Learn the following words and word combinations by heart:

assume arbitrary line   connecting rod   be sufficient belong to	допускать; полагать; представить; пусть; произвольно выбранная, произвольно проведённая линия соединительная тяга; соединительная штанга; шатун быть достаточным принадлежать …
case         general ~	обычный случай; пример; факт    общий случай
special  ~	особый случай
hence	следовательно, в результате; с этих пор
instead of	вместо (чего-либо)
move in the same way	перемещаться точно так же, таким же образом
otherwise	иначе, иным образом, иным способом
plane motion	движение в плоскости, двухмерное движение
pole reciprocating engine resolution of sth into…	полюс поршневой двигатель разложение (вектора); разложение (на составляющие);  разложение (сил);
successive	последующий; следующий один за другим; последовательный
whatever	какой бы ни, любой

Plane motion of a rigid body is such motion in which all the points  of a body move parallel to a fixed plane P (Fig. 14). Many machine parts have plane motion, for example, a wheel running on a straight track or the connecting rod of a reciprocating engine. Rotation is, in fact, a special case of plane motion.

Let us consider the section S of a body produced by passing any plane Oxy parallel to a fixed plane P (see Fig. 14). All the points of the body belonging to line MM' normal to plane P move in the same way.

 

Fig. 14                                    Fig. 15

              Therefore, in investigating plane motion it is sufficient to investigate the motion of section S of that body in the plane Oxy. Here we shall always take the plane Oxy parallel to the page and represent a body by its section S.

The position of section S in plane Oxy is completely specified by the position of any line AB in this section (Fig. 15). The position of the line AB may be specified by the coordinates xA and yA of point A and the angle φ between an arbitrary line AB in section S and axis x.

The point A chosen to define the position of section S is called the pole. As the body moves, the quantities x_A, y_A, and φ will change and the motion of the body, i.e., its position in space at any moment of time, will be completely specified if we know

                        x_A = f1 (t), y_A = f2 (t), f = f3 (t).                                (48)

Eqs. (48) are the equations of plane motion of a rigid body. Let us show that plane motion is a combination of translation and rotation. Consider the successive positions I and II of the section S of a moving body at instants t₁, and t₂ = t₁ +∆t (Fig. 16). It will be observed that the following method can be employed to move section S, and with it the whole body, from position I to position II.

Fig. 16

 

Let us first translate the body so that pole A occupies position  (line A₁B₁, occupies position A₂B₁') and than turn the section about pole A₂, through angle ∆φ₁. In the same way we can move the body from position II to some new position III, etc. We conclude that the plane motion of a rigid body is a combination of a translation, in which all the points move in the same way as the pole A, and of a rotation about that pole.

The translational component of plane motion can, evidently, be described by the first two of Eqs. (48), and the rotational component by the third.

The principal kinematics characteristics of this type of motion are the velocity and acceleration of translation, each equal to the velocity and acceleration of the pole () and the angular velocity and angular acceleration of the rotation about the pole. The values of these characteristics can be found for any time t from Eqs. (48).

In analyzing plane motion, we are free to choose any point of the body as the pole. Let us consider a point C as a pole instead of A and determine the position of the line CD making an angle φ₁ with axis x (Fig. 17). The characteristics of the translatory component of the motion would have been different, for in the general case  and  (otherwise the motion would be that of pure translation). The characteristics of the rotational component of the motion remain, however, the same. For, drawing CB, parallel to AB, we find that at any instant of time angle φ₁ = φ –a, where a = const. Hence,

                          , ,

This result can also be obtained from an examination of Fig. 16: whatever point is taken as the pole, to carry section S from position I to position II line A1B1 must be made parallel to A2B2, i.e., the section must be rotated around any pole through the same angle ∆φ₁, equal to the angle between the two lines. Hence, the rotational component of motion does not depend on the position of the pole.

  

Fig. 17

Comprehension  check.