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, Sept. 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.
HW #2: Due Friday, Sept. 12: Problems 2.16, 17, 32. (If you can show 1∓2μcosθ - 2sinθ = 0 for prob. 32, that will
be good enough. The upper sign in the ∓ is for the case of the block moving up the slope. If it moves downward,
the friction has the opposite direction. I'll consider extra points if you want to come up with an expression for θ.)
"Dailies":
Tuesday, Sept. 2: Consider two different masses having the same kinetic energy. Show mathematically which has
the greater momentum.
Thursday, Sept. 4: Describe a non-inertial reference frame other than a vertically accelerating or a rotating one.
Show that Newton's First Law is not valid.
Tuesday, Sept. 9: Draw a FBD for Problem 2.16.
WEEK 2: GO TO LECTURE 2, 3
LECTURE 2: SECTIONS 2.3-4
2.3 THE VALIDITY OF NEWTON'S LAWS
When can we apply Newton's laws? When are they valid? You may expect me to say "all the time", but that's not
always the case. Newton's laws are only valid in what we shall call "Inertial Reference Frames", or "IRFs".
2.3 REFERENCE FRAMES
A reference frame is nothing more than the perspective of a particular observer. She will probably consider her
own position as the center of coordinates, (Don't we all think of the Universe as revolving around us, more or
less?) and will think of the directions in front, to her left, and above as the x, y, and z axes, and will measure time
according to her own wristwatch. Relativity is the study of how different observers describe the same phenomena.
In this course, we will focus on classical relativity, a good approximation for speeds much less than that of light.
INERTIAL AND NONINERTIAL REFERENCE FRAMES
Let us define an inertial reference frame as a reference frame in which Newton's Laws hold. After all, inertia is
resistance to change of motion. An object at rest will remain at rest unless acted on by a net nonzero force only if
it is measured in an inertial reference frame [IRF].
Consider a pair of fuzzy dice hanging from a car's rear-view mirror. If the car is at rest, they hang straight down.
The tension in the strings cancels the pull of their weight. Without changing those two forces, we begin to
accelerate the car. The dice will appear to accelerate toward the rear of the car, because it has ceased to be an
"IRF".
An important theorem says that any two inertial reference frames must move with a constant relative motion. In
other words, if the Earth is an inertial reference frame (and it is close enough for our present purposes), then only
reference frames moving with a constant speed and direction relative to it are IRFs.
Now what can we say about what is experienced in a non-inertial frame? Consider how you feel in an elevator
which is traveling downward and picking up speed. You feel lighter. It's as if there's an extra force opposing
gravity pulling up on you, which happens to be the opposite direction of the direction in which you are
accelerating. If you are in a car making a sudden turn around a curve, you feel yourself being "pulled" towards the
outside of the circle that you're moving in. Again it feels like you are experiencing an additional force in the
opposite direction of the direction of the centripetal acceleration. (i.e. in the direction opposite the direction that
points towards the center of the circle) This is known as "centrifugal" force, as opposed to the actual "centripetal"
force that causes you to travel in a circle.
In general, these "pseudoforces", the forces that you feel but that don't correspond to actual forces --- artifacts
that demonstrate that you (or your reference frame) are accelerating --- are related to the actual acceleration of

the frame, A , by


F   mA .
Pseudoforce due to an accelerated reference frame
It is a good thing that Newton's laws hold in reference frames that are not at rest because otherwise we would
have to throw out our textbooks when considering motion on the Earth, which is rotating at about 700 mph at our
latitude and hurtling through space around the Sun, which is moving at 43000 mph relative to our galaxy, which is
speeding at 600,000 mph relative to nearby galaxies. Now, because the rotation of the earth about its axis and its
revolution around the sun are circular motions, the Earth is not strictly an inertial reference frame. This means that
we should be able to witness certain noninertial pseudoforces. One of these is the Coriolis force, which causes
hurricanes to rotate in one direction north of the equator and in the opposite direction south of the equator, but
does not have much effect on the vortex of a toilet flushing, despite the Simpsons' episode about swirling toilets
with which you may be familiar. We will discuss the Coriolis force in Chapter 10, at the very end of this semester.
CENTRIPETAL AND CENTRIFUGAL FORCES
This is as good a time as any to review the differences and similarities between two terms -- centripetal and
centrifugal force -- that are both invoked in describing circular motion. Centripetal force is the real force, the net
force that obeys Newton's laws. It is the force as observed by someone at rest relative to the circling object.
Centrifugal force is the "pseudoforce", the force that the accelerating object feels. (That force pushing you to the
outside of the car, or pushing the laundry to the outside of the drum during the spin cycle. It is a pseudoforce
because it is experienced by an observer (the spinning thing) in a non-inertial reference frame (its own frame, the
frame that is circling). Both are related to the centripetal acceleration, ac
ac = v2/r,
where v is the tangential velocity of the thing moving in a circle, and r is the radius of the circle in which it moves.
The centripetal force is just mass times ac; the centrifugal force is the negative of mass times ac. Notice that the
centripetal acceleration points toward the center of the circle. (Draw the tangential velocities of the second hand
of a clock at the 12:00 position and at the 3:00 position, and convince yourself that the difference vector of the
velocities points toward the 7:30 position.)
SECTION 2.4: EQUATIONS OF MOTION FOR A PARTICLE
OK, enough of the overview. Let's take Newton's second law and transform it into a differential equation that we
can solve.


F  mr
This is a vector equation, and so we have to break it down into its component parts. This is most easily done in
Cartesian coordinates, that is, in x-, y-, and z-coordinates. It is easiest to do that in these coordinates and that is
because the equations are identical for each of these coordinates.

 
 

Fx  mx , Fy  my , Fz  mz
What about in other systems of coordinates? Well the other two systems that we often use in physics are
cylindrical and spherical coordinates. Since motion is often in a plane, that means that either of those other two
systems can be reduced to two dimensional circular motion. We will consider 2D motion in Chapter 8 when we
look at the general case of central forces, that is, forces that act on a line between two objects and depend only
on the distance between them. Even in Cartesian coordinates, most of the time we can resort to simple twodimensional motion.
UNIFORMLY ACCELERATED MOTION IN ONE DIMENSION
Since we will start out with problems in which the force, and therefore the acceleration, is constant, we can
recycle the mathematics of uniformly accelerated motion that we used in Intro Phys. There are four basic
equations that link the basic variables of motion:
v  v0  at
x  x0  v avg t
Four kinematics equations
for objects in uniformly accelerated motion
x  x0  v0  12 at 2
v 2  v02  2ax
𝑑𝑣
𝑑𝑥
The first two of these can be derived from 𝑎 = and 𝑣 = :
𝑑𝑡
𝑑𝑡
Integrate. 𝑣 = ∫ 𝑎𝑑𝑡. If 𝑎 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, then 𝑣 = 𝑣0 + 𝑎𝑡.
1
Integrate again. 𝑥 = ∫ 𝑣𝑑𝑡 = ∫(𝑣0 + 𝑎𝑡)𝑑𝑡 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2
2
The second and third can be derived through substitutions. Prove to yourself you can do them. Note that this
suite of equations is valid only if the acceleration is constant.
To solve a problem involving uniformly accelerated motion, first draw a picture of the problem and label it with all
the variables that you know, as well as whichever ones you are looking to find. If any of the four equations above
has only one quantity that you don't know, it’s a candidate to be the right equation for solving the problem. That's
all there is to it. Of these equations, notice that the third one might actually require you to solve a quadratic
formula. Review the quadratic formula if necessary.
FREE FALL:
One application of the mathematics of uniform acceleration is to problems involving free fall. Whenever the only
force acting on an object is gravity -- nothing else is touching the object, and there is negligible wind resistance -the object accelerates downward with an acceleration of
a = -g = -32ft/s2 = -9.8m/s2
acceleration due to free fall
Notice that this downward acceleration is the case for rising objects as they slow down as well as for objects
falling and picking up speed. NOTE: I will use the convention of "g" as a positive quantity, and downward
acceleration as negative.
LECTURE 3: SECTION 2.4, these notes. (Return to top.)
GENERALIZING TO 2D:
To generalize one-dimensional motion to two or three dimensions, we need to define the components of
position (x and y), velocity (vx and vy) and acceleration (ax and ay), and use them instead of the onedimensional quantities we used in the four equations above.
PARABOLAS:
We will show why projectiles follow parabolic arches, not just parabolas in y vs t, but also in y vs x.
Examples: rocket launch, fountains.
OVERVIEW OF THE NEXT FEW CLASSES:
We have two different chores for the next couple of classes. First, we need to develop a set of procedures
for analyzing a force problem, and second, we need to develop an inventory of different types of forces
and types of problems that we can analyze. The difficulty in doing all that in these notes is that these
notes, and class itself, are "linear", while these types of forces and problems are not. There is no natural
order of progression to follow. What I will try to do in these notes is begin by talking in general terms
about how to set up these problems, then identify the types of forces we will deal with, and then talk about
specific "scenarios", that is, special kinds of problems that require special considerations, such as inclined
planes, pulleys, problems involving multiple objects, and projectile motion.
Well, we have the theory for using force to analyze motion. But we need some help in order to actually
apply this to real problems. Here are some of the tools that we we use in doing these problems. First we
will look at problems in which the force is constant. This was as far as we got in Introductory Physics. The
simplest of these types of problems are problems in which the total force is zero, that is, "statics"
problems. I will introduce in class four kinds of forces and then various scenarios, and we will gradually
work up to looking at ever more complicated examples. Later we will look at non-constant forces. We will
need to develop new mathematical tools to deal with them.
PROBLEM-SOLVING TECHNIQUES: FREE-BODY DIAGRAMS
The first task in applying Newton's laws to any problem is to draw a picture. One reason for doing this is
that it allows you to begin to translate a word problem into a mathematics problem. Draw a picture of
whatever situation you are considering and label it with everything that you are given, converting the word
descriptions into short mathematical expressions. For a problem that you wish to consider in terms of
forces, you will want to convert the simple picture into what is called a "free-body diagram ". Consider an
object in this picture which is experiencing more than one force. Ah, this is a very important point. All
forces are exerted on specific objects. In Newton's third law, the action and the reaction force act on
different objects. Anyway, consider an object in your picture which is experiencing more than one force.
We wish to consider only those forces acting on that object. Begin by isolating that object pictorially. Do
this by drawing a dashed circle around the object. Next, draw an arrow for each force, beginning the
arrow at the center of this object and pointing it in the direction of the force. For now we will pretend that
the object is like a point particle, and we will draw the forces radiating from its "center of mass".
When you have finished identifying and marking all of these forces, then it is time to apply Newton's

second law in component form --- Fx
 
 

 mx , Fy  my , Fz  mz --- to the object of interest. From here
on in, this becomes a mathematics problem rather than a physics problem.
"FUNDAMENTAL" PHYSICAL FORCES
There are four fundamental forces in nature, at least that we know about. They are, in order of decreasing
strength, the strong nuclear force, the electromagnetic force, the weak nuclear force, and the gravitational
force. In our world, for scales larger then the size of the atomic nucleus, the only two of these forces that
really matter are the electromagnetic force and gravitational forces. The gravitational force results in
something that we call weight. The electromagnetic force accounts for most other macroscopic forces that
we observe, such as friction, "stickiness" and all other contact forces. (We'll talk about them shortly.)
WEIGHT
The weight of an object is related to its "quantity of matter", aka mass. What is the mass? It is the amount
of "stuff " in an object. We can define the mass practically by measuring how much force gravity pulls it
with. But we should not confuse the the mass with that force, which is called the weight. If your instructor
has a mass of 65 kg, or about 140 lbs, he will still have a mass of 65 kgs on the moon, but will weigh
about 23 lbs there. That is, his mass is universal, regardless of where he is, but his weight depends on
where he is and what other astronomical objects are pulling on him. We can describe this connection
between mass and weight by the following equation


W  mg
where the vector on the left is the weight, m is the mass, and the vector on the right is a factor that depends on
whether you are on the moon, on the Earth at sea level, or in say Denver. For our purposes, we will always use g
= 9.8 m/s2 for its magnitude, unless the problem states otherwise. Well, that is the magnitude of this force. The
direction of this force is "downward", where we define "down" as being the center, more or less, of the Earth.
(More about that in Ch. 10)
A reminder: another historical word for "mass" is "inertia". Inertia means the resistance of an object to being
budged, which depends only on how much "stuff" it contains, i.e. its mass.
OTHER FUNDAMENTAL FORCES AS ELECTROMAGNETIC FORCES
I said that the only forces that we really notice in our macroscopic world are gravitational or electromagnetic. Most
of the electromagnetic forces are not what you would immediately identify as electromagnetic. For example, when
you drag a sled along the ground and it experiences a frictional force, that frictional force is due to the
electromagnetic interaction between individual molecules in the sled and in the ground. This is called friction.
Furthermore, but less obvious, the force that keeps that object from falling through the ground is also
electromagnetic. This comes from the fact that the molecules in the sled repel the molecules in the ground (and
vice versa) if they get too close. Because this force acts perpendicular to the surface between the two objects, it is
called the "normal" force. The third hidden electromagnetic force is the force that acts along the rope with which
you are pulling this sled. The force here is conveyed within the rope by interatomic forces, which are
electromagnetic.
EQUATIONS FOR THESE OTHER FORCES:
THE NORMAL FORCE
Of these three electromagnetic forces, only one of them has a simple expression for magnitude. But I'm getting
ahead of myself. Let's start with the normal force. The normal force, as I mentioned before, has a direction
perpendicular to the two surfaces. It acts to keep the atoms of each object from falling through each other. In fact
it is nothing more than the electron clouds repelling each other. If you have any questions at all about whether it
points out of the surface or into the surface, ask yourself what would happen if you remove that surface, or if it
were made of thin paper that would tear immediately. The magnitude of this force is
[ There is no equation for the magnitude of the normal force* ]
*although it is always less than the maximum that the surface (table, chair, floor) can support: a simple cardboard
box will not support the weight of an SUV.
FRICTION
Now for the frictional force. As for its direction, the direction is in whichever direction hinders the relative motion of
the two surfaces. Often, when the instructor gets sloppy, s/he will say that the frictional force acts in the direction
opposite the motion. This is usually true, but there is a very interesting counterexample. Imagine that you are
standing in a bus as it starts to accelerate from rest. If the floor is dry and the acceleration is not too sudden, you
find yourself accelerating as well. How come? Well, it is the bus that is pushing you forward, through the
interaction between your feet and the floor. Is this friction? Well, what happens if someone oils the floor so that
you no longer have a good grip on it? You stay in place while the bus moves out from under you. So friction can
act in the direction of motion, but in this case it doesn't. It always acts in a direction opposite to the relative motion
of the two surfaces.
As for the magnitude of the frictional force, there is not always an expression for it:
[In the static case, there is no equation for the magnitude of friction*]
*although it is always less than the maximum friction those two surfaces can exert. In fact, in the kinetic case,
when the two surfaces are passing each other, experimental evidence shows that we can model the frictional
force as proportional to the normal force between the two objects:
fK = μK N
where fK is the kinetic frictional force, μ is "the coefficient of friction", a quantity characteristic of the two surfaces in
contact, and N is the normal force between the two surfaces. If we want to get technical, the coefficient of friction
depends on whether or not these two surfaces are moving relative to each other or not, so we can put a K
subscript on μ, to make it the kinetic coefficient of friction. In general, we can say that
f  N
TENSION
The force of tension is along the direction of the rope, chain, cord, etc. which experiences the tension. (For the
sake of argument, let's call this a 'rope'.) This rope pulls the objects on either end of it with this same quantity, T,
of tension, although usually in different directions. From Newton's third law, we see that the cord is in turn pulled
with a tension T by these two objects, with these two forces on the rope cancelling. The formula for the magnitude
of this force is
[ There is no formula for the magnitude of this force*]
*although it is always less than the maximum that the 'rope' can provide. It will match the amount of force with
which you pull on it, until it snaps. (something to consider before you strap yourself to that bungee cord)
One thing to be careful of: if more than two strings are knotted together, the different strings may have different
tensions within them, but if you have a single piece of string, it has the same tension throughout. Watch out.
INCLINED PLANE
Galileo used inclined planes as a way of "diluting" gravity. A block that falls 16ft in its first second of free falls less
than a foot on a frictionless plane inclined at 3 degrees from the horizontal. (Lucky thing, since Galileo had only
his own pulse or sense of rhythm to help him keep time.) The most important part of doing inclined plane
problems, whether statics or not, is to divide this problem into separate x-coordinate and y-coordinate
components, and solve each coordinate separately. We usually define the x-axis to point along the surface of the
plane (let's assume up the plane), and the y-coordinate to be perpendicular to the surface, and define θ as the
angle of elevation of the plane, that is, the angle between its inclination and the horizontal. For resolving the
weight into its components, this means that
Wx  mg sin 
W y  mg cos 
Notice that usually in physics we try to define angles relative to the positive x-axis, so that Fx = F cosθ and Fy = F
sinθ. The only reason the cosines and sines are backwards for this particular example is that θ is defined relative
to the vertical axis.
SUMMARY OF FORCES:
Force
⃗⃗⃗
Weight, 𝑊
Magnitude
𝑊 = 𝑚𝑔
Direction
“center of Earth”
⃗
Normal, 𝑁
<none>
perpendicular to surface
friction, 𝑓
𝑓 ≤ 𝜇𝑁
opposing relative motion
Tension, ⃗𝑇
<none>
along rope, chain, etc.
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