For audience from foreign students
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Physics
The science of the
simplest and the most
common matter motion
forms and their relative
transformations
The science that studies
the general properties and
the laws of substance and
field movements
The simplest matter
motion forms are present
in all the most complex
forms of it
A substance and a field
are forms of the matter
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The matter
The world around us, all that exist around us and revealed
by us per sensations
The motion is an essential matter property and its
existence form
The motion is various matter changes: from the simple
transfer to the most complex mentality processes
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Kinematics Basis
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The mechanics
Part of physics, which studies the motion laws and reasons that cause or
change this motion
Studies the motion
laws of macroscopic
bodies with
velocities that are
comparable with
the speed of light
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The branches of mechanics
Kinematics
Studies the objects movement without the causes that lead to this
movements
2. Dynamics
Studies the motion laws and causes that induce and change this
motion
3. Statics
Studies equilibrium laws of bodies system
1.
If the motion laws are known, then the equilibrium laws could be
established
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Physical models in mechanics
Physical models are used in mechanics for
describing bodies motion. This model depends
on specific tasks terms
Mass point is a body having a mass the size of
which can be disregarded in this task. It is an
abstract concept
Mass point system. Random macroscopic object
or bodies system can be represented as a mass
point system. In this system mass points interact
among themselves.
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Physical models in mechanics
A perfectly rigid body
Is a body, that can not become deformed at any terms. The
distance between two particles of that body remains
constant at any terms.
A perfectly elastic body
Body deformation obeys the Hooke’s law.
After the termination of external forces the body takes its
original size and form.
A perfectly inelastic body
Body keeps strain state after termination of external forces.
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Mechanical motion
Base, or frame
z
r
z
0
x
y
x
y
An indication body
The position of moving body is
determined relative to randomly
selected body (indication body).
Coordinate system
Is a system associated with an
indication body.
Base, or frame
Is a combination of the indication
body, the coordinate system and
the clock synchronized among
themselves.
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Mechanical motion
Kinematic equation of mass point movement
x=x(t)
y=y(t)
z=z(t)
or
z
r
z
0
x
y
x
r = r (t)
y
The position of mass point А at frame is described by three
coordinates x, y, z or radius-vector r. When mass point moves, its
coordinates change within time (t). So this movement is described
by the scalar equation system or the equivalent vector equation.
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Trajectory
Trajectory is a line describing by moving mass point
(body) relative to selected base.
Depending on trajectory form there are several motion
types:
1. Rectilinear, or straight-line
2. Curvilinear
3. Circular motion and etc.
The trajectory form depends on a mass point motion
character and a base.
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Path length and displacement
vector
z
r1
A
∆r
B
0
r2
x
y
Displacement vector
It is a vector from the start
position of the moving point
to its position at this
moment.
Path length
It is a length of path AB
passed by mass point at a
given period of time.
∆s=s(t) – a scalar time function.
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Forward motion
It is a motion, when some straight line, inflexibly connected
with a moving body and drown between two random points,
stays parallel to itself.
In this case all body points are moving equally. Thus, the
forward motion of this body can be characterized by the
motion of some random body point (e.g., by body mass
center motion)
A’
A
C’
A’’
C’’
C
B
B’
B’’
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Rotatory motion
It is a motion, when all body
points are moving in circles,
which centers are lying on the
equal straight line called axis
of rotation.
The different body points are
moving differently, that is why,
its rotatory motion cannot be
characterized by a motion of
any point.
O
O’
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Velocity
∆s
A velocity is a vector quantity
A
that determines, from one
hand, the motion speed and
∆r
v
from other hand, its direction
B
at a given period of time.
r1
A mean velocity is a vector
r2
quantity that is determined
O
by ratio of the point radiusvector increment ∆r to the
The mean velocity vector
time interval t within which
direction
coincides
with
the
this increment has occurred.
direction ∆r
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Velocity
One meter per second means the velocity
of forward and uniformly moving point,
with which this point in 1 second shifts at a
distance of 1 meter.
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Acceleration
An acceleration is a non-uniform motion characteristic that
determines the velocity change rate in absolute value and in
direction.
A mean acceleration is a vector quantity that is determined
by ratio of velocity change ∆v to the time interval ∆t within
which this change occurred.
An instant acceleration is a vector quantity that is
determined by first derivative of the velocity with time
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Components of acceleration
1. A tangential acceleration
characterizes the velocity change
rate in absolute value. It is
directed along the tangent to
trajectory.
2. A normal acceleration
characterizes the velocity change
rate in direction. It is directed to
the trajectory curvature center.
an
a
aτ
v
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The full acceleration in curvilinear
motion
Geometric sum of the tangential and normal
acceleration component
The full acceleration absolute value
The unit of acceleration
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Motion classification
aτ
0
aτ = a = const
aτ = f(t)
an
Motion
0
Straight-line and uniform
0
Straight-line and uniformly
accelerated
0
Straight-line, with variable
acceleration
0
const
Uniform, in a circle
0
≠0
Uniform and curvilinear
const
≠0
Curvilinear and uniformly
accelerated
aτ = f(t)
≠0
Curvilinear, with variable
acceleration
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Uniform and uniformly accelerated motion
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Uniform motion (v = v0 = const)
Velocity
Acceleration
v
v0
s
0
Passed way
t
s
0
t
s
s0
0
t
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Uniformly accelerated motion
(a = const) a
Acceleration
a
v
0
t
v
Velocity
v
0
s
t
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Uniformly accelerated motion
(a = const)
s
Passed way
0
t
s
s0
0
t
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Free fall
It is a motion that body would commit
only by gravity without air resistance.
If body free falls from low altitude h
(h<<R, R – the Earth radius), then it will
move with equal acceleration g, which
directs straight down.
g = 9.81 m/s2. It is free fall acceleration.
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Kinematic equation of motion
The common vector equation, that determines a body
motion with equal acceleration g and initial velocity v0
from the point r0, can be presented as follow:
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Projection of kinematic equation
on the axis y
The axis directs straight down. The indication point is
put at the motion beginning point. t0 = 0
0
g
y
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Free fall
The path passed by body in free fall at the moment t.
Free fall without initial velocity
1. Free fall currency
2. Velocity
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Motion of body thrown straight up
The body moves straight up with initial speed
v 0.
Without air resistance body acceleration a in
any motion period of time is equal to free fall
acceleration g (a = g).
To the highest rise point the motion is
uniformly decelerating, and, after, this is a free
fall without initial velocity.
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Kinematic equations
The common vector equation, that describes a body motion
with equal acceleration g and initial velocity v0 from the point
r0, can be presented as follow:
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Projection of kinematic equation
on the axis y
The axis directs straight up. The indication point is put at the
motion beginning point. t0 = 0
y
vt1=0
h
v0
0
v
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Motion of body thrown straight up
Rise time
Height of lift, or lift
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Motion of body thrown straight up
The common time of motion
Time of falling
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Motion of body thrown straight up
Finite velocity of motion
The finite motion velocity in absolute value is
equal to the initial velocity. The “minus” means
that the finite motion velocity is directed against
an axis y, i.e. straight down.
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Elementary rotation angle
Elementary, or infinitesimal, turns
are considered as vectors.
An absolute value of vector dφ is
equal to the rotation angle. Its
direction coincides with forward
motion direction of screw edge.
Screw head goes round in point
motion direction in a circle. In
other words, it obeys the rule of
the right screws.
dφ
∆φ
0
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Angular velocity
Vector ω is directed along the
rotation axis according to the
rule of the right screws.
An angular velocity is a
vector quantity that is
determined by the first
derivative of the body rotation
angle with time.
v
ω
dφ
v
0
R
0
R
ω
v
0
R
0
v
R
ω
dφ
ω
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Connection between linear and
angular velocity
Absolute value
dφ
∆φ
0
∆s
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Connection between linear and
angular velocity
Vector
Vector product coincides
with vector v direction. Its
absolute value can be
presented as:
ω
R
α r
v
0
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Unit of angular velocity
1 radian per second is an angular velocity of an uniformly
revolving body. All its points in 1 second turn relative to the
axis on angle by 1 radian.
Dummy vector, or pseudovector
It is a vector whose directions are connected with rotation
direction. In this case this vectors can be delayed from any
point on the rotation axis.
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Uniform motion of mass point in a
circle
It is a motion when a mass point, or body, passes
equal circular arc lengths within equal time periods.
An angular velocity:
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Rotation period T
Rotation period is equal to
the time when a mass point
commits the full cycle in a
circle, i.e. the mass point
turns by angle 2π.
Rotation frequency
Rotation frequency is equal
to the number of turns that a
mass point commits in one
second.
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Peculiarity of uniform motion in a
circle
Uniform motion in a circle is a
special case of curvilinear
motion.
A mass point moves with
velocity that is constant in
absolute value.
But the direction changes
during time.
Thus, the motion is accelerated.
v
an
v
an
0
an
R
v
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Acceleration
Normal acceleration component
is directed along the radius to
the circle centre and
perpendicularly to the velocity
vector.
A mass point acceleration in
any point of the circle is
centripetal.
v
an
v
an
0
an
R
v
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Angular acceleration
An angular acceleration is a vector quantity that is
determined by first derivative of a angular velocity with
time. An angular acceleration vector is dummy vector.
ω2
ω2
ε
ω1
ω1
0
0
ε
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Angular acceleration
1 radian per second squared is equal to angular
acceleration of uniformly rotating body, that
changes angular acceleration on 1 radian per
second within 1 second.
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Inertial frames. Mass and pulse. Force
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Newton’s first law
 Every object in a state of
uniform motion tends to remain
in that state of motion unless an
external force is applied to it.
 There are such frames for which
forward moving bodies save
their velocity constant, unless
another bodies act on them.
 Newton’s first law content
composes statement about
inertial frames existence.
Sir Isaac Newton
1642-1727
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Newton’s first law
 Inertial frame is a frame for
which a mass point, free
from external impacts, stays
at rest or moves straight and
uniform.
 Noninertial frame is a frame
that moves with acceleration
relative to inertial frame.
 It was empirically
established that the
heliocentric frame could be
considered as inertial system.
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Inertia, or force of inertia
 Inertia is a force specified
by accelerated base motion
relative to measurable base.
 Inertia is generated by
accelerated base motion, not
by reaction between bodies.
Thus, this forces do not
obey to Newton’s laws,
because there is no
antagonistic force while
force of inertia effects to
any body.
Galileo Galilei
(1564-1642)
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Mass
 Inertness is a inherent property. There




is resistance to velocity change in
absolute value and in direction.
Mass is a physical quantity that
determines inertial and gravity matter
properties.
Currently there is an evidence that
inertial and gravity mass are equal.
Mass is an additive quantity (composite
body mass is equal to mass sum of its
parts); mass does not change with
motion, i.e. it is constant quantity.
1 kilogram is a mass of international
kilogram prototype.
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Pulse of mass point
 It is a vector quantity that is equal to product of body mass
to its velocity. Pulse vector direction coincides with
velocity vector direction.
 1 kilogram-metre per second is equal to body pulse with
mass in 1 kilogram and it moves with velocity in 1 metre
per second.
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Force
 Vector quantity. It is a measure of mechanical impact from
other bodies or fields that results in accelerated motion or
changes of form and size.
 At every time moment, the force is characterized by:
 numeric value,
 direction in space,
 application point.
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Common formulation
 Mass point pulse change velocity is equal to
the acting force.
 This equation also is known as a mass point
motion equation.
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Formulation with acceleration
 An obtained mass point acceleration is in proportion to a
force, that generates it, and inversely proportional to its
mass.
 An acceleration direction coincides with the force.
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Force unit
 1 Newton is a force that imparts to the mass of 1 kilogram
an acceleration of 1 metre per squared second along force
direction.
The second law is correct only in inertial bases.
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Principle of superposition
 If few forces have an effect on mass point, then each of
them will impart an acceleration to the mass point
according to Newton’s second law, as if other forces do
not existe.
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Force components
Fn
F
a
an
aτ
Fτ
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Formulation. Forces
 Forces, with which mass
points effect to each other,
are always equal in absolute
value, but direct inversely.
They effect along straight
line drown between these
two points.
 These forces, when applied
to different mass points,
always effect in pairs. They
are the forces of the same
nature.
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Friction types
64
Frictional force
 It is tangential force that occurs from contact between
body surfaces and prevents their displacement.
 Frictional forces depend on comparative bodies velocities.
They can have different nature, but result in
transformation of mechanical energy into the internal
energy of adjoining bodies.
 Forces of friction are directed along tangent line to the
friction surfaces, or layers. They oppose the relative
surfaces displacement.
65
Friction
Friction of
rest
External, or
dry, friction
Sliding
friction
Internal
friction
Rolling
friction
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Friction types
 An external friction is a friction that occurs in the plane of
two bodies contacted with their relative displacement.
1.Sliding friction occurs in the case when a body slides
on the base surface.
2.Rolling friction occurs when a body rolls on the base
surface.
 Friction of rest is a friction without relative displacement
of two bodies contacted.
 An internal friction is a friction between parts of the same
body, e.g. between different layers of liquid or gas; while
their velocities are changed from one layer to another.
There is no friction of rest.
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Friction of rest
 The relative body motion occurs when an external friction
F > (Ffr0)max
(Ffr0)max – rest friction limit; μ0 – the rest friction coefficient;
N – normal pressure force.
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Sliding friction
 Sliding friction is
proportional to normal
pressure force N with which a
body acts on another one.
 μ – the sliding friction
coefficient that depends on
properties of surfaces contact. It
is dimensionless quantity.
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Basic concepts
 A mechanical system is a mass points set, considered as a
whole.
 Internal forces act between mass points of mechanical
system.
 External forces act upon mass points of mechanical
system.
 A closed system is a mechanical system without external
forces acting.
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Momentum conversation
 There is a mechanical system that includes n bodies. Their
mass and velocity are equal: m1, m2, … , mn and v1, v2, …
, vn respectively.
 Closed system pulse is conserved, i.e. it does not change
within a time.
 This law is an universal fundamental law of Nature.
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Mass center
 It is an imaginary point C that characterizes weight
deposition of that system, or body.
 Mass center radius-vector
 Mass center velocity
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Law of mass center motion
 System center mass moves as mass point that contains the
whole mass system. Forces, that act on it, are equal to
geometric sum of all external forces.
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Energy. Work.
 An energy is universal measure of different motion forms
and their interaction.
 There are different forms of energy: mechanical power,
heat, electromagnetic energy, nuclear power, etc.
 Work is a quantitative characteristic of energy interchange
between interacting bodies.
F
α
Fs
s
76
Work
Fs
2
dr
α
1
F
 α – angle between F and dr; ds=|dr| - elementary path.
 Work is a scalar quantity.
77
Work
Fs
1
dA
2
A
s
ds
 Geometric sense. Required work is determined by
shaded figure area on the chart.
 1 joule is a work committed by force, that is equal to 1
Newton, in the way 1 meter.
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Power
James Watt
1736 – 1819
 It is a physical quantity that is
characterizing by the velocity
of the commitment work.
 Power, developed, by force F
is equal to product of force
vector by velocity vector.
 Power is a scalar quantity.
 1 Watt is a power at which the
work 1 Joule is committed
within 1 second .
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Kinetic energy
 It is an energy of mechanical system
motion.
 Kinetic energy is determined by the work
that should be committed to impart the
velocity.
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Characteristics of kinetic energy
1. It is always positive
2. It varies in different
inertial bases
3. It is system state
function.
82
Linkage of work and kinetic energy
 The kinetic energy increment of mass point on the
elementary displacement is equal to the elementary work
on the same movement.
 The force F, acting on a body at rest and calling its
motion, commits the work; moving body energy increases
by expended work amount.
 The work dA of force F on the way, that the body passes
within a time of velocity increment from 0 to v, spends on
kinetic body energy increment.
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Kinetic energy
 The kinetic energy of the body with mass m, moving with
velocity v, is determined by work, which should be
performed in order to impart the velocity to the body.
Work of force when body moves
from point 1 to point 2
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Theorem of kinetic energy
 The kinetic energy increment of mass point at a certain
displacement is equal to algebraic sum of all the forces
acting on a mass point in the same movement.
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Potential field
 A field, in which the work performed by forces,
when the body moves from one position to
another, does not depend on what trajectory
occurred this motion, it depends only on the
initial and final positions.
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Conservative forces
 It is a force, whose work does not depend on the path
on which the body moves from the first point to
another one, but only depends on the initial (1) and
final (2) body position. E.g., gravity.
1
2
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Dissipative force
 It is a force, whose work depends on the body
displacement trajectory from the first point to another one.
E.g., frictional force or resisting force.
89
Work of conservative forces along
a closed path
 Works A1b2 and A2a1,
they do not depend on
displacement trajectory.
 They are equal in
absolute value.
 They differ only by
signs.
a
2
1
b
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Potential energy
 It is a mechanical energy of bodies
system, that is determined by their
relative position and character of
forces interacting between them.
 Specific form of the function
depends on the force field nature.
In the case of a bow and arrow, the energy is converted from the potential
energy in the archer's arm to the potential energy in the bent limbs of the bow
when the string is drawn back. When the string is released, the potential energy
in the bow limbs is transferred back through the string to become kinetic
energy in the arrow as it takes flight.
92
Work of conservative forces and
potential energy
 Work of conservative forces does not depend on the trajectory.
 It is equal to zero on any closing path.
 A change in potential energy is equal to the amount of work.
 This change does not depend on the trajectory as well.
 Therefore, the potential energy content is determined only by
the initial and final system configuration.
 The work of conservative forces is equal to the potential energy
increment with a “minus” within the elementary (infinitesimal)
system reconfiguration, as the work is committed by the loss of
the potential energy.
93
Characteristics of potential energy
 The potential energy is considered equal to zero in a
certain position. In other words, zero reference level is
selected.
 The body energy is counted relative to zero.
94
Conservative force and potential
energy
- scalar gradient U;
i, j, k – unit vectors of the
axes
95
Potential energy of the body with
mass m at a height h
 Potential energy is equal to the gravity work when a body
is falling from the height h on the Earth surface.
 Height is measured from zero, for which U0=0 and g –
free fall acceleration.
96
Potential energy of the elastic
deformation (spring)
 Force work of spring deformation
spends on the potential energy
increment of it.
 k – elasticity coefficient.
97
Total mechanical energy
 It is a mechanical motion and interaction
energy, or kinetic and potential energy
sum.
98
Law of energy conservation
 The total amount of energy in an isolated system remains
constant over time.
 For an isolated system, energy can change its location
within the system, it can also change form within the
system, but energy is neither created nor destroyed.
99
Elastic and inelastic collision
100
Collision
 It is a collision of two or more bodies, when the
interaction takes a very short time.
 Interaction forces between the colliding bodies (striking,
or instantaneous forces) are so large that the external
forces, acting on them, can be neglected.
 Thus, the bodies system can be considered approximately
as a closed system during their collision. In that case the
laws of conservation are applied to this system.
101
Essence of strike
 Kinetic energy of the colliding bodies is converted into
elastic energy in a short space of time.
 There is an energy redistribution between the colliding
bodies during collision.
 Observations show that the relative velocity does not
reach its original value after the collision.
 This occurs due to the fact that there is no perfect elastic
bodies and perfect smooth surfaces.
102
Coefficient of restitution
 It is a ratio of the normal bodies velocity components after
(vn’) and before (vn) collision.
 In a case if a body has the restitution coefficient ε that is equal
to 0 , body will be considered as absolutely inelastic.
 In a case if a body has the restitution coefficient ε that is equal
to 1, body will be considered as absolutely elastic.
103
 Line of impact
is a straight line passing through the bodies contact point
and the normal to the contact surface.
 Central impact
is a impact, when bodies move along a straight line passing
through their gravity center.
104
Central absolutely elastic collision
 It is a collision of two bodies, when both interacting bodies
have not deformed.
 All kinetic energy, that the bodies possessed before collision,
turns again into kinetic energy.
 Velocity vector projections on the
v1
v2
straight line passed through the
m1
m2 balls centers are equal to velocity
module.
 Their directions are taken into
account by such signs as:
v’1
“positive” value is a motion to
v’2
the right;
“negative” value is a motion
to the left
105
Laws of conservation
 Conservation of momentum
 Conservation of mechanical energy
106
Velocities after elastic collision
107
Special cases
 m1=m2 => v’1 = v2; v’2=v1.
Balls with equal mass exchange their energy.
108
Special cases
109
Special cases
110
Special cases
111
Special cases
112
Central absolutely inelastic
collision
 It is a collision of two bodies, when both bodies move
forward together as a unit.
113
Conservation of mechanical energy
 There are forces between balls in the central absolutely
inelastic collision.
 These forces do not depend on the deformation, but on the
velocities only.
 Therefore, the forces are similar to friction forces, and
conservation of mechanical energy is not observed.
 Kinetic energy ‘loss’ is a result of deformation. Energy
transforms into heat or other energy forms.
114
Difference between the kinetic
energy before and after collision
115
Special case
 If the immobile body mass is very large (m1>>m2), then
v<<v1 and almost all the kinetic energy transforms into
other energy forms.
116
Inertia. Kinematic energy of rotation.
117
118
Body inertia relative to fixed axis
 It is a physical quantity that is
equal to the sum of products of
elementary masses and the
distance squares from mass
point to the axis.
 Inertia is a additive value: inertia
is a body parts inertia sum.
m2
r1
r2 m
3
r3
m1
119
Inertia in the case of persistent
mass distribution
 Integrals are taken in a whole body volume, and the
quantities ρ and r are point functions.
 ρ – body density at a given point;
 dm= ρdV – mass of small body element with volume dV
that is distance r away from rotation axis.
120
Inertia of solid cylinder
 There are separate hollow concentric
cylinders with an infinitesimal
thickness dr, inner r and outer r + dr
radius.
121
Parallel axis theorem,
or Huygens–Steiner theorem
 The theorem determines the moment of inertia J of a rigid
body about any given axis, given that moment of inertia JC
about the parallel axis through the center of mass C of an
object and the perpendicular distance a between the axes.
122
Inertia moments of homogeneous
rigid bodies
123
Inertia moments of homogeneous
rigid bodies
124
The kinetic energy of a rotating
rigid body
 Body rotates around a fixed axis
z.
 Assume that the body is divided
into elementary mass m1, m2, … ,
mi, … at a distance r1, r2, … , ri,
….
 Rotating solid volume elements
with mass mi perform motion
along circles of different radii ri.
 Angular rotation velocity of all
elements is identical.
z
ri
ω
vi
mi
125
The kinetic energy of a rotating
rigid body
126
The kinetic energy of the body in
planar motion
 The kinetic energy is composed of the forward motion
energy with a velocity, that is equal to the mass center
velocity, and rotation energy about an axis passing
through the body mass center.
 m – body mass; vC – mass center velocity; JC – body
inertia moment relative to the axis passing through the
mass center; ω – angular velocity.
127
128
Torque relative to a fixed point
 Torque is a physical quantity, that is defined by the vector
product of the radius-vector, drawn from purchase 0 to A,
by the force F.
 M is a pseudovector. Its direction coincides with the
forward motion direction of right-hand screw.
M
0
F
r
l
A
α
129
Torque relative to a fixed axis
 Torque is a scalar quantity
that is equal to the torque
vector M projection on the
axis z.
z
F
Mz
M
A
0
r
130
Dynamics equation of rigid body
rotation
 The body is solid.
 Work of this force is
equal to the work that is
done at the whole body
turn.
131
132
Angular momentum of a mass
point relative to a fixed point
 Angular momentum is a
physical quantity that is
defined by the vector product
of the mass point radius-vector
r, drawn from point 0, by the
mass point pulse.
 Li is a pseudovector. Its
direction coincides with the
right screw forward motion
direction.
Li
0
ri
l
mi
pi
α
133
Angular momentum of a mass
point relative to a fixed axis
 Angular momentum is a
scalar quantity, that is equal
to the angular momentum
z
vector projection on the axis
z.
pi
Liz
Li
mi
0
r
134
Angular momentum of a rotating
rigid body point
 When a rigid body revolves about a fixed axis z, every
separate body point moves in a circle of constant radius ri
with a certain velocity vi.
 Velocity and pulse is perpendicular to this radius, i.e.
radius is a vector arm mivi.
 Angular momentum direction is defined by right screw
rule.
135
Angular momentum of a rigid body
relative to a fixed axis
 Angular momentum is a separate particles angular
momentum sum relative to the same axis.
 It is equal to the product of body angular momentum
relative to the same axis by angular velocity.
136
137
Dynamics equation of rotation
motion
 Derivative of a rigid body angular momentum vector
is equal to the external forces moment.
138
Conservation of an angular
momentum
 Angular momentum of a closed system is conserved, i.e. it
does not change within a time.
 The law of angular momentum conservation is a
fundamental law of nature.
 The law of angular momentum conservation is a
consequence of the space isotropy.
 Space isotropy is the invariance of physical laws relative
to the frame axis direction choice, or relative to the closed
system rotation in space at any angle.
139
Types of deformation
140
Deformation
 These are shape and size changes of solids under the
external forces action.
141
Compliance
 It is a deformation that
disappears with a cessation of
the external forces action.
142
Plastic deformation
 It is a deformation that is
prolonged with the
termination of the external
forces action.
 Deformation of the real
body is always plastic, as
they never completely
disappear after the
termination of the external
forces action .
 However, if the residual
deformations are small,
they can be neglected.
143
Strain
 It is a quantitative measure that characterizes the
deformation degree. This measure is determined by the
ratio of the absolute deformation ∆x to the quantity x.
 Strain specifies the initial body size and shape.
144
Longitudinal strain
 Longitudinal strain is the relative change in the rod length.
145
Transverse tension (compression)
 Tension (compression) is the relative change in the rod
diameter.
146
147
Stress, tension, or strain
 It is the force which is acting on the area unit of body
cross-section.
 If the force is perpendicular to the surface, the stress is
normal; if the force is tangential to the surface, the stress
is tangential.
 The tensile strain ε and stress σ are in the direct proportion
to each other for small deformations.
148
Young's modulus
 Young's modulus, also known as the tensile
modulus or elastic modulus, is an elastic material stiffness
measure and is a quantity used to characterize materials.
 Young's modulus is determined by the stress causing an
elongation that is equal to one.
149
Hooke’s law
 The body extension is in direct
proportion with the load applied to it.
 Hooke's law holds only for elastic
deformations.
 In the case where l is
the displacement of the spring's end
from its equilibrium position, F is the
restoring force exerted by the spring
on that end, and k is a constant called
the rate or spring constant.
Robert Hooke
1635 – 1703
150
Potential energy of the stretched
(compressed) rod
 Potential energy is equal to the work, that is committed by
external forces during the deformation.
151
Kepler’s laws. The law of gravity
152
153
The first Kepler’s law
 The orbit of every planet is an ellipse with the Sun at one




of the two foci.
Note that the Sun is not at the center of the ellipse, but at
one of its foci.
The other focal point, marked with a lighter dot, has no
physical significance for the orbit.
The center of an ellipse is the midpoint of the line
segment joining its focal points.
A circle is a special case of an ellipse where both focal
points coincide.
154
155
The first Kepler’s law
 The perihelion is the point in the orbit of




a planet, asteroid or comet where it is nearest to the sun.
The aphelion is the point in the orbit of
a planet or comet where it is farthest from the Sun.
Earth comes closest to the sun every year about January 3.
It is farthest from the sun every year about July 4.
The difference in distance between Earth's nearest point to
the sun in January and the farthest point from the sun in
July is not very great.
Earth is about 147.1 million kilometers from the sun in
early January, in contrast to about 152.1 million
kilometers in early July.
156
The second Kepler’s law
 A line joining a planet and the
Sun sweeps out equal areas
during equal intervals of time.
157
The third Kepler’s law
 The square of the orbital period of a planet is
directly proportional to the cube of the semi-major axis of
its orbit.
 Kepler's third law as applied to the planets and satellites,
allows, in particular, to calculate the planets masses.
158
The law of gravity
 Every object in the Universe attracts every other object
with a force directed along the centers line for the two
objects that is proportional to the product of their masses
and inversely proportional to the separation square
between the two objects.
 Fg is the gravitational force
 m1 & m2 are the two objects
masses
 r is the separation between the
objects
 G is the universal gravitational
constant (6,67 * 1011 N*m2/kg2)
159
Gravitational field
 It is the field,
through which
the gravitational
interaction occurs
between the
bodies.
 This field is
generated by the
bodies and is a
matter existence
form.
160
Generalized law of Galileo
 All the bodies fall with the same acceleration (free fall
acceleration) in the same gravitational field.
 The free fall acceleration on Earth, near the Earth surface
is 9.8 m/s2. On the moon it is 1.6m/s2
 But as the object moves further and further away from the
Earth, or any massive body, the acceleration due to gravity
decreases by the inverse square law.
 Thus rather than being a constant value, the acceleration
infinitely approaches zero as the object infinitely
approaches infinity.
161
162
Gravity
 Gravity is the force acting on any body located near the
surface.
 It is directed straight down.
163
Gravity force and gravitation
 This formula is valid if as we ignore the daily rotation of




the Earth and the body height above the Earth.
m is the body mass;
M is the Earth mass;
R is the Earth radius;
G is the universal gravitational constant.
164
Body weight
 Weight is the force with which the body effects on the
support, or suspension due to the Earth gravitation, that
keeps the body from free fall.
 Gravity force always exists. Weight is revealed, when
there are other forces except the gravity force, acting on
the body.
 Thus, the body moves with acceleration a, not with free
fall acceleration g.
165
Weightlessness (imponderability)
 It is a body state in which it is moving only by gravity.
 If the body is moving freely in the gravitational field
along any path along any direction, the acceleration is
equal to the free fall acceleration and the weight is equal
to zero, i.e. body is weightless.
166
167
168
Distinctive peculiarity of liquids
and gases
 Gas molecules commit a random and chaotic motion.
They are not connected or are very poorly connected by
the interaction forces. Therefore, they are moving freely.
As a result of collisions they tend to fly away in order to
fill all given amount, i.e. amount of gas is determined by
the volume of the vessel.
 Liquid has a certain amount. And it takes the shape of the
vessel in which it is contained. But in liquids, in contrast
to the gas, average distance remains almost constant
between the molecules. Thus, the liquid has almost the
same amount.
169
Liquids and gases
 The liquids and gases behavior is defined by the same
parameters and identical equations in a number of
mechanical phenomena. Thus there is a single term
“liquid”.
 Incompressible fluid is a liquid or gas, where the
density dependence of the pressure can be neglected.
170
Fluid pressure
 If the lamina is placed into the static fluid, the fluid parts
from different sides will act on each element with forces,
that are equal to the absolute values and their directions
are perpendicular to the plate.
 Fluid pressure is a physical quantity, that is defined by
forces acting from the fluid side on the area unit in a
perpendicular direction to the surface.
171
Pressure unit
 1 Pascal is the pressure generated by the force 1 Newton
that is distributed along the normal to the surface with
area 1 meter squared.
172
173
Pascal’s law
 Pascal's Laws relates to pressures in fluids - liquid or
gaseous state:
 if the weight of a fluid is neglected the pressure throughout
an enclosed volume will be the same
 the static pressure in a fluid acts equally in all directions
 the static pressure acts at right angles to any surface in
contact with the fluid.
 E.g., hydraulic lift.
174
Archimedean principle
 A body immersed in a fluid
undergoes an apparent loss in
the weight equal to the weight
of displaced fluid.
Archimedes
Thoughtful by Fetti (1620)
175
Hydrostatic pressure
 Fluid pressure in equilibrium horizontally always the
same, otherwise an equilibrium does not exist.
 Therefore, free surface fluid at rest is always horizontal in
a distance from the vessel walls.
 If the fluid is incompressible, its density will not depend
on the pressure.
 The pressure varies linearly with height.
176
177
Basic concepts
 Current is a liquid motion.
 Stream is a totality of particles in a moving fluid.
 Stream-line is a line, in which every point has a tangential
line that coincides with the velocity vector direction at a
given time. Stream-line pattern can characterize direction
and absolute value of velocity at different space points, i.e.
it characterizes the fluid state.
 Fluid tube is a fluid part within a stream-line.
 Sustained stationary current is a fluid current, when the
shape and location of the stream-lines and velocities
values do not change within a time at each point.
178
Equation of continuity for
incompressible fluid
 Product of an incompressible fluid velocity on the tube
cross-section is constant for a given tube.
179
180
Basic data
 Perfect fluid is an imaginary fluid, where there are no
internal friction forces (physical abstraction).
181
Total fluid energy
 Total energy derived from the kinetic and potential energy
in the gravity field.
 According to the law of energy conservation, the change
in the total energy of perfect incompressible fluid is equal
to external forces work to move the fluid mass between
the sections.
182
Work of external forces
Volume occupied by liquid
 According to the continuity equation for an
incompressible fluid Sv=const, the volume occupied by
liquid remains constant.
183
The Bernoulli equation
 Bernoulli performed his
experiments on liquids, so his
equation in its original form is
valid only for incompressible
flow and for real fluids with small
internal friction force.
Daniel Bernoulli
8 February 1700 –
17 March 1782
184
185
Viscosity (internal friction)
 It is a quantity of real liquids that describes the resistance
to displacement of the first fluid part relative to another
one.
 When one layer of a real fluid moves relative to another
layer, there are internal friction forces directed along the
tangential to the layers surface.
 These forces consist in the fact that:
 the layer, moving quickly, effects on the layer, moving
slowly, with accelerating force.
 the layer, moving slowly, effects on the layer, moving
quickly, with inhibitory force.
186
Velocity gradient
 It is a quantity that shows
how fast velocity changes
within a passage from the
first layer to the next layer
in the direction x, that is
perpendicular to the layers
motion direction.
187
Internal friction force
 Internal friction force
depends on the fact, how
much the velocity changes
within a passage from the
first layer to another one.
188
Dynamic viscosity
 It is defined by internal friction force, that is acting at the
layer surface unit, when there is a velocity gradient, which
is equal to one.
189
Unit of dynamic viscosity
 If a fluid with a viscosity of 1 Pascal-second is placed
between two plates, and one plate is pushed sideways with
a shear stress of 1 Pascal, the fluid moves a distance equal
to the thickness of the layer between the plates in
1 second.
190
191
Laminar flow
 Laminar flow (or streamline flow) occurs when a fluid
flows in parallel layers, with no disruption between the
layers.
 Laminar flow occurs when a fluid moves with lower
velocities.
 The external layers are immovable because of molecular
adhesion forces.
 The following layers velocities are increasing with
increment of a distance to the pipe surface.
192
Turbulent flow
 In fluid dynamics, turbulence or turbulent flow is a flow
regime characterized by chaotic and stochastic property
changes.
193
Reynolds number
 The Reynolds number is a dimensionless measure that
determines fluid current character.
 If Re≤1000, the flow will be laminar.
 1000 ≤ Re≤2000 there is a passage from laminar flow to
the turbulent.
 If the pipe is smooth and Re=2300, the flow will be
turbulent.
194
Reynolds number
 For the same Re a flow is
the same for different
fluids in pipes with
different section.
Osborne Reynolds
(1842–1912)
195
196
Stokes’ technique
 This technique is based on the
velocity measure of small
slowly moving spheres in a
fluid.
Sir George Gabriel Stokes
(1819–1903)
197
Stokes’ technique
 Gravity
 Archimedes force
 Resistance force
 ρ is a ball density; r is a ball radius; ρ’ is a fluid density; g
is a free fall acceleration; v is a ball velocity.
198
Stokes’ technique
199
Pouiseuille’s technique
 This technique is based on the
laminar flow in a fine capillary.
 There is a capillary with radius R
and with length l. There is mentally
divided cylindrical layer with
radius r and thickness dr.
Jean Louis Marie Poiseuille
(1797–1869)
200
Pouiseuille’s technique
 Internal friction force acting on the
lateral layer surface dS
 Internal friction force is balanced by
pressure force acting on its base.
201
Pouiseuille’s technique
 Fluid particles velocities
 Koefficient of dynamic
viscosity
202