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Lesson 2: Newton’s Mechanics
I.
Particle
A particle is an object with constant mass and no physical dimensions
(i.e. It is a point).
No physical objects actually fit the definition of a particle although
lepton’s have physical dimensions of no more than 10-18 m. The
concept of a particle is a mathematical model which simplifies the
description of real physical systems. It is useful for problems in which
the physical dimensions of an object are small compared to other
physical dimensions in a problem and where one neglects any
rotational motion.
II.
A.
Newton’s First Law (For a Particle)
Words
A particle will continue in a straight line at constant speed unless
acted upon by a net external force.
B.
Mathematically


If  Fext  0 then v constant .
C.
What Does Newton’s First Law Tell Us?
1.
2.
3.
It tells us that the cause of acceleration is a net external force.
It states that the natural motion of a particle is constant velocity.
We can use these facts to determine when a net external force is
present!!
Newton’s Fist Law isn’t used to compute anything, but its importance
philosophically is immense!!
III.
A.
Newton’s 2nd Law (For a Particle)
Words
The acceleration of a particle is in the direction of the net external
force on the particle and the magnitude of its acceleration is
proportional to the magnitude of the net external force.
B.
Mathematically

C.


Fext  m a
Inertia & Mass
The tendency of a body to continue at constant velocity is called
inertia. Mass is the measure of a body’s inertia and is the
proportionality constant in Newton’s 2nd Law. You should also
remember that mass appears as the source of the gravitational force in
Newton’s Universal Law of Gravity. The fact that a body’s
gravitational mass was the same as a body’s inertial mass (a
seemingly different concept) was an amazing discovery. The
explanation of this observation remained a mystery until Einstein’s
solution in the early 20th century.
IV.
A.
Newton’s 3rd Law (For a Particle – Weak Form)
Words
If particle A applies a force of some type upon particle B then particle
B must apply the same type of force upon particle A that is equal in
magnitude and opposite in direction.
B.
Mathematically


FAB   FBA
C.
What Should You Get From The Third Law?
1. Particle don’t just act they interact!! There must be two objects and
two forces (force pair) for every interaction.
2. The total net force acting upon a system composed of all of the
particles must equal zero!!
3. Essential for connecting forces in different free body diagrams.
Example: What is the force of the string on the horizontal beam shown in the
diagram below?
10 kg
Solution: Many students will answer 98 N downward without thinking. This
is an easy problem even for an introductory physics course and using the full
strength of Newtonian Mechanics may seem like over kill. However, our
goals are conceptual understanding and procedures for handling more
challenging problems so I urge you to slow down and really think through
the problem.
Fact #1: The weight of the block is the gravitational force applied by the
earth upon the block and is not one of the forces acting upon the beam!!
Fact #2: I can think of instances in which the string will apply a force less
than 98 N downward upon the beam and instances in which the force is
greater than 98 N.
Let us analyze the system in a methodical manner by drawing free body
diagrams of the block and beam.
T1
P1
1
y
+
W1
W2
L/2
L/2
T1
x
Even this rather trivial problem where I have assumed a string that can’t
stretch and has no mass, we have multiple free body diagrams with several
forces and an applied torque by the wall. You should note that many of the
forces and the applied torque do not have their partner forces and torques
(according to the Newton’s third law) in the diagrams. This is because I have
only drawn free body diagrams for the beam and the block. For example, the
partner force for the weight of the block is the upward gravitational force
applied by the block upon the Earth. This force would appear on the free
body diagram of the Earth. You should also note that I have used Newton’s
3rd Law in dealing with the interaction between the block and the beam. The
force on both diagrams has the same magnitude T1, are the same type of
force (tension forces) and the directions of the arrows have been chosen to
fulfill the requirement in the third law that the forces are in opposite
directions. You should also note that the forces must be in two different free
body diagrams since the 3rd Law talks about two bodies interacting (one
force per each object’s free body diagram). Finally, you should note that one
can not find the tension in the rope without knowing the acceleration of the
block. Applying Newton’s 2nd Law, we have for the block
F M a
y
1
y
T1  M1 g  M1 a y
T1  M1 ( a y  g )
Thus, the tension is zero if the block and beam are accelerating downward in
free fall (i.e. ay = -g) while the tension is 196 N downward if the whole
system is in an elevator accelerating upward at 9.8 m/s2 (i.e. ay = g). The
student who immediately writes down 98 N downward is only correct if they
know that the system is not accelerating!! (Note: The system can be moving
and still have no acceleration.)
V.
Inertial Reference Frames
Since all motion is described by the position vector and its derivatives,
the mathematical description of the motion of any particle depends
upon your reference frame (coordinate system).
Newton was aware that the results of his Laws depended on the
reference frame of the observer.
Consider a boy and a girl doing an experiment with a box on a merrygo- round. They place the box at the outer edge of the merry-goround. The girl sits down in the center of the merry-go-round and
stretches out her arm pointing it toward the box. The boy stands on
the ground beside the merry-go-round and pushes the merry-go-round
until it reaches a good speed. The boy and girl then attempt to use the
motion of the box to and Newton’s Laws to determine the forces
acting upon the box. According to the girl, at every instant in timer the
box is located at the same distance from her position along the x’ axis
(direction that her arm points). Thus, the box is stationary and
therefore it is also not accelerating. In her frame of reference, there is
no net external force applied to the block according to Newton’s 1 st
Law. However, the situation is very different to the boy who sees the
block travel in a curved path. He believes that the block is
accelerating and that there is a net external force being applied.
x’
x
x
x’
Time #1
Time #2
The reason that the boy and girl disagree on the forces acting on the
box is that they are accelerating with respect to each other. If the two
observers are not accelerating with respect to each other then they will
agree on the acceleration of the box. Galileo realized that fact and it is
called Galilean Relativity:
All frames in uniform motion are equivalent.
This would seem to imply that the trick to applying Newton’s Laws
using only real forces lies in finding the correct reference frame (i.e. a
non-accelerated reference frame). This is the idea behind the concept
of an inertial reference frame. By definition, an inertial reference
frame is a frame of reference in which Newton’s 1st Law is valid.
However, we are quickly left in a predicament unless we can correctly
identify a point in space that is not accelerating. After all, the boy is
on a planet that spins about its axis (big merry-go-round) so he is
accelerating. The planet is also orbiting the sun which is rotating in
the Milky Way galaxy, etc. To a physicist prior to the 1900’s the
solution was to attach a reference frame to the luminiferous eather
(the invisible, stationary fluid that filled all of space). However, the
Michelson-Morley experiment showed that the eather does not exist.
Thus, Newton’s mechanics are fundamentally flawed in their need for
special reference frames. Einstein removed this flaw in his theory of
relativity.
VI.
A.
Linear Momentum
Definition:
The linear momentum of an object is defined as the product of the
object’s mass and its velocity.


PMV
This is an example of multiplication of a vector by a scalar. The
velocity and linear momentum vectors must always point in the same
direction!!
B.
Linear momentum of an object depends on the reference frame since
the object’s velocity depends on the reference frame. In analyzing
collisions, Einstein realized that mass must also depend on the
reference frame in order to conserve momentum.
VII. Newton’s Laws Revisited (Mass can vary)
A.
Newton’s 1st Law
If
B.



Fext 0 then P  constant
Newton’s 2nd Law



dP
Fext 
dt
This gives us the following




d(Mv) dM 
dv
Fext 

vM
dt
dt
dt



dM 
Fext 
vMa
dt
For a particle (constant mass), the first term on the right hand side is
zero and we have our previous result. The first term is often called the
rocket term since it explains how rockets can accelerate in outer space
even without an external force. However, the term has important
consequences in many mechanical systems on Earth where material is
added or removed from conveyor belts or other transportation
systems.