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Файл:A General Course of Physics. Mechanics. Textbook
.pdf
A General Course of Physics
MECHANICS
Textbook
St.-Petersburg
2017
ISBN 978-5-94211-777-1
УДК 531(075.80)
ББК 22.2 О288
Reviewers:
Professor Yulia M. Sischuk, Doctor of Philology, the Head of Foreign Languag-
es Department, St. Petersburg Mining University
Professor Nikolay A. Timofeev, Doctor of Physical and Mathematical Sciences,
the Head of the Department of Optics, St. Petersburg State University
Professor Vladimir S. Sukhomlinov, Doctor of Physical and Mathematical
Sciences, the Institute of Technical Physics, Russian Academy of Sciences
Mustafaev, A.S., Filyasova, Yu.A. A General Course of Physics. MECHANICS:
Textbook. St. Petersburg, 2017. 134 p.
The textbook is written in accordance with the syllabus in physics for higher
educational institutions and is intended for self-study of the course «A General Course of
Physics. MECHANICS». The book contains explanations of fundamental concepts and
descriptions of laws of dynamics, universal gravitation, hydrostatics and hydrodynamics,
the theory of relativity, among others. The material represented presumes knowledge of
modern mathematical tools; an emphasis is put on the physical nature of phenomena.
There are test questions and special tasks at the end of each section for self-control and
the monitoring of academic progress in theoretical and practical aspects of the course.
The present textbook is the first edition in the English language and is written in
conformity with the St. Petersburg Mining University Development Program for the pe-
riod until the year 2018. The authors’ attention is focused on achieving priority goals in
the university international activities such as the monitoring of global educational trends,
best foreign practices analysis, learning resources development and internationalization in
accordance with the global trends in teaching foreign students and postgraduates, and
creation of a multilingual academic environment for productive international collaboration. The textbook meets the requirements specified for bachelor’s and master’s degree
students of technical universities.
Science editor: Professor Alexander S. Mustafaev, Doctor of Physical and Ma-
thematical Sciences, the Head of General and Technical Physics, St. Petersburg Mining
University
ISBN 978-5-94211-777-1
St.-Petersburg Mining University, 2017
© Mustafaev A.S., Filyasova Yu.A., 2017
3
1. KINEMATICS .............................................................................................................................. 4
C O N T E N T S
1.1. Fundamental concepts ......................................................................................................... 4
1.2. Velocity and acceleration .................................................................................................... 5
1.3. Acceleration for curvilinear motion – tangential and normal acceleration ........................... 8
1.4. Kinematics of circular motion ............................................................................................. 9
2. DYNAMICS OF TRANSLATIONAL MOTION ......................................................................... 16
2.1. Newton’s Laws of Motion .................................................................................................. 16
2.2. Momentum and Impulse Connection .................................................................................. 19
2.3. The Law of Momentum Conservation................................................................................. 20
2.4. Motion of a Body with Variable Mass. Reactive Motion .................................................... 22
2.5. The Center of Mass and the Law of Motion ........................................................................ 25
3. WORK AND ENERGY ................................................................................................................ 30
3.1. Work and power .................................................................................................................. 30
3.2. Conservative and non-conservative forces. Potential field of force ..................................... 33
3.3. Energy. Potential energy ..................................................................................................... 35
3.4. Kinetic energy. The law of energy conservation ................................................................. 39
3.5. Relationship between potential energy and force ................................................................ 41
3.6. Central collision of spheres ................................................................................................. 43
4. GYRODYNAMICS ...................................................................................................................... 50
4.1. Rotational kinetic energy. Moment of rigid body inertia ..................................................... 50
4.2. The moment of inertia for bodies with simple geometric forms .......................................... 52
4.3. Principal axes of inertia ....................................................................................................... 55
4.4. Moment of force ................................................................................................................. 56
4.5. Fundamental equation of solid gyration dynamics .............................................................. 59
4.6. Work of external forces while solid body rotation .............................................................. 62
4.7. Angular impulse. The law of angular impulse conservation ................................................ 63
4.8. Gyroscope. Gyrocompass ................................................................................................... 66
4.9. Non-inertial reference frames .............................................................................................. 68
5. UNIVERSAL GRAVITATION .................................................................................................... 75
5.1. Kepler’s laws ...................................................................................................................... 75
5.2. The law of universal attraction ............................................................................................ 75
5.3. Gravity force and the weight of a body. Weightlessness. .................................................... 77
5.4. Cosmic velocities ................................................................................................................ 79
6. OSCILLATORY MOTION .......................................................................................................... 83
6.1. Harmonic motion ................................................................................................................ 83
6.2. Physical and mathematical pendulums ................................................................................ 85
6.3. Velocity, acceleration and energy of harmonic oscillations ...................................................... 88
6.4. Summation of oscillations with equal direction and frequency ........................................... 90
6.5. Beats ................................................................................................................................... 92
6.6. Addition of mutually orthogonal oscillations. Lissajous figures .......................................... 94
6.7. Damped oscillations ............................................................................................................ 98
6.8. Forced oscillations. Resonance ........................................................................................... 101
7. THE ELEMENTS OF HYDROSTATICS AND HYDRODYNAMICS ....................................... 106
7.1. Principal laws and relationships in hydrostatics .................................................................. 106
7.2. Hydrodynamics of ideal fluids: Principal laws .................................................................... 109
8. FUNDAMENTALS IN THE THEORY OF RELATIVITY.......................................................... 116
8.1. Galileo’s principle of relativity. Galilean transformations ................................................... 116
8.2. Einstein’s principle of relativity .......................................................................................... 118
8.3. Lorentz transformations ...................................................................................................... 121
8.4. Relativistic kinematics ........................................................................................................ 123
8.5. Relativistic dynamics ................................ .......................................................................... 127
Appendix ........................................................................................................................................... 131
Subject Index …………………………………………………………………………………………132
4
1. KINEMATICS
kzjyixr
kji ,,
rOM
x
y
z
O
M
z
y
O
0
r
Fig. 1.
1.1. Fundamental Concepts
The subdiscipline of mechanics, which studies motions of objects both in time and space without being concerned with the forces,
causing the motions, is called kinematics.
Motions of one and the same body may be different, depending on the body which they refer to. Consequently, in order to describe
a motion, it is necessary to indicate with respect to what other bodies
the given motion occurs. The body determined for this purpose is
called a reference body. The coordinate system directly associated
with this body, together with the unit of length and clocks, forms a
reference frame.
The simplest object in classical mechanics is a point particle. Ac-
cording to its definition, it is a body whose dimensions may be disregarded
in the context of a given problem. Spatial position and motion of a particle
can be determined in the Cartesian coordinate system, which specifies the
body, unaffected by forces.
The position of particle M in this system is characterized by the
coordinates x, y, z (Fig. 1):
,
in this case,
are unit axes;
is the radius-vector of the par-
ticle.
Instead of x, y, z coordinates, it is
possible to set the length of the radiusvector r and two angles and , which
make up the radius-vector with the axes y
and z.
Now let us consider the motion of a point particle. The sum total
of all successive positions of the point
particle during its motion forms a line
5
in space – its trajectory, or path (Fig. 2).
1
r
tt
2
r
r
12
rrr
r
tr /
av
v
av
v
r
av
v
dt
rd
t
r
t
lim
0
v
2
r
O
M
1
M2
S
r
1
r
Fig. 2
Assume that a point particle at the
moment of time t is in position М1 and its
radius-vector is
of time
its radius-vector is
, whereas at the moment
it is in position М2, and now
.
The length of the radial arc
М1М
= S is the distance traveled by the
2
particle during the time t. The vector,
directed from point М1 to point М2 equal to
is called the displace-
ment vector:
.
The displacement vector shows the distance travelled by the particle
in space. The displacement vector characterizes both direction and distance
of movement. The displacement vector magnitude |
| equals the chord
length, which embraces the radial arc, although it is not the same as S, and
the difference between them is less than t.
1.2. Velocity and Acceleration
The vector
gives the average displacement velocity
during the time t. The vector
chordwise М
(Fig. 2). The magnitude and the direction of the vector
1М2
, as well as the vector
, is directed
depend on the t value.
Turning our attention to the limit (t → 0), we get the instanta-
neous velocity vector at the given point М1:
.
Consequently, the instantaneous velocity vector (or velocity vec-
tor) is the derivative of the radius-vector with respect to time.
6
As the chord at the extreme coincides with the tangent line, the
v
kji
zyx
vvvv
dt
dz
dt
dy
dt
dx
zyx
vvv ,,
222
||
zyx
vvv v
|| r
dt
dS
t
S
dt
rd
t
r
tt
limlim
00
vv
|| vv
dtdS v
S t
dtdSS
0 0
v
constv
tS v
instantaneous velocity vector
at each moment of time is directed along
the tangent line to the trajectory.
In the Cartesian coordinate system,
projections are represented in the following way:
Taking this into account, we can determine the instantaneous ve-
, velocity
locity magnitude:
.
The velocity magnitude can also be specified in a different manner.
Let us assume that the time interval of motion is t 0. Then, the difference
between the arc distance S and the length of the embracing chord
will diminish (Fig. 2); therefore, at the extreme we will get the following:
. (1)
It is clear that the instantaneous velocity magnitude
can
be determined either as the derivative of displacement with respect to
time or as the derivative magnitude of the radius-vector with respect to
time. Both formulas are correct.
As it follows from formula (1),
,
and after taking the integral:
.
If the velocity direction does not change, the motion is rectilinear. If both velocity magnitude and direction remain the same, then the
motion is called rectilinear and uniform motion.
When motion is uniform,
. In this case,
,
7
viz. S is the distance traveled by the body; v and t are velocity and time of
a
v
2
2
0
lim
dt
rd
dt
d
t
a
t
vv
v
dt
d
t
aa
v
v
lim
adtd v
tv
v
adtd
0
0
v
at0vv
at0vv
0
v
motion, respectively.
The latter formula is true for any uniform (and not necessarily
rectilinear) motion.
Now let us consider the type of motion, which does not remain
equal, i.e. accelerated motion.
The acceleration of the body
between the increment velocity vector
is defined as the limit whose ratio
and the time interval t at-
tends to, on condition that t 0:
.
As a result, the acceleration vector can be determined either as
the first derivative of the instantaneous velocity vector with respect to
time or as the second derivative of the radius-vector with respect to time.
In the case of the rectilinear motion, the direction of acceleration
coincides with the vector direction
:
.
If acceleration is constant and equidirectional with velocity, then
the velocity magnitude increases. Such motion is called uniformly accelerated motion. Provided that the velocity vector and the acceleration vector
are oppositely directed, then the velocity diminishes, and the motion uniformly decreases.
For the rectilinear accelerated motion:
;
;
;
, (2)
specifically,
and v stand for the velocities at the initial and final mo-
ments of time, respectively. The acceleration a included in formula (2) is
an algebraic value: a 0 for uniformly accelerated motion and a 0 for
uniformly decelerated motion.
8
The distance traveled by the body moving with uniform accelera-
t t
at
tdtatdtS
0
2
0
0
0
2
vvv
aS2
2
0
2
vv
dt
d
a
t
v
tion within the time interval from 0 to t can be specified via the integration of equation (2) with respect to time:
. (3)
Having excluded time from equations (2) and (3), we obtain the
formula necessary for solving a number of problems:
.
1.3. Acceleration for curvilinear motion: tangential and normal
acceleration
Velocity is a vector; therefore it is characterized by magnitude
and direction. During the curvilinear motion, velocity can change with
time both in magnitude and in direction. As a result, acceleration is
represented by two components (Fig. 3).
The tangential acceleration аt characterizes changes in the velocity modulus and equals the derivative of the instantaneous velocity modulus with respect to time:
.
The tangential acceleration is directed along the tangent line to the trajectory (as well as the velocity vector). If the
velocity modulus increases, then the tangential acceleration is positive, whereas if
the velocity modulus diminishes, the tangential acceleration is negative.
9
R
a
n
2
v
22
nt
aaa
The normal (centripetal) acceleration аn indicates changes in the
velocity direction:
,
specifically, R is the radius of the trajectory curvature.
The normal acceleration vector is Fig. 3
perpendicular to velocity and directed towards the center of the curvilinear trajectory (Fig. 3). As it is given in the picture, this complex acceleration can be determined by using Pythagoras theorem:
.
The complex acceleration is directed in the same way as the force
acting upon the body and is determined by Newton’s second law of motion.
Let us consider a few individual cases.
1. Uniform rectilinear motion. The velocity magnitude and direction remain constant. Therefore, аt = 0, аn = 0 and, as a result, а = 0
also.
2. Non-uniform rectilinear motion. The velocity magnitude varies but the velocity direction remains the same. Therefore, аt = const,
а
= 0 and а = аt.
n
3. Uniform circular motion. The velocity magnitude remains
constant but the velocity direction varies. So, аt = 0, аn = const and а = аn.
1.4. Kinematics of circular motion
Let us examine point particle M, rotating about its axis OO (Fig.
4). Assuming that at a certain moment of time t, the particle occurs in
position M, and at the time (t + dt) – in position М'. For the short time
10
period dt, the radius-vector will turn through the angle d, and the point
rddS
dt
d
r
dt
dS
dtdS/v
dt
d
rv
particle will travel the distance dS circumferentially.
The magnitude dS can be expressed
via the angle d and the circle radius r:
.
We will divide both parts of the equ-
ation by dt:
. (4)
The left-hand part of the equation (4) represents the modulus of
linear velocity
. The derivative of with respect to time t in
the right-hand part (4) is called angular velocity:
, (5)
it shows the rotational speed of a point particle and is measured in rad/s.
For the complete specification of the rotational motion, it is insufficient to know only the numerical value of the angular velocity, but it
is also necessary to set a position of the rotation axis in space and the direction of rotation about the axis, i.e. it is necessary to regard the angular
velocity as a vector variable. The direction of the angular velocity vector
is related to the line depicting rotation by the right-hand screw rule. According to the rule, the line should be directed in such a way that when
looking at it, we can see the clockwise rotation (when rotating the head of
the right-hand screw clockwise, we cause it
to move away from us) (Fig. 5).
Equations (4) and (5) point at the relation between the instantaneous v and angular velocities:
.
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