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Файл:A General Course of Physics. Mechanics. Textbook
.pdf
31
In particular, it takes place during rectilinear motion, when the
F
kxFF
2
2
0
kx
kxdxdlFA
x
S
l
2
2
kx
A
Fig. 10
F
x
kx
kx2/2
force F constant in magnitude makes up a constant angle with the direction of motion.
Work is an algebraic quantity. Therefore, when 90º > > 0,
A > 0, work is performed by the applied force itself; when 90, A < 00,
work is fulfilled against the force applied; when = 90, A = 0.
Centripetal force is always directed perpendicular to the velocity
(displacement), А = 0. In this case, it does not perform any work. An example of variable force can be the elastic force of a spring.
Let us determine the work done during the spring’s extension. In
order to extend the spring, it is necessary to apply the amount of force
equal in magnitude to the elastic force.
The elastic force is F = –kx, where х is the spring elongation, k is
the elastic coefficient.
is the force applied to the spring when stretched.
.
The work of the elastic force is
;
.
This formula for work can be obtained graphically (Fig. 10):
numerically, the work equals the area
of the crosshatched triangle.
The unit of work for the elastic force in the International Measurement System (SI) is the joule:
1 J = 1nm.
In practice, it is not only the
work done that is important, but also
the time period during which it is
performed:
32
For this reason, another term should be introduced – power
t
A
P
dt
dA
t
A
P
t
lim
0
Fv
dt
dl
F
dt
dA
P
input
eff
input
eff
P
P
A
A
which shows the work intensity per unit time – for the specification of
mechanisms designed for performing operations. Power P is a quantity
equal to the ratio of work А to the time period t during which it is
performed:
.
If unequal amounts of work are performed over equal time periods, then the power is time-varying. In these circumstances, the term of
instantaneous power is introduced:
.
Let us consider the equation for power in a different form:
dA = FdS.
Then,
.
The unit of power in the International System of Units (SI) is the
watt: 1 W = 1 J/s.
If a mechanism is designed for performing mechanical work (such
as lifting heavy weights), then the total input work is not usually effective
work, because some of the amount is spent on doing work against friction. In
this connection, the term efficiency ratio is used:
,
where А
and Р
eff
are effective work and power, respectively; А
eff
input
, Р
input
– are the input work and power, respectively.
33
3.2. Conservative and Non-Conservative Forces.
1
2
а
b
dl
F
Fig. 11
Potential Field of Force
Any object is exposed to the action of surrounding bodies. Together with the change of the object in space, the force of the surrounding
bodies acting upon the object will change too. At every point in space the
object is exposed to the action of a particular force which is characteristic
of the point. As a result, it is considered that the object is in the field of
forces. For instance, an object near the Earth’s surface is exposed to the
force of gravity. The Earth itself is under the influence of attraction
forces, created by other celestial bodies. In the space among other motionless electrically charged bodies, there is an electrostatic field of
forces.
There are conservative (or potential) and non-conservative
forces.
The forces, acting on an object, are called conservative (potential), if the work of these forces does not depend on the trajectory of the
object displacement, but is only determined by the initial and final positions of the object in space. In this case, the field of forces is called potential.
The forces acting on an object are called non-conservative, if the
work of these forces depends on the object displacement trajectory.
Let us show that the work of conservative forces along any
closed trajectory equals zero. We will divide the closed trajectory with
points 1 and 2 into two parts: a and b
(Fig. 11). The work along the whole
trajectory is
A = (A12)a+(A21)b, (19)
where (A12)a is the work aimed at the
object displacement from point 1 to
point 2 along the trajectory a; (A21)b is
the work aimed at the object displacement from point 2 to point 1 along the
trajectory b.
Let us denote the work aimed
at the object displacement from point
34
1 to point 2 along the trajectory b as (A12)b. At any elementary segment of
0ldlF
2
1
12
2
1
h
h
F
FdhFdlA
Fig.12
1
2
h
1
h
2
dh
dl
mg
+
the trajectory where the object is displaced in the opposite direction, the
angle between the force direction F and the displacement vector dl will
be acute in one case and obtuse in the other case. Consequently, the work
(A21)b and the work (A12)b differ only in one sign:
(A21)b = –(A12)b. (20)
Having added (19) to (20), we will get
A = (A12)a – (A12)b = 0.
But in the potential field, work does not depend on the trajectory,
i.e. (A12)a = (A12)b, and so А = 0.
Therefore, the potential field of forces can also be defined in a
different way: it is a field of such forces, the work of which equals zero
along any closed trajectory. Mathematically, it is expressed in the following way:
.
An example of the potential field is the field of gravity. Let us
prove it.
Assume that an object is moving from point 1, being at the
height of h1, to point 2, being at
the height h2, along an arbitrary
trajectory (Fig. 12). The force acting upon the object at any point of
the trajectory has the same magnitude mg and is directed vertically
downwards. Here, it is better to
express the work in the form:
in particular, dlF=dh is the displacement vector projection dl on
the force direction; F = –mg = const.
, (21)
35
Then, formula (21) will have the form:
)(
2112
2
1
hhmgdhmgA
h
h
lFlFA
frfr
fr
F
l
AWW
21
.
This equation does not depend on the trajectory. Consequently,
the field of gravity is potential in its nature. Attraction fields, electrostatic
fields and fields of elastic forces are also referred to as potential fields.
Now we will prove that friction forces are non-conservative, and
as a result, their field is not potential.
As work in the potential field over a closed trajectory equals zero, then forces perform positive work at certain parts of the trajectory and
negative work – at other parts of the trajectory. The work of friction
forces
always remains negative because
and
have the opposite directions. Thus, the work of friction forces
over a closed trajectory will not be equal to zero, and therefore, friction
forces are non-conservative.
3.3. Energy. Potential Energy
The state of a mechanical system at any point and at any time can
be characterized by the following parameters – coordinates x, y, z and the
velocity projections on these coordinates vx, vy, vz.
It is also possible to specify the state of a system in a different
way – using one quantity, which is called energy W. Energy is the function of the system state and depends on coordinates and velocity:
W = f (xi,vi). If coordinates or velocity vary, energy changes too. The unit
of energy measurement is work. Assume that the system alters from state
1 with the energy W1 into state 2 with the energy W2. Then,
. (22)
If W1 > W2, then A > 0. In this case, the system itself performs
work over the external bodies by means of reducing its own energy. It
means that energy indicates the ability of objects to perform work.
36
If W1 < W2, then A < 0. In such a case, work is done on the system
by external forces. Due to this, energy in the system increases.
The total mechanical energy can often be represented by two additive components:
W(x, y, z, vx, vy, vz) = Wp(x, y, z) + Wk(vx, vy, vz)
where Wp(x, y, z) is the potential energy. It depends on the relative posi-
tions of bodies in the system or on relative positions of separate parts of the
same body (e.g. a compressed spring); Wk(vx, vy, vz) is the kinetic energy; it
depends on velocities of the system bodies. The potential and kinetic energies, as well as work, are measured in the SI system in joules (J).
Firstly, let us examine the potential energy. First of all, it is worth
mentioning that the notion of the potential energy can be applied only to
the potential field of forces. We will illustrate it.
Let us consider a body (or a system of bodies) in the potential field
of forces. Assume that the body is travelling from point M along a closed
trajectory and then returns back to point M.
Then, due to the connection between energy and work, we will
get formula (22).
But as we saw earlier, the work along a closed trajectory in the
potential field of forces equals zero. Consequently,
W1 – W2 = 0; W1 = W2.
Thus, going about a closed trajectory and returning to the initial
point, we get the same energy value, i.e. this quantity is the function of
the body position. It is this quantity that is called the potential energy.
In a non-potential field of forces, the work about a closed trajectory does not equal zero and, hence, it is not possible to introduce the notion of the potential energy.
Now we will derive equations of the potential energy for various
potential fields.
1. Gravity field. If a body with a mass of m is moving from level
h1 into level h2 (Fig. 13), the force of gravity F = –mg will perform the
work
37
2
1
2
1
h
h
h
h
mgdhFdhA
21
mghmgh
2
r
mM
GF
21
22
11
2
1
2
1
2
1
rr
GmM
r
dr
GmMdr
r
mM
GFdrA
r
r
r
r
r
r
21
r
mM
G
r
mM
GA
const
r
mM
GW
ï
h
1
h
2
W
p1
W
p2
F = –mg
+
Fig. 13
. (23)
On the other hand, the work can be presented as the difference
between the potential energies at these levels:
А = W
– Wp2. (24)
p1
By comparing (23) and (24), we get
Wp = mgh + const.
The potential energy is determined by selection criteria of the
energy reference-starting point. Wp = 0 is generally accepted when h = 0.
Then, const = 0 and Wp = mgh.
2. Attraction field. According to the law of universal gravitation,
the force of attractive interaction is acting between the bodies with the
masses m and M, being placed at a distance of r from each other.
,
where G is the gravitational constant.
If the distance between the bodies changes from r1 to r2 , this
force will perform the work
;
On the other hand, work is re-
lated to energy (24).
By comparing (24) and (25)
we get:
. (25)
,
38
It is evident that when bodies are at a very long distance from each
r
mM
GWï
2
21
r
qq
kF
r
qq
kW
ï
21
22
2
2
2
1
2
1
2
1
kxkx
kxdxFdxA
x
x
x
x
const
2
2
kx
Wp
other, they almost never interact. The potential energy of interaction must be
zero, i.e. when r→ Wp→0. For such a kind of reference-starting point selection for energy, const = 0 and
.
3. The field of Coulomb forces. According to Coulomb’s law, the
force of repulsive interaction acting between two similar charges q1 and
q2, located at a distance of r from each other is:
,
specifically, k is the electric constant.
By analogy with the attraction field, we get
4. The field of elastic forces. In conformity with Hook’s law, the
elastic force is the compression (extension) of a spring; k is
F = –kx,
where x is the spring compression (extension); k is the spring tension.
The work, done during compression or extension of a spring
from x1 till x2, is
. (26)
On the other hand, work is related to energy (24).
By comparing (24) and (26), we come to
.
The value x = 0 corresponds to the equilibrium position (of the
spring). It is generally believed that in equilibrium, the potential energy
39
equals zero. In such cases of reference-
2
2
kx
Wp
dtFFdldA v
dt
d
mmaF
v
dt
dt
d
mdA v
v
vvdmdA
2
1
2
2
2
2
1
vv
v
v
m
mvdvA
22
2
1
2
2
vv mm
A
k12k
WWA
m
v
Fig. 14
starting point selection for energy,
const = 0 and
.
In conclusion, the expression of
the potential energy will be different for
various potential fields.
3.4. Kinetic Energy. The Law of Energy Conservation
The kinetic energy of an object is the energy that depends on the
velocity at which the object is moving. Let us derive the equation for kinetic energy.
Assume that a body with the mass m is moving at the velocity v
at this moment and is acted upon by the force F with the same direction
as the velocity (Fig. 14). Then, during the time dt this force will perform
the work dA upon the body:
;
;
;
The total work, done for a limited time t, during which the velocity of the body changes from v1 to v2,
;
(27)
The work, performed by an external force over the body, increases its velocity and, as a result, increases its kinetic energy from Wk1 to
Wk2. As the work is a unit of measuring energy, then it is possible to write
. (28)
Having compared (27) and (28), we will get
.
40
const
2
2
vm
W
k
.
2
2
vm
Wk
pk
WWW
AWW
12
When v = 0 the kinetic energy is considered to be zero. Then
. (29)
As opposed to the potential energy, whose formula depends on
the potential field type, the formula of the kinetic energy (28) remains
unchanged. Only in the theory of relativity does it have a different form.
When considering the potential energy, we defined work A as the
work, performed by the system itself over external forces.
In this particular case, in formula (28) work A is the work, performed by external forces over an object.
It follows from formula (28) that the work, performed over the
object, equals the kinetic energy increment, gained by the object. If
А > 0, then the kinetic energy increases, if А < 0, it decreases.
Let us consider the law of mechanical energy conservation
through the example of a system of bodies with internal conservative
forces (such a system is called conservative). Assume that external forces
also act upon the system.
In such a case, the body (or the system of bodies) may possess
both the kinetic and potential energies simultaneously. The sum of these
energies makes up the total mechanical energy of the body (or the system
of bodies):
.
We saw that when work is done by external forces over a system,
the kinetic energy increases. In a conservative system, work done by external forces enables the increase of both the kinetic and potential energies, i.e. the increase of the total mechanical energy. For instance, when
the body is lifted up with acceleration, both the potential and kinetic
energies increase. Consequently, if a system acted upon by an external
force enters into state 2 with the energy W2 from state 1 with the energy
W1, then
or W = A.
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