Michelson and Morley showed that there was no such difference.
The speed of light appeared the same no matter what.
Einstein asked us to suppose, in his light postulate, that the speed
of light, usually abbreviated as c, was constant no matter what frame of
reference it was measured from. But yet, if the light was transmitted inside
a moving object, one would expect it to appear slower to one observer inside
the object, and faster to another observer outside the object. The only
way to make the light postulate work is to speculate that time actually
slows down inside the object, and that the object's length becomes
shorter
in the direction of motion.
A result of this system is that time slows down further and further the
closer the object gets to the speed of light, (v=c). In the limit, as the
speed approaches that of light, lengths go to 0, and times go to infinity.
Thus, it is impossible to ever accelerate an object to this speed. This
means that more and more force will be required to accelerate the object,
approaching infinite force, as the object approaches light-speed. Since
F=ma, this means that the object's mass will appear to approach infinity
as well:
It follows, then, that energy of motion actually weighs something,
and has mass! A rat running around in circles inside a box will
weigh more on a scale than the same box with a rat just sitting still.
Of course, since most speeds of normal people and rats are well below that
of light, this increased mass is tiny, but it is real and would
be felt if the rat was moving at relativistic speeds, close to that of
light. As we said earlier, in the unit on potential and kinetic energy,
potential energy is just stored kinetic energy, like the rat running around
in the box. In fact, since kinetic energy has mass all on its own, it is
likely that mass itself -- even rest mass -- is simply stored kinetic energy.
In principle one should be able to free all that energy (like letting the
rat out of the box). In fact, if a particle collides with its antiparticle
(exactly
alike, but with opposite charge), their rest mass disintegrates, and they
produce pure kinetic energy (i.e., light -- light is pure kinetic energy
because there is no rest mass; a photon is all energy of motion). An example
is an electron meeting a positron (or anti-electron) -- they disappear
and are replaced by a single photon. The equation that relates mass and
energy is:
E = mc2
However, it is more natural to just set c = 1, since Einstein's theory
is usually interpreted to mean that time is just another dimension of space.
That means distance and time are measured in the same units, and all speeds
turn into unitless ratios (the slope of a line on a d-t- graph). If c=1,
then the energy-mass equation just says that energy is equal to mass:
E = mc2 = m (12) =
E = m
So time is a fourth dimension of space. Objects contract when they are
in motion, because their speed corresponds to having a slant, or slope,
in spacetime, and the object looks contracted for the same reason a pencil
looks contracted if you rotate it away from you. However, be careful of
this visualization. Einstein's spacetime is not Euclidean. If it
were, the speed of light would be infinite, and not constant for all observers.
In fact, Einstein's speed of light is a 45° slope in spacetime (d-t).
This makes his geometry non-Euclidean and very difficult to visualize.
Twin Paradox: when one twin goes on a space trip and the other stays
home, both twins think the other is moving, so they both see each other
as moving slowly. However, when the travelling twin decelerates, reverses
and heads back home, his entire frame of reference changes, due to acceleration,
and this speeds the twin at home up rapidly, during a very short time,
so when the twin returns to Earth it is very clear which twin was travelling
and which stayed home -- the twin that was accelerating is years younger
than the one that stayed home. The twin who accelerated in a sense "took
a short-cut" through spacetim