LECTURE NOTES FOR PHYSICS 307: CLASSICAL MECHANICS
TEXT: THORNTON AND MARION'S CLASSICAL MECHANICS OF PARTICLES AND SYSTEMS
SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors
ASIGNMENTS: (Subject to change: check back often.)
HW #1: Due Friday, September 5: Problems 2.5, 2.6, plus the following problem:
As a moving car plows through the air in front of it, that air exerts a force on the car: the wind resistance. Let's
imagine that during 1 second, we change the velocity of a mass of air, equivalent to the volume the car plows
through, from zero to 25m/s (55mph). Air has a density of 1.3kg/m 3 and the car scoops out a volume of air having
a cross-sectional area of 3m2. Show that the force the air exerts on the car varies as the square of the car's
speed, and that in this case it equals about 2500N or 550lb. (This is why your fuel efficiency drops above about
55mph.)
"Dailies":
Tuesday, Sept. 2: Consider two different masses having the same kinetic energy. Show mathematically which has
the greater momentum.
LECTURE 1: SECTIONS 2.1-2
OVERVIEW
You can think of the curriculum for physics majors as following a "spiral" path, one that returns to cover the same
material again, but each time in greater detail. In Physics 151/152, you studied the major topics in physics --mechanics, thermal physics, electricity and magnetism, optics, relativity, nuclear physics and quantum
mechanics. In Physics 221/2, you studied relativity, nuclear physics, and quantum mechanics in greater depth, as
well as being introduced to a few new topics. In this course, we will look more closely at mechanics, the study of
motion. We will start by reviewing some of the material that you learned before, but we will study it in greater
depth. For one thing, we will use the calculus of differentiation and integration to deal with examples of motion
that are more complex than what we were able to deal with freshman year.
2.1-2 INTRODUCTION: NEWTON AND MOMENTUM
We will start with Newton's three laws of motion from his Principia, but with a slight twist. Newton stated his laws
in terms of "the alteration of motion". We can replace "motion" with "momentum". Why would he consider
momentum the principal quantity of motion, rather than say velocity? Well, consider two objects colliding from
opposite directions. If we knew nothing about motion, we might say that the one with more "motion" will be the
one which causes the other to reverse its direction after the collision. OK, what determines which object will
continue to travel on the same direction? The momentum. A bullet fired at a charging rhinoceros is unlikely to
change the rhinoceros's direction from forward to back, despite the fact that the bullet has a larger speed (AND
KE?). Just for the record, momentum is defined as


p  mv


p  mv
for non relativistic objects;
for relativistic objects.
2.2 NEWTON'S FIRST LAW
Let's state Newton's first law in terms of momentum: If the net force acting on an object is zero, its motion is
constant. In other words,

F
 0
means

p  const .
This is the case either for an object at rest, or for an object traveling with a constant speed and direction (and thus
"uniform motion"). We are so used to this equation that we often forget what it means. Let's begin by breaking in
two and consider the two parts separately. First of all,
An object at rest remains at rest unless acted on by an outside force.
("Newton Ia")
Pretty harmless, eh? But what does it really mean? We can think of this law as providing a formal definition of the
term 'force'.
A force is anything that can alter the motion (momentum) of an object.
(Formal definition of force: alternate interpretation of Newton Ia)
This is a formal definition, not of much practical use. A more practical definition might read: "A force is anything
that can be measured in pounds (lb), or in Newtons." (N=kg.m/s2; 1lb = 4.45N.)
Notice that I said "anything that can alter the motion". If the net force is zero, as when, for example, the force of
gravity pulling you downward is exactly matched by your chair pushing up on you, then neither force changes the
motion of the object, even though either one could.
The second half of Newton's First law is
An object in motion remains in motion with constant motion (momentum) unless acted on by an
outside force.
("Newton Ib")
One way of interpreting this half of Newton's first law is to compare it to earlier theories of motion. Aristotle and his
followers believed that objects required external forces to continue moving, that is, that in the absence of an
external force, objects come to a halt. This is not a surprising conclusion to draw by living in our universe, full of
friction, in which things inevitably come to a halt. What happens if you don't keep your foot on the gas pedal on a
level road? The car stops. Of course, if the road were icy, so that there was less friction under the tires, it would
take longer. All we have to do is extrapolate these results to a frictionless universe to come up with Newton's
First.
I bring this up to remind you of what appears to be common sense: that an object in motion will come to rest
unless you continue to provide it with some impetus. The reason we believe this, by the way, is that in our world it
is nearly impossible to avoid friction, and friction acts to slow down almost all moving things, with the exception of
say electrons circling in atoms and molecules colliding with each other, so in this common-sensical case, the total
force does not equal zero.
Aside: This is another reason why physics appears to be so hard: we have been making observations of the
Universe around us since we were born, and usually we have been observing very complicated systems (which
include friction, for example), which make it hard to generalize properly. The conclusions we draw from our
highchairs congeal into that which we call common sense, and common sense is notoriously difficult to alter.
2.2 NEWTON'S SECOND LAW
Well, what happens if the force is not equal to zero? Newton's second law gives us the answer and provides us
with the foundation for most of mechanics. Stated in terms of momentum,
 dp
F
  dt
Notice that this reduces to our familiar
Newton's Second Law (in terms of momentum)


F
  ma if we assume that the mass is constant. In the relativistic limit
of course this breaks down. It also breaks down for cases in which the mass of the object is changing with time,
such as for a rocket which loses mass as its fuel as it is ejected. In fact we will look at this specific case later on in
Section 2.7.
2.2 NEWTON'S THIRD LAW
This is one of those laws of physics that is so well-known that if were only to say the words "For every action ... ",
you should be able to complete the phrase. Like the very famous laws of science, Einstein's E=mc2, the Second
law of Thermodynamics, and Darwin's Law of Natural Selection, we think we understand, but we could probably
do with a little reminder. In terms of momentum, if we have what we call a "system" of two particles, which may
exert forces on each other, then if there is no net external force,

ptotal  const ,
or


dp1
dp 2
,

dt
dt
Newton's Third Law (in terms of momentum)
where


p1 and p 2 are the momenta of the individual particles.
YSBATs (You should be able to...") In these notes I will try to sum up by listing what you need to memorize,
what you need to familiarize yourself with, the skills you need to master, etc.
From this lecture, for example, you need to memorize Newton’s 3 Laws, the definition of momentum (both
relativistic and classical), both definitions of force. In addition, you should be able to explain what the 3 laws
mean, and know what quantity is always conserved in a collision.
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