Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:A General Course of Physics. Mechanics. Textbook
.pdf
11
As the linear and angular velocities are vectors, the relation
r
v
dtd
t
dtd
00
t
2
2
T
between them can be expressed as the vector multiplication:
.
Now let us consider the uniform rotational motion.
This kind of motion suggests that the angular velocity remains
constant = const, and formula (5) implies that
(6)
In order to find the relation between and for a finite time period, it is necessary to integrate the equation (6):
.
As = const, it is possible to remove it from the integral, and as
a result of integration we get:
, (7)
Formula (7) is true only for the rotational motion around the circumference with the constant angular velocity.
For such a motion, it is possible to introduce the period of revolution T. It is the time during which the body completes one revolution, i.e.
it rotates through 360 degrees. The value reciprocal to the period is frequency v. In the International System of Units (SI), the period T is measured in seconds (s), while the frequency – in hertz (Hz). One hertz equals
one revolution per second.
During the time period T, the point particle completes one revolution, i.e. it rotates through the angle = 2 (rad). Then, it follows from
equation (7) that
viz. is the rate of rotation, Hz.
,
12
If the rotation is non-uniform, then the rate of change in the angu-
dt
d
r
dt
d
r
dt
rd
dt
d
a
r
T
r
r
a
n
2
2
2
2
4
v
const
t
0
2
2
0
tt
0
t
lar velocity can be characterized by the angular acceleration (radians per
second squared):
.
The tangential acceleration of a point particle moving around the
circumference is related to the angular acceleration in the following way:
.
The normal acceleration (centripetal acceleration) is expressed as
follows:
.
If the angular acceleration does not change over time (
then
; (8)
, (9)
specifically,
and
are the rates of rotation at the initial time moment
and at the time moment t, respectively.
The angular acceleration is an algebraic value: 0 for uniformly accelerated rotation and 0 for uniformly decelerated rotation.
Formulas (8) and (9) completely describe the rotational motion
with the angular acceleration constant in magnitude and are similar to the
formulas of velocity and distance for the rectilinear uniformly accelerated
motion:
),
13
Rectilinear
uniformly
accelerated motion
Rotational
uniformly
accelerated motion
velocity
v = v0+at;
angular velocity
= 0+t;
distance
2
2
0
ta
tS v
angle of rotation
2
2
0
tt
v
v
r
t
a
n
a
Test Questions
1. What is the difference between the distance travelled by a point
particle and its displacement?
2. What is the term for the average displacement velocity of a
point particle moving arbitrarily, without any particular direction?
3. What is the instantaneous velocity direction
of the vector
moving arbitrarily and curvilinearly in relation to the motion
trajectory?
4. What is the spatial correlation between the instantaneous velocity vector
and the radius-vector
of a point particle?
5. What is the tangential acceleration direction
of a point par-
ticle moving arbitrarily and curvilinearly?
6. What is the direction of the normal acceleration
of a point
particle moving arbitrarily and curvilinearly?
7. What is the complex acceleration direction of a point particle
moving curvilinearly?
8. What is the relation between the tangential acceleration at of a
point particle moving curvilinearly and its instantaneous velocity?
14
9. What is the relation between the normal acceleration an of a
v
point particle moving curvilinearly, the instantaneous velocity
and the trajectory curvature radius R?
10. What is the relation between the complex, normal and tangential accelerations of a point particle moving curvilinearly?
11. Define the term angular velocity
and specify its units of
measurement.
12. Define the term angular acceleration
and specify its units of
measurement.
13. What is the relation between the linear velocity v of a point
particle moving around the circumference with the radius R
and the angular rate of rotation ?
14. A wheel with the radius 1 m is rotating in such a way that the
dependence of its rotation angle on time is = 1 + t + t2
( = 2 rad/s, = 1 rad/s2). What is the tangential acceleration
of the point particles on the wheel rim one second after the rotation onset?
15. If the normal acceleration of a point particle on a rotating
wheel rim has increased four-fold, what fold increase will occur for the linear velocity of the particle?
16. What is the tangential acceleration modulus value of the body
point at a distance of 10 cm away from the rotation axis if the
angular acceleration of the rotating body is 5 rad/s2?
17. A vehicle, moving rectilinearly and with uniform acceleration,
has increased its speed from 36 km/h to 72 km/h over a distance of 150 m. What is the vehicle’s acceleration, its travel
time over the distance and the average velocity?
18. A point particle is moving around a circumference with a radius of 1 m and by the end of its motion it completes one revolution. What is the distance traveled by the point particle and
its displacement?
15
19. A point particle is moving along the sides of a square with a
velocity, constant in magnitude (the length of one square side
is 1 m). What will be the displacement of the point particle, if it
travels two square sides? What is the distance traveled by the
body?
20. A body begins rotating with uniform acceleration and makes 200
complete revolutions, having gained a velocity of 20 rad/s. What
will be its angular acceleration? How long did the acceleration
last before the body gained the speed of 20 rad/s?
21. A point particle with a mass m = 1 kg is moving around a circumference with a radius R = 2 m. The angle of rotation with
respect to time is determined by the formula = 2t + 0,5t2.
What is the value of the angular velocity two seconds after the
motion commenced? What are the values of the tangential and
normal accelerations one second after the motion started? What
is the value of the force applied to the body?
22. A speedboat, moving down a river, outruns a drifting wooden
raft. One hour later, t = 60 min., the speedboat turns back and
meets the raft again 6 km down the river. Calculate the river’s
flow rate if the speedboat’s motor was working at a uniform
speed in both directions.
23. A point particle is located on the rim of a wheel with the radius
R = 0.5 m which is rolling without sliding along a level surface
at a speed v = 1 m/s. Find the acceleration modulus and direction of the point particle.
16
2. DYNAMICS OF TRANSLATIONAL MOTION
2.1. Newton’s Laws of Motion
Dynamics is a subdiscipline of mechanics, concerned with the
study of forces and torques and their effect on motion.
Classical mechanics is based on three laws of dynamics defined
by Newton in 1687. Newton’s laws resulted from a generalization of a
great deal of experimental facts.
Until the beginning of the 20th century, it was considered that any
physical phenomenon could be reduced to a mechanical process which
obeyed Newton’s laws.
However, Albert Einstein advanced relativistic mechanics in
1905 which studied objects moving at high velocities.
Quantum mechanics, concerning with laws determining the motion of elementary particles, was developed in the 1920-s. Classical mechanics is a branch of both relativistic mechanics and quantum mechanics.
Newton’s First Law. An object at rest stays at rest and an object in
motion stays in motion moving at the same speed and in the same direction
unless acted upon by an unbalanced force.
However, in reality there are no objects, unaffected by other objects. Therefore, in practice an object stays at rest or moves uniformly or
rectilinearly if the forces applied are balanced.
The statement referred to Newton’s first law is not immediately
obvious. It was known by daily experience that if a force stops acting,
then motion ceases as well. Consequently, it is natural to suppose that
motion requires a force and remains until the force is applied. This idea
was proposed by Aristotle. Galileo (1564-1642) disproved this concept as
a result of analysis and a number of experiments.
Newton’s first law is not true for all reference frames, but only for
inertial systems. Actually, this law admits that such systems exist. So, Newton’s first law can be defined in another way: there are such reference frames
in which objects remain at rest or move rectilinearly at a constant velocity
unless acted upon by a net force.
Such frames of reference are called inertial reference frames.
There can be an infinite number of them, and any frame of reference be-
17
ing in uniform motion with respect to an inertial frame is also an inertial
const....
3
3
2
2
1
1
a
F
a
F
a
F
aFm /
mFa
frame.
The resistance of any physical object to any change in its state of
rest or uniform and rectilinear motion is called inertia. Therefore, Newton’s first law is also referred to as the law of inertia.
Newton’s Second Law. To characterize the degree of impact on an
object, a special physical quantity is introduced – force. The unit of force in
the International System of Units (SI system) is the newton. As the force is
always directed, therefore the force is a vector variable. Forces, applied to an
object, can change either its state of motion (changing its velocity) or its
shape and size (causing deformation).
Let us apply a force to an object and measure the resultant acceleration. It turns out that when forces F1, F2, F3, act upon an object, its accele-
ration equals а1, а2, а3, … . But the ratio remains the same:
.
Subsequently, F/a may serve as an object characteristic. This
physical quantity is called the mass of a body:
(10)
The mass of a body determines its inertial properties, being its
measure of inertia or, in other words, the mass of a body is a response characteristic to the force acting upon the object with certain acceleration.
The unit of mass in the International System of Units (SI) is the kilogram (kg). The unit of mass alongside the unit of length (the meter) and the
unit of time (the second) is a base unit in the International System of Units
(SI) which has a measurement standard.
Let us write down expression (10) in the following way:
. (11)
Ratio (11) is an analytic form of Newton’s second law.
Newton’s second law states the following: The acceleration of an
object is directly proportional to the magnitude of force acting upon it,
and inversely proportional to the mass of the object.
18
As both the force and acceleration are vector variables, then
mFa
amF
amF
i
i
1221
mmaa
2112
FF
2112
FF
F
12
F
21
m
1
m
2
a
1
a
2
Fig. 6
Newton’s second law can be expressed in the vector form:
or
.
If several forces act upon an object, then the acceleration will be
a result of the total force:
Newton’s Third Law. Any force that acts upon an object results from interactions with another object: if an object m1 acts upon an
object m2 with the force F21, then the object m2, in turn, acts upon the
object m1 with the force F12.
Consider the following example (Fig. 6):
Two objects with the mass of m1 and m2, isolated from external
forces, attract each other due to the fact that they are electrically charged.
Under the action of forces F12 and F21, the objects acquire accelerations
a1 and a2, respectively. The acceleration values are inversely related to the
mass of the objects:
,
consequently, m1·a1=m2·a2 and, therefore,
.
Clearly, the directions of the forces are opposite:
Newton’s third law is stated as follows: Any forces, exerted by
objects on each other, result from interaction; the forces are always equal
in magnitude but opposite in direction.
The forces F12 and F
21
are equal in value and opposite in
direction, but they are applied to
different objects, so they cannot
balance each other.
19
2.2. Momentum and Impulse Connection
F
F
dtF
2
1
t
t
dtF
v
mp
dt
mdF)v
(
dt
pd
Any change in the motion of an object (its velocity magnitude
and direction) is determined not only by the force
, but also by the du-
ration of its action. The change of motion is proportional both to the force
and the time interval dt during which the force was applied. For this
reason, there is a concept of momentum.
Momentum is a physical quantity, measured by multiplying force
by the time of its action
. This equation is only true for short time periods and constant force. In a general case, if the force vector doesn’t re-
main constant and the force action time is long, then the momentum between the moments t1 and t2 will be specified by the integral:
.
The measurement unit of momentum in the International System
of Units (SI) is 1 Ns = 1 kgm/s.
Apart from momentum, there is also a concept of impulse.
The impulse of an object is a physical quantity, measured by
multiplying the mass of an object by its velocity:
.
In terms of measurement units, impulse is equivalent to momen-
tum.
The impulse of a body is a vector variable and is directed in the
same way as momentum.
The concepts momentum and impulse allow for expressing New-
ton’s second law using another mathematical formula:
. (12)
In this form, the law is called change in momentum, according to
which the rate of impulse change equals the total force applied to the
body.
20
Multiplying both parts of the equation (12) by the differential dt, we
pddtF
)( vmd
pd
dtF
ij
f
i
F
dt
d1p
Ffff
N
111312
dt
pd
2
222321
Ffff
N
will attain one more formula for Newton’s second law:
.
According to the formula, the law of impulse change can be
written in the following way: The change of impulse
during the time
interval dt equals momentum
, acting upon the body during the time
interval.
Consequently, Newton’s second law can be expressed with the help
of several mathematical formulas. All of them are interchangeable as they
describe one and same law. Practical application of any formula depends on
the problem to be solved.
2.3. The Law of Momentum Conservation
Let us consider a system, consisting of several objects. Forces, acting upon objects of the system, can be classified into internal and external.
Internal forces are those with which individual parts of the system act upon
each other, while external forces are those, exerted by exogenous bodies
which do not belong to the system.
If external forces are absent, the system is called closed, or iso-
lated. If the sum total of all external forces equals zero, then such a system is quasi-closed (quasi-isolated).
Let us consider a physical system, consisting of N objects. Let us
number the objects from i = 1 to i = N. Assume that internal forces
act
upon objects of the system interactively, and apart from that, external
forces
from bodies not belonging to the system are exerted on the ob-
jects of the system. Let us write Newton’s second law for each object of
the physical system:
;
;
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]