или , (6.24)где – момент инерции фи

или
, (6.24)
где – момент инерции физического маятника относительно оси вращения.
Рассмотрим малые колебания, при которых . Уравнение (6.24) примет вид:

или
, (6.25)
где . Полученное уравнение совпадает с уравнением (2.2), описываю-щим свободные колебания материальной точки в среде без сопротивления. Таким образом, малые свободные колебания физического маятника являются гармоническими
,
где постоянные А и α определяют из начальных условий, а период этих коле-баний находят по формуле
.
Вопросы для самоконтроля
1. Как определить кинетический момент материальной точки и механической системы?
2. Чему равен кинетический момент твердого тела относительно неподвижной оси вращения?
3. Как сформулировать теоремы об изменении кинетического момента мате-риальной точки и механической системы?
4. Как сформулировать закон сохранения кинетического момента механиче-ской системы?
5. Как записать дифференциальное уравнение вращательного движения твердого тела?
Лекция 7. ТЕОРЕМА ОБ ИЗМЕНЕНИИ КИНЕТИЧЕСКОЙ ЭНЕРГИИ
7.1. Кинетическая энергия материальной точки и механической системы
Рассмотренные в лекциях 5 и 6 меры механического движения – количество движения и кинетический момент – не описывают движение системы, проис-ходящее под действием внутренних сил. Приведем в качестве примера систе-му, состоящую из двух одинаковых шаров, соединенных пружиной (рис. 7.1).

Сожмем пружину и отпустим шары без начальной скорости, поместив их на гладкую горизонтальную плоскость. Под действием внутренних сил (сил упругости пружины) шары будут совершать колебательное движение, причем в любой момент времени их скорости будут равны по величине и противоположны по направлению . Количество движения и кинетиче-ский момент этой системы относительно произвольной неподвижной точки О тождественно равны нулю и не отражают движение системы:
;
.
Этого недостатка не имеет рассматриваемая в данной лекции динамическая характеристика – кинетическая энергия.
Кинетическая энергия материальной точки – это скалярная мера механиче-ского движения, равная половине произведения массы точки на квадрат ее скорости:
. (7.1)
Единица измерения кинетической энергии в системе СИ – 1 Дж.
Кинетическая энергия механической системы – это сумма кинетических энергий всех материальных точек, образующих систему:
. (7.2)
Кинетическая энергия является неотрицательной величиной, она равна нулю только в том случае, если неподвижны все точки системы. Однако и кинетиче-ская энергия не является универсальной мерой движения, так как, будучи ве-личиной скалярной, не отражает направление движения.

7.2. Кинетическая энергия твердого тела
Получим формулы для определения кинетической энергии твердого те-ла в различных случаях его движения.
7.2.1. Поступательное движение
При этом виде движения скорости всех точек тела одинаковы (рис. 7.2). Из формулы (7.2) следует
, (7.3)
где m – масса тела.
Таким образом, кинетическая энергия твердого тела, движущегося по-ступательно, равна половине произведения массы тела на квадрат его скоро-сти.
7.2.2. Вращательное движение
При вращении твердого тела вокруг неподвижной оси с угловой скоро-стью ω скорость точки (рис. 7.3), находящейся на расстоянии от оси вращения, . Из формулы (7.2) получим
.
Здесь – момент инерции тела относительно оси вращения.
Теперь окончательно запишем
, (7.4)
т.е. кинетическая энергия тела, вращающегося вокруг неподвижной оси, рав-на половине произведения его момента инерции относительно оси вращения на квадрат угловой скорости тела.
7.2.3. Плоское движение
Пусть фигура движется в плоскости xOy с угловой скоростью ω, а ее мгновенный центр скоростей (м.ц.с.) находится в точке Р (рис. 7.4). Тогда скорость точки , находящейся на расстоянии от м.ц.с., будет равна .
Из формулы (7.2) получим
,
где – момент инерции тела относительно оси , проходящей через м.ц.с. перпендикулярно плоскости движения.
Таким образом, кинетическая энергия тела при его плоском движении
. (7.5)
Непосредственное использование этой формулы осложнено тем, что момент инерции в общем случае является переменной величиной. Это связано с изменением положения м.ц.с. плоской фигуры в процессе ее движения.


Таким образом, кинетическая энергия тела при его плоском движении
. (7.5)
Непосредственное использование этой формулы осложнено тем, что момент инерции в общем случае является переменной величиной, Это связано с изменением положения м.ц.с. плоской фигуры в процессе ее движения.
Поэтому получим другое соотношение, содержащее постоянный мо-мент инерции, для чего проведем через центр масс С плоской фигуры ось и используем формулу (4.12)
. (7.6)
Из формул (7

Результаты (английский) 1:

or (6.24)where is the moment of inertia of a physical pendulum relative to the axis of rotation. Consider small variations in which. Equation (6.24) takes the form: or (6.25)where. The resulting equation is the same as equation (2.2), describe the prevailing free-oscillation of a material point in Wednesday without resistance. Thus, small oscillations of the pendulum physical free are harmonic ,where the constant a and α define the initial conditions and period of these stake-bath found by the formula .Questions for self-monitoring1. How to determine the angular momentum of the material point and mechanical system?2. What is the angular momentum of a rigid body with respect to the stationary axis of rotation?3. How to formulate theorems about changing kinetic moment mate-an equatorial point and mechanical system?4. How to formulate the law of conservation of kinetic moment of the Vienna system?5. How do I write the differential equation of rotational motion of a rigid body?Lecture 7. KINETIC ENERGY CHANGE THEOREM7.1. The kinetic energy of a material point and mechanical systemCovered in lectures 5 and 6 measures mechanical motion-momentum and angular momentum-do not describe the movement system, origin-powered under the influence of internal forces. Let's give as an example the MU system, consisting of two identical balls connected by a spring (fig. 7.1). Going to compress the spring and let the balls without initial velocity by putting them on a smooth horizontal plane. Under the influence of internal forces (spring elasticity) balls will make oscillating motion, and at any time their speed will be equal in magnitude and opposite in direction. Number of motion and kinetic energy-sky point of this system about an arbitrary fixed point on identically equal to zero and do not reflect the movement of the system: ; .This shortcoming is not dealt with in this lecture, dynamic characteristic is the kinetic energy.The kinetic energy of a material point is a scalar measure of the outer movements, equal to half of the works of mass points on the square of its velocity: . (7.1)The unit of measurement of kinetic energy in SI-1 j. The kinetic energy of a mechanical system is the sum of the kinetic energies of all material points forming a system: . (7.2)Kinetic energy is a non-negative value, it is zero if and only if all fixed point system. However, kinetic energy-energy there is not a universal measure movements as being guises ve scalar does not reflect the direction of movement. 7.2. The kinetic energy of a rigid body Get the formula to determine the kinetic energy of the solid ones-la in various cases. 7.2.1. The progressive movement This type of movement speed of all points of the body are the same (fig. 7.2). From the formula (7) should be (7.3)where m is the mass of the body. Thus, the kinetic energy of a rigid body moving-stupatel'no, equal to half their body weight by the square of the piece it soon. 7.2.2. Rotational movement When the rotation of a rigid body around a fixed axis with angular velocity ω Stu soon point (fig. 7.3), located at a distance from the axis of rotation. From the formula (7.2) get .Here is the moment of inertia of the body about an axis of rotation. Now finally we write (7.4)i.e. the kinetic energy of the body, rotating around a fixed axis, Rav-half works its moment of inertia about an axis of rotation on the square of the angular velocity of the body. 7.2.3. Flat movement Let the figure moves in a plane with an angular velocity ω xOy and its instantaneous Center of velocity (m.c.s.) is in a point p (fig. 7.4). Then the speed of a point at a distance from the m.c.s. will be equal.From the formula (7.2) get ,where is the moment of inertia of the body about an axis passing through the m.c.s. perpendicular to the plane of motion.Thus, kinetic energy of the body with its flat movement . (7.5)Direct use of this formula, complicated by the fact that in the General case, the moment of inertia is variable. This is due to the changing of position of the m.c.s. flat figure in its movement. Thus, kinetic energy of the body with its flat movement . (7.5)Direct use of this formula, complicated by the fact that in the General case, the moment of inertia is variable, it is bound to change provisions of the m.c.s. flat figure in its movement. So get another ratio that contains permanent Mo-ment of inertia, which draw through Center of mass with flat shapes and use the formula (4.12) . (7.6)Of formulas (7

Результаты (английский) 2:

or
, (6.24)
where - the. moment of inertia of a physical pendulum with respect to the axis of rotation
Consider small oscillations in which. Equation (6.24) takes the form:

or
, (6.25)
where. This equation coincides with the equation (2.2) describing the conductive-free oscillations of a material point in a medium without resistance. Thus, small free oscillations of a physical pendulum are harmonic
,
where the constants A and α are determined from the initial conditions, and the period of these oscillations are the stake-on formula
.
Questions for self-control
1. How to determine the angular momentum of the material point and a mechanical system?
2. What is the angular momentum of a rigid body about a fixed axis of rotation?
3. How to formulate the theorem of change of angular momentum mate-rial point and a mechanical system?
4. How to formulate the law of conservation of angular momentum mechanically-tion system?
5. How to write a differential equation of the rotational motion of a solid?
Lecture 7. Theorem TO AMEND THE KINETIC ENERGY
7.1. The kinetic energy of the material point and the mechanical system
considered in lectures 5 and 6 measure mechanical movement - the amount of movement and the angular momentum - do not describe the motion of the system, prois-propelled by the action of the internal forces. As an example, the system-mu, consisting of two identical balls connected by a spring (Fig. 7.1).

Compress the spring and release the balls without initial velocity, placing them on a smooth horizontal plane. Under the influence of internal forces (spring elastic force) the balls will make an oscillating movement, and at any time of the velocity are equal in magnitude and opposite in direction. The momentum and kinetic-ing point of this system with respect to an arbitrary fixed point O are identically zero and do not reflect the motion of the system:
;
.
This disadvantage is not considered in this lecture dynamic characteristic - kinetic energy.
The kinetic energy of a material point - a scalar measure mechanical-sky movement, which is equal to half the product of the mass points on the square of its speed:
. (7.1)
. The unit of measure of the kinetic energy in the SI system - 1 J.
The kinetic energy of the mechanical system - this is the sum of the kinetic energies of all the material points making up the system:
. (7.2)
The kinetic energy is a non-negative value is zero only if all fixed points of the system. However, the kinetic-energy Skye is not a universal measure of the movement as being ve-mask of a scalar does not reflect the direction of movement.

7.2. The kinetic energy of a rigid body
obtain a formula for the determination of the kinetic energy of rigid body la in different cases of its movement.
7.2.1. Translational motion
At this type of motion of all the points of the body are the same speed (Fig. 7.2). Formula (7.2) implies
(7.3)
where m -. Body weight
Thus, the kinetic energy of a rigid body moving in translational, equal to half the product of body weight by the square of his soon-sti.
7.2.2. The rotational movement
During the rotation of a rigid body around a fixed axis with angular velocity ω Stu-point speed (Fig. 7.3), at a distance from the axis of rotation. From (7.2) we get
.
There - the moment of inertia with respect to the axis of rotation.
Now, finally, we write
(7.4),
ie, the kinetic energy of a body rotating around a fixed axis, equal, at half time the product of inertia about the axis of rotation on the square of the angular velocity of the body.
7.2.3. Surface Movement
Let the figure moves in a plane xOy with an angular velocity ω, and its instantaneous velocity center (m.ts.s.) located at the point P (Fig. 7.4). Then the velocity of a point at a distance from m.ts.s., will be equal.
From (7.2) we obtain
,
where - the moment of inertia about the axis passing through m.ts.s. perpendicular to the plane of movement.
Thus, the kinetic energy of the body during its movement plane
. (7.5)
The direct use of this formula is complicated by the fact that, in general, the moment of inertia is variable. This is due to a change in position m.ts.s. flat figure during its movement.


Thus, the kinetic energy of the body during its movement plane
. (7.5)
The direct use of this formula is complicated by the fact that, in general, the moment of inertia is variable, This is due to the change in position m.ts.s. plane figure in the process of movement.
Therefore, we obtain a different ratio, containing a permanent mo-moment of inertia, which draw through the center of mass with a flat figure axis and use the formula (4.12)
. (7.6)
From (7

Результаты (английский) 3:

or(6.24)where is the moment of inertia of a physical pendulum about an axis of rotation.consider the small fluctuations in which. equation (6.24) takes the form:or(6.25)where. the equation coincides with the equation (2.2), describing the relevant free fluctuations of a material point in the environment without resistance. thus, the small free fluctuations of physical pendulum are гармоническими,where the constant a and various sets of initial conditions, and the period of the colet - баний are as follows.questions for self-control.1. how to determine the kinetic moment of the point and the mechanical system.2. what is the kinetic moment of rigid body about the fixed axis of rotation.3. how to formulate a theorem on the change of the kinetic moment of materials риальной point and mechanical systems?4. how to formulate the law of conservation of kinetic moment механиче by system?5. how to write differential equation of rotational motion of rigid body.lecture 7. on the change of kinetic energy theorem7.1. the kinetic energy of a material point and mechanical systemthe in class 5 and 6 measures the mechanical movement is the movement and the kinetic moment is not to describe the system, has a walking under the action of internal forces. take as an example the mu system consisting of two identical balls, the spring (figure 1). 7.1).compress a spring and release the balls without the initial speed by placing them on a smooth horizontal plane. under the action of internal forces (forces of the springs) balls perform the swinging motion, and at any time their velocities are equal in magnitude and opposite in direction. the number of users and кинетиче a moment of this system on the arbitrary fixed point o is equal to zero and does not reflect the movement of the system.;.this deficiency was not considered in this lecture, the dynamic characteristic is the kinetic energy.the kinetic energy of a material point is a scalar measure механиче and movement, equal to half of the mass point on the square of its speed.. (7.1)the unit of measurement of kinetic energy in the system b is 1 joule.the kinetic energy of a mechanical system is the sum of the kinetic energies of all material points within the system.. (7.2)the kinetic energy is неотрицательной size, it is zero if and only if the fixed all the points system. however, and кинетиче nations energy is not a universal measure of movement, as in ve as a scalar, reflects the direction of movement.7.2. the kinetic energy of rigid bodyget the formula to calculate the kinetic energy of the la in different cases of its movement.7.2.1. the progressive move