Newtonian Kinematics and Dynamics

Mechanics

Nov 13, 2013 (4 years and 5 months ago)

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Chapter 2
Newtonian Kinematics and
Dynamics
In this chapter we will:

study the central role of inertial
frames in Newtonian mechanics

introduce Newton’s three laws of motion

discuss the limitations of Newtonian mechanics
Physical phenomena often show a high degree of regularity.Our very notion of
time has been inﬂuenced by the periodic apparent motion of the sun and the moon
in the sky.Since the ancient times,natural philosophers aimed to devise empirical
laws in order to describe and predict the occurrence of such phenomena.
In ancient Babylonia (modern day Iraq),observations of the sun and the moon
during the 7th century BC established rules for predicting even complicated phe-
nomena,such as the 6585.322 day Saros cycle (about 11 years) that describes the
time between nearly identical solar and lunar eclipses.This cycle was used by
Thales of Miletus to predict a solar eclipse that occurred in 585 BC and o!ered as
an omen to stop the war between the Lydians and the Persians.By the time of
Ptolemy (2nd century AD),a very detaile
d set of empirical laws had been devised
to describe the motions of all objects in the solar system known at the time.
The Ptolemaic system of planetary motions placed the earth in the center of the
universe,considered the circle as the fundamental geometrical shape of planetary
motions,and required circles traveling aro
und circles,called epicycles,to describe
the motion of each planet.This empirica
l model of the solar
system epitomized
the two main characteristics of a physical law that were contributed by the ancient
philosophers:it was
predictive
and
consistent with observations
.It was based on
mathematical equations and allowed for quantitative predictions to be made for the
location of the planets at any point in the future.It had also been painstakingly
compared to observations and its parameters had been adjusted to achieve the
highest possible degree of c
onsistency with the data.However,it lacked a third
35
36
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
characteristic that would have made it equivalent to the laws of physics that we
study today.It did not o!er a
unifying framework
under which all the motions of
planets (or of other objects for that matter) could be explained with a simple set
of rules or equations.
In 1687,the British scientist sir Isaac Newton made the next revolutionary step
in the study of physics with his three-volume work
Philosophiae Naturalis Principia
Mathematica
,which is often referred to simply as
Principia
.Based on ideas by
Galileo Galilei and other natural philosophers of his time,he set forth a very small
number of laws of physics,from which he could derive not only the equations that
govern the motions of planets but also those that describe the motion of any other
object in the universe.Moreover,he dem
onstrated that his equations agreed in
Galileo Galilei
(1564–1642)
detail with the empirical laws of planetary motions discovered earlier in the 17th
century by Johannes Kepler based on the ideas of Nicolaus Copernicus and on
accurate observations made by Tycho Brahe.
Newton’s laws remained the cornerstone of physics until the formulation of the
theory of relativity and of quantum mechanics in the beginning of the 20th century.
Even though they are now known to be ina
ccurate when describing objects that
move at very high velocities or approach each other at very small distances,they
still o!er a very useful tool in making predictions for phenomena that we encounter
in every day life.In this chapter,we will study Newton’s three laws of motion
as a foundation of our study of mechanical systems.A thorough discussion of
the evolution of methods in the study of the physical universe can be found in
two excellent books:
The Sleepwalkers
[1]
by A.Koesler and
The Scientists
[2]
by J.
Gribbin.
2.1 The Galilean Principle of Relativity
In Chapter 1,we asserted that physical laws may not depend on the position and
orientation of an observer performing an experiment.Requiring that the physical
laws are invariant under translations and rotations of the coordinate system led us
to the study of vector analysis and calculus.In this paragraph,we will extend our
set of requirements and investigate how physical laws depend on the velocity of the
observer.
Consider two observers
O
and
O
!
,with the primed observer moving at a velocity
!
u
with respect to the unprimed one.Each observer makes measurements using a
Cartesian coordinate system.Without loss of generality,we can assume that the
two coordinate systems have parallel axes,that they overlap at time
t
= 0,and
that they are oriented in such a way that the relative motion of the two observers
is along the
x
1
axis.
!
"
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t
u
v
w
x
y
z
{
|
}
~

O
d=u
·
t
x'
x'
x'
x
x
x
1
2
3
2
3
1

O'
After a time"
t
,the distance between the two observers is
d
=
u

t
.The
coordinates of a given point
P
measured by the two observers is given by transfor-
mation (1.1),i.e.,
2.1.THE GALILEAN PRINCIPLE OF RELATIVITY
37
x
!
1
=
x
1
!
ut
x
!
2
=
x
2
(2.1)
x
!
3
=
x
3
Galilean Transformation
We call these equations the Galilean transformation,for reason that will become
apparent shortly.
We will now consider that the point
P
corresponds to the location of a moving
object with position vector
!
r
=
3
!
i
=1
x
i
ˆ
e
i
and velocity
!
v
=
d!
r
dt
=
3
!
i
=1
dx
i
dt
ˆ
e
i
in the unprimed frame.We can then use the Galilean transformation (2.1) to derive
a relation between the components of the velocity measured by the two observers
as
v
!
1
=
v
1
!
u
v
!
2
=
v
2
v
!
3
=
v
3
or in a more compact vector form
!
v
!
=
!
v
!
!
u.
Note that this last expression is valid in general,independent of the relative orien-
tation of the velocity vector between the t
wo observers and the coordinate axes.
Our short discussion of the Galilean transformation so far demonstrates that
the velocity of a moving object depends on the velocity of the observer that makes
the measurement.This is,of course,somet
hing we experience in our everyday life.
When we are standing on a street,the trees on
its side have zero velocity.However,
when we observe the same trees from inside our moving car,they appear to move
with a velocity opposite to that of the car.
We can go one step further and deﬁne the a
cceleration vector of moving object,
as the time derivative of its velocity vector,i.e.,
!
a
"
d!
v
dt
=
d
2
!
r
dt
2
.
If the relative velocity of the two observers remains constant in time,then it is
straightforward to show,by taking the time derivatives of both sides of equa-
tions (2.1),that each observer measure
s the same acceleration for the object,
!
a
!
=
!
a.
In the early 17th century,the Italian scientist Galileo Galilei theorized that the
laws of physics are the same between any two frames that are related by transfor-
mations (2.1).
38
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
Galileo introduced this assertion,which we call the Galilean principle of rel-
ativity,in his 1634 book
Two New Sciences
.He based it,according to his own
description,on a set of experiments of the motion of objects on inclined planes.
Legend has it that Galileo proved his assertion by dropping two objects o!the
leaning tower in the city of Pisa.While the leaning tower experiment is almost
certainly just a legend,there is also speculation that Galileo’s experiments with
inclined planes were simply thought exper
iments that were never carried out.In-
dependent of its origin,the Galilean principle of relativity plays a central role in
Newton’s laws of motion that we will now discuss.
Example 2.1:
Acceleration Measured in a Rotating Frame
In this example,we will investigate whe
ther the acceleration measured by two ob-
servers that are rotating with respect to each other is the same.
We consider two Cartesian frames,one for each observer
O
and
O
!
,with a common
point of origin.At some initial time
t
= 0,the coordinate axes of the two frames
coincide.The coordinate system of the primed observer is then allowed to rotate
at a constant frequency
f
around the
x
3
-axis.
!
"
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)
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+
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1
2
3
4
5
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=
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B
C
D
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F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
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a
b
c
d
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f
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j
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l
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o
p
q
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s
t
u
v
w
x
y
z
{
|
}
~
x
x'
1
x'
2
x'
3
x
2
3
x
1

We will measure the relative orientation of the two coordinate systems by the
angle
"
between the axes
x
1
and
x
!
1
.This angle changes periodically from
"
= 0 to
"
= 2
#
in regular time intervals of duration
f
"
1
.As a result,after a time
t
,the
angle between the two axes is
"
= 2
#ft.
We now consider an object with coordinates (
x
1
,x
2
,x
3
) in the frame of the
unprimed observer.In order to calculate its coordinates in the frame of the primed
observer,we will use the transformation (1.4)
"
#
x
!
1
x
!
2
x
!
3
\$
%
=
"
#
cos(2
#ft
) sin(2
#ft
) 0
!
sin(2
#ft
) cos(2
#ft
) 0
0 0 1
\$
%
"
#
x
1
x
2
x
3
\$
%
.
or,equivalently,
x
!
1
= cos(2
#ft
)
x
1
+sin(2
#ft
)
x
2
x
!
2
=
!
sin(2
#ft
)
x
1
+cos(2
#ft
)
x
2
x
!
3
=
x
3
We can now take the ﬁrst and second derivatives of these expressions in order
to calculate the transformation laws for t
he velocity and acceleration of the object
between the two frames.For example,we ﬁnd after a small amount of algebra that
the
u
!
1
component of the velocity vector of the object in the primed frame is
u
!
1
= sin(2
#f
)(
!
2
#fx
1
+
u
2
) +cos(2
#f
)(
!
2
#fx
2
+
u
1
)
and the corresponding compon
ent of the acceleration is
a
!
1
= cos(2
#ft
)
&
!
(2
#f
)
2
x
1
+2
#f
(
u
1
+
u
2
) +
a
2
'
+sin(2
#ft
)
&
!
(2
#f
)
2
x
2
!
2
#f
(
u
1
+
u
2
) +
a
1
'
.
2.2.NEWTON’S LAWS OF MOTION
39
Newton’s Absolute Time and Space
In the
Principia
,Newton appears to be well aware of the inability of any observer
in the Universe to measure absolute distances in space and time.In fact,he
concedes that “there may be no body tru
ly at rest to which places and motions
may be referred”
[3]
.He feels compelled,however,to deﬁne absolute space and
time as a conceptual construction,perhaps related to the ﬁxed stars of the
distant Universe.
The concept of absolute time and space poses many philosophical questions that
have bothered a number of researchers since the time of Newton.Can we deﬁne
absolute time and space in a Universe that is completely empty?What is the
meaning of relative motions in a Universe that contains a single object?Our
inability to answer questions like these imply that even a conceptual construction
of an absolute time and space is a!ected by the description of the Universe as a
whole.In fact,Newton himself used the distant,ﬁxed stars in his deﬁnition.
The concept that inertial frames can only be deﬁned when the properties of the
whole Universe are taken into account is often referred to as
Mach’s Principle
,in
honor of the Austrian scientist and philosopher Ernst Mach (1838–1916).Mach’s
criticism of Newton’s ideas inﬂuenced Albert Einstein and the development of
the theory of General Relativity.
Isaac Newton
(1643–1727)
Ernst Mach
(1838–1916)
Clearly,
a
!
1
#
=
a
1
and the same is true for the othe
r two components of the accelera-
tion.As a result,two observers that are rotating with respect to each other do not
measure the same accelerat
ion for a moving object.
2.2 Newton’s Laws of Motion
In order to express any law of physics in terms of a mathematical equation,we Inertial
Frames
need ﬁrst to deﬁne the reference frame in which it is valid.The Galilean principle
of relativity suggests that all frames related by the Galilean transformation,i.e.,
frames that move with constant velocity wi
th respect to each other,are equivalent.
They are,therefore,good candidates for being the fundamental reference frames on
which the laws of motion can be deﬁned.We will call the set of reference frames
related by the Galilean transformation
inertial frames
.
The deﬁnition of inertial frames is based on the relative motion of one frame
with respect to another.If we know that on
e reference frame is inertial,then we
can verify whether a second reference fra
me is also inertial by conﬁrming that the
two frames are related by the Galilean transformation.This procedure,however,
requires that we know
a priori
that at least one reference frame is inertial.Newton
40
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
considered this absolute reference frame to be the one of the ﬁxed,distant stars.
This idea was reasonable in the 17th centu
ry,since stars were indeed thought to be
ﬁxed in the sky.However,we now know that it is inadequate as a deﬁnition of an
absolute reference frame,given the fact that all stars have considerable velocities
with respect to each other and the Universe itself expands as a whole.The concept
of an absolute inertial reference frame has remained a matter of debate since the
times of Newton and a source of inspiration for Einstein’s theory of general relativity
(see box in the previous page).
Based on this relative deﬁnition of inertial frames,we can now discuss in some
detail Newton’s three laws of motion.
A body will remain at rest or in a uniform motion unless a net force is acting
on it.
Newton’s First Law
The ﬁrst law is often considered to be devoid of any additional physical meaning
or importance.However,it was introduced to distinguish Newton’s theory fromthe
prevailing ideas of the time that the fundamental trajectory of an object is a circle.
Moreover,we can use it to construct a second deﬁnition of inertial frames:an
inertial frame is one in which a body will remain at rest or in a uniform motion
unless a net force is acting on it.Note that a rotating frame cannot be inertial.
The acceleration
!
a
of a body,when measured in an inertial frame,is proportional
to the net force
!
F
acting on it
!
F
=
m!
a.
Newton’s Second Law
As implied by the principle of relativity,the laws of motion of an object cannot
depend explicitly on its velocity but th
ey may depend on its acceleration.We will
show in the following chapters that the simplest of all possibilities,i.e.,that the
acceleration of an object is linearly proport
ional to the total for
ce acting on it (as
opposed to any other function of the total force),leads to predictions that agree
with experiments.The constant of proport
ionality in Newton’s second lawmeasures
the inertia of an object in response to an external force:the higher its inertia,the
slower the object will accelerate.We will ca
ll this constant of pr
oportionality the
inertial mass
of an object.
Two bodies exert parallel but opposite forces on each other.
Newton’s Third Law
The third law guarantees that the mot
ion of an object cannot be altered by
internal forces.If piece
A
of an object exerts a force on piece
B
,then piece
B
also
exerts the opposite force on piece
A
.The net force on the object is zero,and the
second law leads to the concl
usion that the acceleratio
n of the object as a whole is
also zero.The third law is very important in the study of mechanics as it leads to
conservation theorems for the energy an
d momentum of a closed
system of objects.
In the remaining of this chapter,we will study the motion of an object that
experiences a constant force.In Chapters 3 and 5,we will investigate the properties
of various forces that a!ect the motion of mechanical systems.In Chapter 4,we will
2.3.THE MOTIONOF ANOBJECTEXPERIENCINGACONSTANT-MAGNITUDE FORCE
41
The International System of Units (SI)
[4]
Throughout this book,we will use the International System of Units (often
called SI from the French Syst´
eme International).This the most widely used
system of units.It was created in its original form in 1799,during the French
revolution,to take advantage of the beneﬁts of the decimal metric system.In its
present form,it consists of the units for seven fundamental quantities,as well as
the names and rules for deriving the units for 26 additional quantities.The SI
system is summarized in the following tabl
e,created by the National Institute
of Standards & Technology:
In our study of mechanics,we will use the units for time,length,and mass,
which are deﬁned in the following manner:
Time:
The unit of time is one second (s),deﬁned as 9,192,631,770 periods
of oscillation in the electromagnetic wave emitted during the transition of an
electron between the two hyperﬁne levels of the Caessium-133 atomin its ground
state.
Length:
The unit of length is one meter (m),deﬁned as the length traveled by
light in vacuum during 1/299,792,458 of a second.
Mass:
The unit of mass is one kilogram(kg),deﬁned as the mass of the prototype
kilogram made of a platinum-iridium alloy and kept in S´
evres,France.
The units of time and length are based on the values of fundamental constants in
the theory of Quantum Electrodynamics (QED) and in the theory of Relativity,
such as Planck’s constant and the speed of light.The unit of mass,on the other
hand,is based on a particular sample and is not motivated by any physical
theory.This reﬂects the fact that we do not yet have a widely accepted theory
for what determines the masses of fundamental particles in physics.
prove various conservation theorems based on Newton’s laws of motion and discuss
their implications.
2.3 The Motion of an Object Experiencing a Constant-
Magnitude Force
The motion of an object under the inﬂuence of a net force
!
F
is described by Newton’s
second law
d
2
!
r
dt
2
=
1
m
!
F.
42
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
In this expression,
!
r
is the position vector of the object and
m
is its inertial mass.
If the object is free to move in the three di
mensional space,Newton’s second law
is equivalent to three second-order,ordinary di!erential equations,one for each
coordinate of the position vector.
We can obtain a solution to a second-order,ordinary di!erential equation,such
as Newton’s second law,if we specify two boundary conditions for each dimension
of the space in which the object is moving.We may do this in one of two ways:
(i)
as an initial value problem,in which case we specify the position and velocity of
an object at some initial time
t
= 0,or
(ii)
as a boundary value problem,in which
case we specify either the position or the ve
locity of an object both at some initial
time
t
= 0 and at same later time
t
=
t
0
.
An initial value problem in mechanics always has one unique solution.One
the other hand,a boundary value problem may have none,one,or many solutions,
depending on the boundary conditions.If we specify,for example,the location from
which we throwa basketball with a given velocity,i.e.,solve an initial value problem,
we will be able to calculate a unique trajectory for the ball.If we specify instead
the initial velocity of the ball as it leaves our hands and its intended ﬁnal location
(the hoop) then we can typically ﬁnd zero,one,or two trajectories,depending on
the magnitude of the velocity and the distance to the hoop.
The fact that an initial value problemin mechanics always has a unique solution
is a consequence of the Cauchy-Kovalevska
ya theorem,which states in general that
a di!erential equation that involves a derivative of order
n
has a unique solution if
the derivatives up to order
n
!
1 are speciﬁed on a single free surface in the domain
of solution.A physical law that always leads to a unique solution given a properly
deﬁned set of initial values is said to allow for a well posed initial value (or Cauchy)
problem.This is a property we require from all our laws of physics.
2.3.1 Linear Motion
When the motion of an object takes place
along a straight line,Newton’s second
law becomes the single,second-
order di!erential equation
d
2
x
dt
2
=
a,
where the coordinate
x
measures the distance along the line of motion.This equa-
tion is equivalent to the system of ﬁrst-order di!erential equations
dx
dt
=
u
(2.2)
du
dt
=
a.
(2.3)
Under the assumption
that the acceleration
a
is constant throughout the motion
of the object,we can integrate equation (2.3) as
u
=
u
0
+
at,
(2.4)
where
u
0
is the velocity of the object at time
t
= 0.Inserting this solution into
2.3.THE MOTIONOF ANOBJECTEXPERIENCINGACONSTANT-MAGNITUDE FORCE
43
equation (2.2) we obtain
x
=
x
0
+
u
0
t
+
1
2
at
2
,
(2.5)
where
x
0
is the position of the object at time
t
= 0.
If we are solving an initial value problem,we can complete the solution by
specifying the position
x
0
and velocity
u
0
at the boundary
t
= 0.Alternatively,we
may be solving a boundary value problem for which the velocity
u
0
at time
t
= 0 is
unknown,but the position
x
1
of the object at a subsequent time
t
=
t
1
is speciﬁed.
In this case,we will evaluate equation (2.5) at time
t
=
t
1
to calculate the initial
velocity of the object as
x
1
=
x
0
+
u
0
t
1
+
1
2
at
2
1
\$
u
0
=
x
1
!
x
0
t
1
!
1
2
at
1
(2.6)
and insert this value into equations (2.5) and (2.4) to obtain
x
=
x
0
+(
x
1
!
x
0
)
t
t
0
+
1
2
at
(
t
!
t
0
)
u
=
x
1
!
x
0
t
0
+
a
(
t
!
1
2
t
0
)
.
In both the initial value and the boundary value cases,we found a unique solution
to this one dimensional kinematics problem.
Example 2.2:
Pilots Experience High Decelerations during Carrier
Landings
The F/A-18 Super Hornet aircraft,shown here during its ﬁrst landing on the
USS Abraham Lincoln,in July 2002,approaches the runway of a carrier at a speed
of
u
ap
= 135 knots= 69
.
5 m s
"
1
.In this example,we will ca
lculate the deceleration
experienced by a pilot in a Super Hornet,a
s the aircraft stops within the length
d
= 400 ft= 122 m of the ﬂight deck of the carrier.
In order to calculate the average deceler
ation experienced by the pilot,we will
assume that the landing occurs along a str
aight line and that the deceleration is
constant throughout the landing.In reality,the aircraft experiences an initial jolt
when the tailhook that extends under the airplane arrests one of the wires on the
ﬂight deck,after which the deceler
ation slowly decreases to zero.
This is a boundary value problem,since we now the velocity of the aircraft at
the beginning of the landing (
u
=
u
ap
) and at the end (
u
= 0).Contrary to the
boundary value problem we discussed earlier in this paragraph,however,we do
not know the duration of the motion,but rather the total length
d
covered by the
aircraft.
!
"
#
\$
%
&
'
(
)
*
+
,
-
.
/
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
x
d
u
ap
We will denote by
a
the unknown acceleration and us
e equation (2.4) to calculate
the time it took for the aircraft to decelerate to zero speed.Setting
u
= 0 and solving
for the time
t
,we obtain
t
=
!
u
ap
a
.
44
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
We will then insert this time into equation (2.5) to calculate the distance covered
by the aircraft during this time as
x
=
!
u
2
ap
2
a
.
Note that the minus sign appears because o
f the negative sign of the acceleration.
In order for the aircraft to complete a su
ccessful landing,the stopping distance
has to be smaller than the length of the ﬂight deck of the carrier,i.e.,
x
%
d
,from
which we obtain
|
a
| %
u
2
ap
2
d
= 39
.
6
*
u
ap
69
.
5 m s
"
1
+
2
(
d
122
m
)
"
1
m s
"
2
.
Aircraft designers often compare the a
cceleration pilots feel during maneuvers
to the acceleration they would have felt if they were falling freely to the ground
from a small height.The latte
r is called the gravitatio
nal acceleration.As we will
see in the next chapter,its value depends slightly on the location on the surface of
the earth.The commonl
y quoted mean value of the gravi
tational acceleration,also
called
standard gravity
,is
g
= 9
.
80665 m s
"
2
.The acceleration we calculated above
is,therefore,
a
g
= 4
.
0
*
u
ap
69
.
5 m s
"
1
+
2
(
d
122
m
)
"
1
*
g
9
.
81 m s
"
2
+
"
1
.
As an aircraft designer would have said,the pilots experience 4
g
’s during carrier
landings.
2.3.2 Uniform Circular Motion
A di!erent situation in which an object mo
ves in the presence of a constant acceler-
ation is that of uniform rotation.In order to explore this case,we will consider an
object moving at a constant speed
v
on a circle of radius
r
.The object completes a
full revolution in a time
P
,which we will call the period of the motion.The rate
f
at which the object completes the revolutions,which we will call the frequency of
the motion,is simply
!
"
#
\$
%
&
'
(
)
*
+
,
-
.
/
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~

u
r

a
f
=
1
P
.
The velocity vector of an object is always
tangent to its traj
ectory.Therefore,
in a uniformcircular motion,it is always perpendicular to the position vector of the
object with respect to the center of the circ
ular trajectory.In order to calculate the
magnitude of the velocity
!
u
,we consider the fact that the object covers a distance
equal to the circumference 2
#r
of the circle in a time equal to one period
P
,so that
|
!
u
|
=
2
#r
P
= 2
#fr.
We will now deﬁne the angular velocity vector
!
\$
with a magnitude
|
!
\$
|
=
2
#
P
= 2
#f
2.3.THE MOTIONOF ANOBJECTEXPERIENCINGACONSTANT-MAGNITUDE FORCE
45
and an orientation that is perpendicular to the plane of the trajectory,such that
!
u
=
!
\$
&
!
r.
(2.7)
Velocities in
Circular Motion
For the counterclockwise circular motion shown in the margin ﬁgure,the direction
of the angular velocity vector is from the plane of the page towards the reader.For
a uniformcircular motion,the angular velocity vector remains constant in time and
measures the rate of change of the azimuthal angle
"
of the objects from a ﬁducial
point along its trajectory,i.e.,
!
\$
=
d"
dt
.
In order to calculate the acceleration r
equired to keep an object in a circular
motion,we take the derivatives of both sides of equation (2.7) with respect to time:
!
a
=
d!
u
dt
=
!
\$
&
d!
r
dt
=
!
\$
&
!
u.
Using now equation (2.7) to express the velocity vector in terms of the angular
velocity,we obtain
!
a
=
!
\$
&
(
!
\$
&
!
r
)
.
(2.8)
Acceleration in
Circular Motion
The magnitude of the acceleration in a uni
form circular orbit is,therefore,
|
!
a
|
=
\$
2
r
=
u
2
r
and its direction is always towards the cen
ter of the trajectory.It is very important
to remember here that the order of the various terms in the cross products in the
last several equations is not arbitrary,as it determines the signs of the results.
Example 2.3:
Stop or Turn?
A car is traveling at a speed
u
0
towards a wall.At a distance
d
from the wall,the
driver decides to do something in order to avoid the collision.Which action would
require the least amount of accelerat
ion (or deceleration) on the car:
(i)
continue
along the same course and apply a cons
tant deceleration to a full stop,or
(ii)
turn
at a constant speed,following a quarter-circle path.
!
"
#
\$
%
&
'
(
)
*
+
,
-
.
/
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
u
0
d
In order to answer this question,we will
calculate the magnit
ude of acceleration
required to avoid collision in each case separately.
In the ﬁrst case,we would like to bring the car to a full stop,after some time
t
0
,such that the total distance traveled is less than
d
.Assuming that the car is
decelerating at a
constant rate
a
lin
,we ﬁrst calculate the time to full stop by setting
u
= 0 in equation (2.4),as
0 =
u
0
+
a
lin
t
0
\$
t
0
=
!
u
0
a
lin
.
46
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
The minus sign in this equation reﬂects t
he fact that the car will be decelerating,
so that
a <
0.We then calculate the total distance traveled during this time using
equation (2.5) as
x
=
u
0
t
0
+
1
2
a
lin
t
2
0
=
!
u
2
0
2
a
lin
.
We would like this length to be smaller than the distance to the wall,
x
%
d
,and,
therefore,
|
a
lin
|'
u
2
0
2
d
.
In the second case,we need to place the ca
r in a circular trajectory of radius
r
%
d
.We calculate the acceleration
a
circ
required to keep the car in this trajectory
using equation (2.8) as
|
a
circ
|
=
u
2
0
r
'
u
2
0
d
.
Comparing the last two inequalities,
we ﬁnd that the minimum acceleration in
the ﬁrst case is only half the minimum accel
eration in the second case.As a result,
staying on course and bringing the car to a full stop required the least amount of
acceleration.
2.4 Advanced Topic:Newton’s Second Lawin Curvi-
linear Coordinate Systems
Newton’s second law involves the second derivative of the position vector of an
object.When expressed in a Cartesian coordinate system,in which the basis vectors
are independent of position,it translates easily into a set of equations for the second
derivatives of the coordinates of the position vector.The situation is di!erent,
however,when Newton’s second law is expressed in a curvilinear coordinate system.
In a spherical-polar coordinate system,the position vector of an object located
at a point with coordinates (
r
,
"
,
%
) is always given by
!
r
=
r
ˆ
e
r
.
In order to calculate the velocity of the object,we have to take into account the fact
that both the coordinate
r
and the basis vector ˆ
e
r
will be di!erent as the object
moves.As a result,we can write
!
v
=
d!
r
dt
=
dr
dt
ˆ
e
r
+
r
d
ˆ
e
r
dt
.
(2.9)
The basis vector ˆ
e
r
changes with time only because the coordinates of the object
change with time.For this reason,we use the chain rule to write
d
ˆ
e
r
dt
=
&
ˆ
e
r
&r
dr
dt
+
&
ˆ
e
r
&"
d"
dt
+
&
ˆ
e
r
&%
d%
dt
.
2.4.
NEWTON’S SECONDLAWINCURVILINEARCOORDINATESYSTEMS
47
At this point,we need to evaluate the partial derivatives of the basis vector with re-
spect to each coordinate.In order to achiev
e this,we use the expressions we derived
in paragraph 1.6,connecting the basis vectors to those of a Cartesian coordinate
system,i.e.,
ˆ
e
r
= sin
"
cos
%
ˆ
e
1
+sin
"
sin
%
ˆ
e
2
+cos
"
ˆ
e
3
ˆ
e
!
= cos
"
cos
%
ˆ
e
1
+cos
"
sin
%
ˆ
e
2
!
sin
"
ˆ
e
3
(2.10)
ˆ
e
"
=
!
sin
%
ˆ
e
1
+cos
%
ˆ
e
2
.
The basis vector ˆ
e
r
depends only on the
"
and
%
coordinates and,therefore,
d
ˆ
e
r
dt
=
&
ˆ
e
r
&"
d"
dt
+
&
ˆ
e
r
&%
d%
dt
= (cos
"
cos
%
ˆ
e
1
+cos
"
cos
%
ˆ
e
2
!
sin
"
ˆ
e
3
)
d"
dt
+(
!
sin
"
sin
%
ˆ
e
1
+sin
"
cos
%
ˆ
e
2
)
d%
dt
=
d"
dt
ˆ
e
!
+sin
"
d%
dt
ˆ
e
"
.
Inserting this expression into equation (2.9),we obtain
!
u
=
dr
dt
ˆ
e
r
+
r
d"
dt
ˆ
e
!
+
r
sin
"
d%
dt
ˆ
e
"
.
Note that the linear velocity of an object
moving on a circular trajectory at the
equatorial plane of the coordinate system (sin
"
= 1) with a constant angular ve-
locity
\$
=
d%/dt
is
!
u
=
r
d%
dt
ˆ
e
"
=
r\$
ˆ
e
"
,
in agreement with what we found in the previous paragraph.
We calculate the acceleration of the ob
ject in a similar manner.Starting from
the deﬁnition,
!
a
=
d!
u
dt
=
d
dt
(
dr
dt
ˆ
e
r
+
r
d"
dt
ˆ
e
!
+
r
sin
"
d%
dt
ˆ
e
"
)
(2.11)
we see that we also need to calculate the time derivatives of the remaining two basis
vectors,
d
ˆ
e
!
/dt
and
d
ˆ
e
"
/dt
.The ˆ
e
!
basis vector depends on the coordinates
"
and
%
and,therefore,
d
ˆ
e
!
dt
=
&
ˆ
e
!
&"
d"
dt
+
&
ˆ
e
!
&%
d%
dt
= (
!
sin
"
cos
%
ˆ
e
1
!
sin
"
sin
%
ˆ
e
2
!
cos
"
ˆ
e
3
)
d"
dt
+(
!
cos
"
sin
%
ˆ
e
1
+cos
"
cos
%
ˆ
e
2
)
d%
dt
=
!
d"
dt
ˆ
e
r
+cos
"
d%
dt
ˆ
e
"
.
On the other hand,the basis vector ˆ
e
"
depends only on the
%
coordinate,i.e.,
d
ˆ
e
"
dt
=
d
ˆ
e
"
d%
d%
dt
= (
!
cos
%
ˆ
e
1
!
sin
%
ˆ
e
2
)
d%
dt
.
48
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
At this point,we use equations (2.10) to write
sin
"
ˆ
e
r
+cos
"
ˆ
e
"
= sin
2
"
cos
%
ˆ
e
1
+sin
2
"
sin
%
ˆ
e
2
+sin
"
cos
"
ˆ
e
3
+cos
2
"
cos
%
ˆ
e
1
+cos
2
"
sin
%
ˆ
e
2
!
sin
"
cos
"
ˆ
e
3
= cos
%
ˆ
e
1
+sin
%
ˆ
e
2
.
so that
d
ˆ
e
"
dt
=
!
sin
"
d%
dt
ˆ
e
r
!
cos
"
d%
dt
ˆ
e
"
.
Inserting all these expressi
ons into equation (
2.11),we can write t
he acceleration
vector of the object,after a small amount of algebra,as
!
a
=
a
r
ˆ
e
r
+
a
!
ˆ
e
!
+
a
"
ˆ
e
"
,
where
a
r
=
d
2
r
dt
2
!
r
(
d"
dt
)
2
!
r
sin
2
"
(
d%
dt
)
2
a
!
=
r
d
2
"
dr
2
+2
dr
dt
d"
dt
!
r
sin
"
cos
"
(
d%
dt
)
2
a
"
=
r
sin
"
d
2
%
dt
2
+2 sin
"
(
dr
dt
)(
d%
dt
)
+2
r
cos
"
(
d"
dt
)(
d%
dt
)
.
If we consider again the special case of uniform circular motion on the equatorial
plane of the coordinate system,then the above expressions reduce to
!
a
=
!
r
(
d%
dt
)
2
ˆ
e
r
=
!
r\$
2
ˆ
e
r
,
which is consistent with what we derived in the previous paragraph.
The expressions we derived for the positi
on vector,velocity,and acceleration of
an object in a spherical-polar coordinate system,as well as those in a cylindrical
coordinate system (see Problem#6) ar
e summarized in the box in the following
page.
2.5 Beyond Newton:Special Relativity
The Galilean principle of relativity and the concept of inertial frames remained the
cornerstones of physics for more than two
centuries.By the late 19th century,how-
ever,two developments started raising serious questions about the general validity
of these two concepts.
In 1864,the Scotish physicist James Clerk Maxwell published a treaty titled “A
James Clerk Maxwell
(1831–1879)
Dynamical Theory of the Electromagnetic Field”,in which he developed the foun-
dations of the theory of electromagnetism.What was surprising about his theory
was the fact that the relevant equations
for the dynamics of the electromagnetic
ﬁeld were not invariant under Galilean transformations.According to Maxwell’s
equations,two observers in two di!erent Galilean inertial frames would predict two
di!erent outcomes for electromagnetic
phenomena (see Problem 9).Clearly,such
inertial frames were not the fundamental frames on which Maxwell’s theory was
2.5.
BEYOND NEWTON:
SPECIAL RELATIVITY
49
Position,Velocity,and Acceleration Vectors
in Curvilinear Coordinate Systems
Spherical-Polar Coordinates:
Position vector
!
r
=
r
ˆ
e
r
Velocity vector
!
u
=
dr
dt
ˆ
e
r
+
r
d"
dt
ˆ
e
!
+
r
sin
"
d%
dt
ˆ
e
"
Acceleration vector
!
a
=
,
d
2
r
dt
2
!
r
(
d"
dt
)
2
!
r
sin
2
"
(
d%
dt
)
2
-
ˆ
e
r
+
,
r
d
2
"
dr
2
+2
dr
dt
d"
dt
!
r
sin
"
cos
"
(
d%
dt
)
2
-
ˆ
e
!
+
.
sin
"
d
2
%
dt
2
+2 sin
"
(
dr
dt
)(
d%
dt
)
+2
r
cos
"
(
d"
dt
)(
d%
dt
)/
ˆ
e
"
Cylindrical Coordinates:
Position vector
!
r
=
'
ˆ
e
#
+
z
ˆ
e
z
Velocity vector
!
u
=
d'
dt
ˆ
e
#
+
'
d%
dt
ˆ
e
"
+
dz
dt
ˆ
e
z
Acceleration vector
!
a
=
,
d
2
'
dt
2
!
'
(
d%
dt
)
2
-
ˆ
e
#
+
.
'
d
2
%
dr
2
+2
d'
dt
d%
dt
/
ˆ
e
"
+
d
2
z
dt
2
ˆ
e
z
to be deﬁned.It was widely believed at the time that a unique reference frame
exists,that of the “luminiferous aether”,in which Maxwell’s equations are properly
deﬁned.
The idea of such a ﬁxed reference frame was put to test by Albert Michelson
and Edward Morley,who performed an ingenious interferometric experiment at the
basement of their o#ce building in wha
t is now called the Case Western Reserve
University,in Cleveland,OH.With that experiment they were able to measure
the relative speed of light along two perp
endicular axes.The also repeated the
measurement during di!erent times of th
e day,as the velocity of their aparatus
with respect to the “aether” changed because of the rotation of the earth and its
orbital motion around the sun.They reached the remarkable conclusion that the
measured speed of light appeared to be independent of the velocity of their aparatus,
in contradiction to the principle of Galilean relativity.
The resolution to these problems came in 1905 by Albert Einstein.In his special
theory of relativity,Einstein generalized our notions of space and time by suggesting
that the speed of light can be independent of the velocity of the observer only if the
distances and time intervals measured by di
!erent inertial observers depend on the
50
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
observers’ velocities.Based on earlier w
ork by the dutch physicist Hendrik Lorentz,
he suggested that the Galilean transformation between inertial frames is only ap-
proximately valid,when the magnitude of the relative velocity of the two frames
is much smaller than the speed of light.The most general transformation between
inertial frames is the Lorentz transformation,which for the example discussed in
pages 36–37,is given by
t
!
=
(
*
t
!
xu
c
2
+
x
!
1
=
(
(
x
1
!
ut
)
x
!
2
=
x
2
(2.12)
x
!
3
=
x
3
Lorentz Transformation
In this expressions,
u
is the magnitude of the relative velocity of the two observers,
c
is the speed of light,and we encounter for the ﬁrst time the Lorentz factor
(
"
(
1
!
u
2
c
2
)
"
2
.
In Chapter 23,we will explore in more detail Einstein’s theory of relativity
as well as the implications of the validity of the Lorentz transformation.For the
purposes of our current discussion,we just need to emphasize what is perhaps the
theory’s most important consequence for all of physics:that all physical laws,and
not just those that describe the kinematics of objects,are the same in all frames
that are related by Lorentz transformations.This axiom is called the principle of
Local Lorentz Invariance (LLI).
Local Lorentz invariance has been test
ed during the last several decades in
many di!erent settings and with many di!erent experiments
[5]
.Interferometric
experiments,such as the Michelson-Morley experiment discussed above,aimto ﬁnd
di!erences in the speed of light along two orthogonal axes.A di!erent set of exper-
iments,called the Hughes-Drever experiments in honor of Vernon Hughes and Ron
Drever,aim to test weather di!erent law
s of physics remains the same in di!erent
inertial frames,as required by Einstein’s principle of relativity.In one version of
this experiment,Drever and his collaborators studied the splitting of the
J
= 3
/
2
ground state of
7
Li in the presence of an external magnetic ﬁeld.Even though
this state is split into four levels by the magnetic ﬁeld,the spacing of the levels is
the same.As a result,all transitions between the split levels give rise to a single
monoenergetic line of emission.If,howeve
r,there was a preferred direction associ-
ated with the velocity of the inertial frame,then the four split levels of this ground
state would not have equal spacing and the resulting emission line would not be
monoenergetic.
In order to interpret the various experiments we have to allow for the Lorentz
transformations to be violated.We achieve this by assuming that the actual speed
of light
c
light
is di!erent from the speed
c
Lorentz
that enters the Lorentz transforma-
tion (2.12).We then use the experiments in order to place an upper limit on the
2.5.
BEYOND NEWTON:
SPECIAL RELATIVITY
51
violation parameter
)
"
c
2
Lorentz
c
2
light
!
1
.
If local Lorentz invariance is valid and the speed of light is the fundamental speed
that enters the Lorentz transformations,then the parameter
)
has to vanish.The
bounds on possible violations of the Lorentz transformation placed by many exper-
iments over the last century (as described in detail in [5]) are shown in the margin
ﬁgure.The most accurate experiments to date place an upper bound on the value
Bounds on possible violations
of Local Lorentz Invariance
[5]
of the violation parameter
)
of
|
)
| %
10
"
22
.
As we can see,the Einstein principle of relativity has passed a large number of
extremely accurate tests to date.For this r
eason,it is the cornerstone of all modern
theories of physics.For the purposes of this book,we will consider the evidence
shown in the margin ﬁgure as justiﬁcation of our use of the Galilean transformation,
which is an approximation of the more general Lorentz transformation,when the
magnitude of the velocity of an inertial frame is much smaller than the speed of
light.
1.
The Sleepwalkers:A History of Man’s Changing Vision of the Universe
.by A.
Koestler (Penguin)
2.
The Scientists:A History of Science Told Through the Lives of Its Greatest
Inventors
.by A.Koestler (Random House)
3.
Newton’s Philosophiae Naturalis Principia Mathematica
,in the Stanford Ency-
clopedia of Philosophy (2007)
4.
The International System of Units (SI)
,NIST Special Publication 330,edited by
B.N.Taylor and A.Thompson,(2008),available at
http://physics.nist.gov/Pubs/SP330/sp330.pdf
5.
The confrontation between General Relativity and Experiment
,by C.Will,Living
Rev.Relativity,9 (2006),3
Suggested Problems
1.Show that Newton’s second law is not invariant under a transformation to a
frame that is uniformly acceleratin
g with respect to an inertial frame.
2.Consider a general transformation be
tween two observers,which may depend
on the position,time,and relative velocity of the two observers.Show that the
Galilean transformation is the only one among these general transformations
for which
(i)
the two observers measure the s
ame time interval between two
events,
(ii)
the two observers measure the sam
e distance between two points in
space,and
(iii)
Newton’s ﬁrst law is valid in both frames.
52
CHAPTER 2.NEWTONIAN KINEMATICS AND DYNAMICS
3.A red car is at a distance
d
behind a blue car,when the driver of the red car
decides to break in order to avoid collision.Assuming that the driver of the
blue car does not break a
nd that the deceleration
|
a
|
of the red car remains
approximately constant during the event,show that collision will be avoided if
the speeds
u
red
and
u
blue
of the two cars obey the relation
u
red
%
u
blue
+
0
2
|
a
|
d.
4.Electrons in an oscillating electric ﬁeld.An electron of mass
m
e
that is initially
at rest is set to motion by an oscillating electric ﬁeld
!
E
.The electric ﬁeld
imparts an acceleration on t
he electron along a single d
irection with a magnitude
a
=
!
eE
0
sin(2
#ft
)
.
In this expression,
e
is the charge of the electron,
E
0
is the maximum strength
of the electric ﬁeld,and
f
is the frequency of oscillation of the electric ﬁeld.
Show that the displacement of the electron after time
t
is
d
=
!
eE
0
2
#fm
e
t
+
eE
0
4
#
2
f
2
m
e
sin(2
#ft
)
.
Why does the electron acquire an overal
l drift (described by the ﬁrst term in
the sum),even though the acceleration has zero mean?
5.What is the time derivative of the accel
eration,also known as the jolt or jerk,
!
j
=
d!
a
dt
for an object in a uniform circular motion?
6.Prove the expressions for the velocity a
nd acceleration of a pa
rticle in a cylin-
drical coordinate system.
7.Consider a force that is always pointing towards a speciﬁc point
P
in a three
dimensional space and an object with an initial velocity
!
v
such that the point
P
and the velocity vector
!
v
deﬁne a unique plane.Show that the trajectory of
the object under the inﬂuence of this central force remains always on the same
plane.
8.Show that the Galilean transformation (2.1) is a special case of the Lorentz
transformation (2.12),when the magnitude of the velocity of the observer is
much smaller than the speed of light.
9.Maxwell’s equations
!
(∙
!
E
= 0
!
(∙
!
B
= 0
!
(&
!
E
=
!
&
!
B
&t
!
(&
!
B
=
!
1
c
2
&
!
E
&t
describe the evolution of the electric and magnetic ﬁeld,
!
E
and
!
B
,respectively,
as a function of time.Show that these equations are invariant under Lorentz
transformations.