CC BY
4
Mustafakulov A.A., candidate of physical and mathematical sciences
associate professor Jizzakh Polytechnic Institute Uzbekistan, Jizzakh
CONTINUITY OF PHYSICS AND MATHEMATICS
Abstract. In this article, physical problems are solved with the help of differential equations, and thus the interdisciplinarity is shown.
Key words: elasticity, mechanical energy, harmonic oscillatory motion, differential equation, Snell's law, Fermat's principle.
1. The total mechanical energy of a body that vibrates under the influence of elastic force
„ —&2 kx2
E =-+- (1)
2 2 W
is expressed by the equation. According to the law of conservation of total
TYl 2
mechanical energy E = + — = const (2)
is and its derivative with respect to time is equal to. That is,
dE
i = E'=0 (3)
It is known that
—&*l = kxdx = 0 (4) dt dt
dx
a=d (5)
is, and (4) can be written as:
d& dx T dx . .. d& T .
—--+ kx— = 0 eKH — — + kx = 0 (6).
dt dt dt dt
d2 x d&
If a = —- = — = x" considering that, —x"+kx = 0 formed, —x" =-kx and dt dt
k d2 x k
x'=— x or —- = — x (7) originates. This is the differential equation of
— dt —
motion of a body under the influence of an elastic force.
2 k
This is the equation of harmonic motion x' '=-c02x compared to c2 =
Ik ^ 2^ —
0
—
C = J— , T = — = 2^J— we determine the equation, that is, the period of the
V — c0 V k
oscillating movement that occurs under the influence of the elastic force. Based
on the above x = x0 cos cat the equation x = x0 cos Jkt can also be written in the
V m
form
h
Figure 1.
2. Full mechanical energy of a body oscillating under the influence of
mg2
gravity E = + mgh (8) being here h = l(1 - cos«) (9) is defined as (Fig. 1).
l - h i-:-
Because —— = cos a is taken as Substitution formula cosa = v 1 - sin2 a
according to (8) h = l (1 -V1 - sin2 a) (10) we write like
In small corners sina = a can be taken as. In that case h = l(1 -V1 -a2) is
s
formed. 1- Designating the arc AS in the picture as s, a = - if we say, then
h = l(1 -
2
1 -^) will be. After the exchanges h = l-Jl2 -s2 (11) we generate.
l2
From formulas (8) and (11) E = + mg(l-Vl2 -s2)(12) results
—+ —g(l -J?2 -2 2
From this equation, we take the derivative with respect to time and set it equal to zero:
1
—3&+—g(l2 + s2)- 2 ss' = 0 We reduce this equation to m and, taking into account that it is for small
angles, s2 << l2 we get, gss' = 0.
It is known that — = s' =& it is based on it#'+ gs = 0. But it can be
dt l
g
n = $ = s"written because of the acceleration s"+—s = 0. r l
Finally, the differential equation for the motion of a mathematical
pendulum oscillating under the influence of gravity
s" = -gs it follows that (13)
Being here, s ' '= ar it is called the tangential constituent of the swing of a mathematical pendulum.
By comparing this equation with the equation of motion of harmonic oscillationx '' = -a>\x, we determine co\ = — whether c0 = J— it is.
l u \l
2n
If cc = — we take into account that, that is, T = 2n
0 T \
— the formula for
—
finding the oscillation period of a mathematical pendulum is formed. In general, the equation of the swing motion of a mathematical pendulum s = s0 cos c01 or
V
—t will be like.
g
3. It is known that the law of refraction of light was derived by Snellius on the basis of Huygens' principle. According to the Huygens principle, any point on the wavefront can become a new wave source. This condition is fulfilled if the medium in which the wave propagates is homogeneous. If the wave is propagating in an inhomogeneous medium, then the wavefront cannot maintain its original flat or spherical shape. The wave deviates from the law of rectilinear propagation. In such cases, it is recommended to rely on the farm principle. According to this principle, the path of a wave from a certain point A to a point V is a curved line, and it must be in the form that takes the least time.
When a wave is moving from a medium with a low density to a medium with a high density, its propagation speed in the second medium decreases (V1 > V2). The total time taken for the wave to travel from point A to point B is defined as follows. (Fig. 2. A1 B1 is the border of two environments, N is normal).
AO OB
t = t + L =-+-
12 % ¿2
Using the properties of right-angled triangles AA1 O and BB1 O, this time is determined as follows.
■yjh2 + x2 -y]h2 + (— - x)
t= . '12 + x2 | A2
Therefore, time t is a function of the variable x, and it is — necessary to
dx
calculate the derivative and make it equal to zero, because in order for the total time sought to reach the smallest - minimum value, based on the differential calculus, it is required that the derivative obtained by the coordinate of the variable x becomes zero.
dt 2x 2(l - x)
dx 232tJh2 + x2 232,jh22 + (— - x)2
= 0
AA1O and BB1 O , x = sin a, 1 * = = sin ^from triangles; is
Vhj2 + x2 h2 + (I - x)2
generated and written in the form of the above equationsina - = 0. or
sm a
&
THJL; From this equality, the well-known Snellius law is derived:
&
1 2 & sin a
n = — =
& sin p
hi
Figure 2
Also, on the basis of Fermat's principle, phenomena such as atmospheric refraction, i.e. non-homogeneity of atmospheric density, changes depending on height, resulting in a certain degree of bending of sunlight, the observer seeing the sun before it rises, are also explained. When using laser beams in geodesy, the need to make corrections to measurement results due to non-homogeneous temperature and density of air layers near the Earth's surface also comes from Fermat's principle.
References:
1.Glazunov A.G., Nurminsky I.I., Pinsky A.A. Methodology of teaching physics in secondary school. Tashkent "Teacher" 1996.348 p.
2. Axmadjonova, Y. T., & Axmadjonova, U. T. (2021). O'quv dars mashg'ulotlarida didaktik o'yin texnologiyalaridan foydalanish. Science and Education, 2(11), 977-984.
3. Мустафакулов, А. А., Халилов, О. К., & Уринов, Ш. С. (2019). Цель и задачи самостоятельной работы студентов.
4. Juraeva, N. M., & Akhmadjonova, U. T. (2022). Interdisciplinary connection in teaching the subject of curved line movement. Экономика и социум, (5-1 (96)), 80-83.
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