What is work?
by George
E. Hrabovsky, President MAST
A Note to the Reader
You will notice that
I ask numerous questions and number them. I have no
intention of answering these questions in sequence.
The numbering is a bookkeeping device so that I can
keep track of things. The point of this column is to
investigate what heat is, and I will only go so far
from from that goal. This means that some questions
will go unanswered. This is reasonable, and it allows
for future projects based on those unanswered questions.
Feel free to attempt to answer these questions for yourself.
Where We Have Been
We have now answered
all of the questions necessary to address the question,
"What is a system?" Recall that the answer
to this question was, "A region of space or a portion
of matter that has a certain amount of one or more substances
ordered in one or more phases." Now we understand
what a region of space is, what is meant by matter,
and what is meant by a phase. I could address either
the nature of matter, or I could define work. At
this point I think work will bring me closer to the
definition of heat, even though the nature of matter
seems also to be important.
There are several places
to turn for definitions. Every book on thermodynamics
talks about work, but they also refer to mechanical
work. It thus seems reasonable to assume
that a book on mechanics is a good place to start. For
this purpose I will introduce a new reference [1].
This reference reports that work done by forces acting
on a system is equal to the change in the kinetic energy
of that system throughout the interval where the forces
are acting. We can write this,
Here 
is the work done and 
is the kinetic energy.
42. What is a
force?
43. How do forces
produce work?
44. What is kinetic
energy?
According to [1],
we can infer that a force is something that
causes acceleration in an object that has mass. This
acceleration will occur in the direction of the applied
force. It turns out that the magnitude of this
force is equivalent to the product of the mass and the
acceleration of the object. We can write this in
the famous form,
Here 
is the applied force, 
is the mass of the object, and 
is the acceleration of the object.
45. What is mass?
46. What is acceleration?
While the origin of mass is a deep mystery
in physics, its behavior is well known. Mass is
a measure of the quantity of inertia contained in an
object.
47. What
is inertia?
Inertia is the ability of an object to
avoid being accelerated.
Acceleration is the rate of change of
an object's velocity. The average acceleration
of an object is given by the formula,
![Overscript[a, _] = (Δ Overscript[v, _])/(Δ t) .](HTMLFiles/index_8.gif)
Here ![Overscript[a, _]](HTMLFiles/index_9.gif)
is the average acceleration, ![Overscript[v, _]](HTMLFiles/index_10.gif)
is the average velocity, and 
is the symbol for an interval. For example, assuming
we measure the average velocity at two times, and we
write these times as 
and  .
Then the interval is
![Δ Overscript[v, _] = Overscript[v, _] _2 - Overscript[v, _] _1 .](HTMLFiles/index_14.gif)
If you are familiar with calculus, acceleration
becomes the time derivative of velocity.
48. What
is a derivative?
49. What
is velocity?
Book Review: Essential Physics
Frank W. K. Kirk, "Essential
Physics," 2000, http://www.physicsforfree.com/essential.html.
Invariance is a central
thread of this book, that is the principle that the
equations of physics must look the same no matter where
you find them. Kirk begins the essence of the book
by explaining that invariance begins with the Pythagorean
theorem of geometry. This electronic book assumes
that you are familiar with basic calculus.
The first chapter is a
mathematical introduction that describes the algebra
of line segments and provides some ideas about curves
and surfaces. Kirk then introduces the idea of
groups in a very natural way. He goes on to present
the basic notions of vectors. In the next section,
Kirk uses quaternions to develop the Laplacian operator. Then
he develops the calculus of three-dimensional vectors. The
next section discusses the fundamentals of linear algebra
and higher-dimensional vectors. There is a section
on the geometry of vectors. Kirk then introduces
one of the fundamental mathematical objects used in
physics, that of the linear operator and their matrix
representations. He then uses this concept to develop
rotation operators and the effects that rotation have
on vectors.
Chapter 2 applies the
mathematics of the first chapter to the problem of motion. It
begins with the direct application of calculus to velocity
and acceleration. Kirk then develops the notion
of differential equations applied to various scenarios
of motion. Finally, velocity and acceleration are
portrayed in polar coordinates.
Chapter 3 begins with
an introduction to Galilean transformations. Then
Kirk introduces the Lorentz transformations of special
relativity. He then describes the relativistic
principle of invariance of the spacetime interval. Kirk
goes on to show that the Lorentz transformations form
a group and then the rotation group. He goes on
to use these mathematical constructs to deduce time
dilation and length contraction, two of the weird predictions
of special relativity. Then there is a section
on 4-velocity.
Chapter 4 is of more direct
relevance to what we are doing. This is the chapter
on dynamics. Kirk begins with a study of the so-called
law of inertia. This is followed by a summary of
Newton's three laws of motion. He then sets out
the laws of conservation of momentum and the law of
conservation of angular momentum. These ideas are
then extended to an arbitrary number of particles as
a single system. This is then extended to the rotation
of two particles about a single point and then the rotation
of a set of points. Work and kinetic energy are
then developed. This is followed by the principle
of potential energy. He then applies these principles
to collisions. These principles are wrapped up
in an extensive discussion of rigid bodies.
Chapter 5 is all about
invariance and the conservation laws. Kirk begins
by relating the conservation of momentum to invariance
of potential under translations. He does the same
thing with conservation of angular momentum and rotation. This
sets the stage for the application of group theory to
physics.
Chapter 6 covers the dynamics
of special relativity, where Kirk introduces the ideas
of 4-momentum and the energy-momentum invariant. Then
there is a section on the Doppler shift. He then
covers relativistic collisions and the Compton effect. These
ideas are then applied to particle physics.
Chapter 7 is all about
Newtonian gravitation, beginning with a discussion of
particles moving along curved paths. Kirk goes
into great detail about Newton's deduction that the
Moon actually falls around the Earth in its orbit. He
then generalizes this to central forces. He then
talks about bound and unbound orbits. Kirk then
discusses gravitational fields and potentials.
Chapter 8 forms an introduction
to Einstein's theory of gravitation. This begins
with the equivalence principle. Kirk then goes
on to describe length contraction and time dilation
in a gravitational field and the Schwarzschild line
element. He then goes on to develop the metric
in the presence of matter. This is followed by
a presentation of the approximation for the weak gravitational
field and then the refractive index of spacetime and
the deflection of sunlight.
Chapter 9 covers the calculus
of variations and its application to mechanics. Kirk
begins with a discussion of Euler's equation and then
Lagrange's equation. This is followed by a discussion
of Hamilton's equations of motion.
Chapter 10 returns to
the conservation laws in light of the new mechanical
machinery developed. Kirk begins with the development
of the conservation of mechanical energy. He then
covers the laws of conservation of momentum and angular
momentum.
Chapter 11 is an introduction
to the principle of chaos. Kirk begins with a discussion
of the motion of a damped and driven pendulum. Then
he describes the computer solution to differential equations.
Chapter 12 is about waves,
beginning with the basic form of a wave. Kirk follows
this with the general wave equation. He then applies
some notions of special relativity to waves. The
discussion then turns to plane harmonic waves, spherical
waves, and the superposition of waves. The chapter ends
with a discussion of standing waves.
Chapter 13 is on Fourier
analysis and begins with a section on orthogonal. Kirk
then follows with sections on Fourier series and Fourier
coefficients. He then applies this to saw-tooth
waveforms.
The book ends with a discussion
of the solution of ordinary differential equations and
a bibliography.
References
[1]: Frank
W. K. Kirk, "Essential
Physics," 2000, http://www.physicsforfree.com/essential.html.
Created
by Mathematica
(February 17, 2005)  |
