груз 2, масса которого в два раза м

груз 2, масса которого в два раза меньше массы диска (рис. 6.5а.). В некото-рый момент времени груз, являющийся самоходным механизмом, начинает двигаться по хорде DЕ. Радиус диска равен r, СО = 0,5r.
Определить угловую скорость диска при прохождении груза через середину хорды О, если в этот момент скорость груза относительно диска равна (рис. 6.5б). Силами сопротивления пренебречь.
Рассмотрим систему, состоящую из диска 1 и груза 2. На нее действуют внешние силы: сила тяжести диска , сила тяжести груза и составляющие реакций опор . Моменты всех этих сил относительно оси z рав-ны нулю, поэтому главный момент внешних сил и кинетический момент системы относительно оси z не изменяется, т.е. . Следовательно, , где – начальный и конечный кинетические моменты системы, соответ-ствующие положениям груза в точках D и O.
В начальный момент времени диск и груз представляют собой одно те-ло, вращающееся вокруг оси z, поэтому
;

;
. (6.18)
В конечный момент времени кинетический момент системы равен сумме ки-нетических моментов диска и груза:
, (6.19)
где – абсолютная скорость груза, . Относительная и переносная скорости груза в точке О направлены вдоль хорды DE,
;

Из равенства (6.19) следует, что
. (6.20)
Приравнивая правые части равенств (6.18) и (6.20)
,
получим .
6.4. Дифференциальное уравнение вращательного движения твердого тела
Рассмотрим тело, вращающееся вокруг неподвижной оси z с угловой скоростью ω. На него действуют система внешних сил ( ) и реакции опор (рис. 6.6). Чтобы исключить из рассмотрения неизвестные реакции , запишем 3-е уравнение системы (6.14):
. (6.21)
Поскольку в соответствии с формулой (6.9) , из уравнения (6.21) полу-чим
(6.22)
или
, (6.23)
где φ – угол поворота тела.
Сравнивая последнее уравнение с дифференциальными уравнениями движения центра масс (5.23), применяемыми для описания поступательного движения тела, приходим к выводу об аналогичности структур этих уравне-ний. Поскольку масса характеризует инертность тела, совершающего посту-пательное движение, момент инерции является мерой инертности тела, вращающегося вокруг неподвижной оси.
Рассмотрим в качестве примера движение под действием силы тяжести физи-ческого маятника, т.е. тела, имеющего горизонтальную ось вращения, кото-рая не проходит через центр масс тела С (рис. 6.7). Обозначим через силу тяжести физического маятника, – реакцию его оси, h – расстояние от оси вращения до центра масс тела. Совместим ось z с осью вращения тела и за-пишем дифференциальное уравнение вращательного движения

Результаты (английский) 1:

cargo 2, whose mass is less than half the mass of the disc (fig. 6.5). SPAC-ing time cargo is a self-propelled mechanism starts to move on the chord DE. The radius of the disk is equal to r, with = 0, 5r.Determine the angular velocity of the disk with the passage of goods through the middle of the chord on, if at this moment the speed of cargo regarding disk (fig. 6.5). The resistance forces are neglected. Consider a system consisting of disk 1 and cargo 2. On it there are external forces: gravity, gravity drive cargo and reactions. Moments of all these forces z-axis Rav-zero, so the main point of external forces and angular momentum of the system on the z axis does not change, i.e.. Consequently, where the initial and final kinetic points system, the relevant provisions of the load at the points D and O. In the initial time and disk load is one those Lo, rotating around the z axis, so ; ; . (6.18)In the end time of the system is the sum of the kinetic moment of Ki-neticheskih drive and moments of cargo: (6.19)where is the absolute speed of the cargo. The relative and the portable cargo speed at the point of aim along the chord DE, ; Of equality (6.19) that . (6.20)Equating the right parts of the equations (6.18) and (6.20) ,get.6.4. The differential equation of rotational motion of a rigid body Consider a body rotating around a fixed axis z with angular velocity ω. He operated a system of external forces and reactions of supports (fig. 6.6). To exclude from consideration the unknown reaction, write 3-e equation systems (6.14): . (6.21)Since, in accordance with the formula (6.9), from equation (6.21) semi-CHIM (6.22)or (6.23)where φ is the angle of rotation of the body. Comparing the last equation with the differential equations of motion of the center of mass (5.23) used to describe the motion of the body, we conclude on the likeness of these structures level. Since the mass inertia of the author's body characterizes the post-patel'noe movement, the moment of inertia is a measure of the inertia of the body, rotating around a fixed axis.For example, consider the motion under gravity Fizi-pendulum, i.e. body having a horizontal axis of rotation, which does not pass through the center of mass of the body (fig. 6.7). Relabel gravity physical pendulum is a reaction of its axis, h is the distance from the axis of rotation to the center of mass of the body. Compatible z axis with the axis of rotation of the body and write the differential equation of rotational motion

Результаты (английский) 2:

load 2, the weight of which is half the mass of the disk (Fig. 6.5a.). In nekoto-ing point of time the goods, which is self-propelled mechanism begins to move along the chord DE. The radius of the disk is equal to r, CO = 0.5R.
Determine the angular velocity of the disk during the passage of cargo through the middle chord Oh, if at this point the speed of the load on the disk equal to (Fig. 6.5b). . Resistance Forces neglected
Consider a system consisting of disc 1 and the load 2. In her acting external forces: gravity drive, the force of gravity and ground reaction components. Moments of these forces with respect to the z-axis equal-are zero, so the main point of the external forces and the angular momentum of the system relative to the z-axis does not change, ie, . Consequently, where - starting and ending moments of the kinetic system, the compliance of relevant provisions of cargo at the points D and O.
Initially, the drive and the goods are those one-lo, rotating around the z axis, so
;

;
. (6.18)
in the final time the angular momentum of the system is the sum of ki-kinetic moments of the disk and load:
(6.19)
where - the absolute speed of the cargo. Relative speed and carrying load at the point O are directed along the chord DE DE,
;

from (6.19) it follows that
. (6.20)
Equating the right sides of equations (6.18) and (6.20)
,
we obtain.
6.4. The differential equation of the rotational motion of a solid
Consider a body rotating around a fixed z axis with an angular velocity ω. On it are the system of external forces (), and supports the reaction (Fig. 6.6). To exclude from consideration the unknown reaction, we can write the third equation of the system (6.14)
. (6.21)
Since in accordance with the formula (6.9) from (6.21) semi-cpm
(6.22)
or
(6.23)
where φ -. The angle of rotation of the body
Comparing the last equation with the differential equations of motion of the center of mass (5.23) applied for describe the translational movement of the body, we come to the conclusion that a similar structure of these equations-tions. Since the mass characterizes the inertia of a body, committing post-translational motion, moment of inertia is a measure of the inertia of the body rotating around a fixed axis.
For example, consider the motion under gravity physics-cal pendulum, ie, body having a horizontal axis of rotation, koto-paradise does not pass through the center of mass of the body C (Fig. 6.7). Let gravity of a physical pendulum, - reaction to its axis, h - the distance from the axis of rotation to the center of mass of the body. Compatible with the z-axis rotation of the body axis and write-differential equation of rotational motion

Результаты (английский) 3:

the load 2, the mass of which is twice less than that of the disc (figure 1). 6.5а.). some would point load, a self-propelled mechanism starts to move on хорде de. the radius of the disk is equal to r, = 0,5r.to determine the угловую speed drive with the passage of goods through the хорды of, if at this time, the speed of the disk is equal to (figure 2). 6.5б). resistance force is neglected.consider a system consisting of a disk 1 and a load 2. to apply external forces: gravity drive, gravity load and the components of the reactions of supports. moments of all these forces on the axis z of the rav is zero, so the main point of external force and the kinetic moment of the system on the axis z does not change, i.e. so, where is the start and end points of a kinetic system, the appropriate provisions of the goods at the points d and o.in the initial time, drive and load represent one of the luo, rotating around an axis z, so;;. (6.18)at the time the kinetic moment system is of key нетических moments of drive and load.(6.19)where is the absolute speed of the goods. the relative speed and a load at a point of focus along the хорды de;the equality of (6.19) should be that. (6.20)with the right part of the равенств (6.18) and (6.20),we get.6.4. differential equation of rotational motion of rigid bodyconsider a body rotating around fixed axis z with the angular speed ω. to operate the system of external forces () and the reaction support (figure 2). 6.6). to exclude unknown reaction record 3rd equation system (6.14).. (6.21)in accordance with the formula (6.9) in equation (6.21) half as soon as(6.22)or(6.23)where φ - angle body.comparing the last equation with the differential equations of the centre of gravity (5.23), used to describe the progressive movement the body, come to the conclusion on the similarity structures of these уравне. since the mass of the inertia body is engaged in post - пательное movement, the moment of inertia is a measure of inertia body rotating around fixed axis.consider, for example, movement under the action of gravity of the pendulum, fizi, i.e. the body with a horizontal axis of rotation, which passes through the center of mass of the body (figure 1). 6.7). let through gravity physical pendulum is the reaction of the axis, h is the distance from the axis of rotation to the center of mass of the body. compatible axis z with the axis of rotation body and the roll rotational motion differential equation