Unit 10. The Path and the Velocity of a Point of a Body.

Learn the following words and word combinations by heart:

body under consideration	рассматриваемое, изучаемое тело
computation	вычисление; расчёт; подсчёт; исчисление
conclusion	вывод
consequently	следовательно
determination	определение
determine	определять
eliminate time from equation	исключить значение времени из уравнения
evolve from	выводить из, развивать из, возникать из
in plane	на плоскости
let’s consider	давайте рассмотрим
parametric form prove the theorem	параметрическая форма доказать терему
retard	задерживать , замедлять движение
rotation	вращательное движение

Part 1. Determination of the Path of a Point of a Body

Let us now investigate the motion of individual points of a rigid body, i.e., determine their paths, velocities and accelerations. For this, as has been shown, it is sufficient to analyze the motion of the points lying in section S. We shall begin with the determination of the paths.

Consider a point M of a body whose position in section S is specified by its distance b = AM from the pole A and the angle BAM = α (Fig. 18). If the motion of the body is described by Eqs. (48), the x and y coordinates of point M in the system Oxy will be

(49)

where xA, yA,φ are the functions of time t given by Eqs. (48).

Fig. 18

Eqs. (49) describes the motion of point M in plane Oxy and at the same time gives the equation of the point’s path in parametric form. The usual equation of the path can be obtained by eliminating time t from Eqs. (49).

If the body under consideration is part of a mechanism, the path of any point M of the body can be determined by expressing the coordinates of the point in terms of a parameter specifying the position of the mechanism and then eliminating that parameter. In this case the equations of motion (48) are not necessary.

Comprehension check.

Ex.1. Put the questions to the following answers.

1. To analyze the motion of the points lying in section S.

2. By the distance of a point M of a body from the pole A and the angle α .

3. It describes the motion of point M in plane Oxy.

Part 2. Determination of the Velocity of a Point of a Body

Plane motion of a rigid body is a combination of a translation in which all points of the body move with the velocity of the pole and a rotation about that pole. Let us show that the velocity of any point M of the body is the geometrical sum of its velocities for each component of the motion.

The position of a point M in section S of the body is specified with reference to the coordinate axes Oxy by the radius vector (Fig.19), where is the radius vector of the pole A, is the vector which specifies the position of point M with reference to the axes Ax'y' that perform translational motion together with A (the motion of section S with reference to those axes is the motion about pole A). Then , .

Fig. 19 Fig. 20

In this equation is equal to the velocity of pole A; the quantity is equal to the velocity of point M at ., i.e., when A is fixed or, in other words, when the body (or, strictly speaking, its section S) rotates about pole A. It thus follows from the preceding equation that

. (50)

The velocity of rotation of point M about pole A is

, (51)

where ω is the angular velocity of the rotation of the body. Thus, the velocity of any point M of a body is the geometrical sum of the velocity of any other point A taken as the pole and the velocity of rotation of point M about the pole.

The magnitude and direction of the velocity are found by constructing a parallelogram (Fig. 20).

Ex.2. Complete the following sentences with the information from the text. (Part 2)

1. Plane motion of a rigid body is a combination of…….

2. The velocity of any point M of the body is……

3. is the vector which specifies…..

4. The magnitude and direction of the velocity are found….

Part 3. Theorem of the Projections of the Velocities of Two Points of a Body

The use of Eq. (50) to determine the velocities of the points of a body usually leads to involved computations. However, we can evolve from Eq. (50) several simpler and more convenient methods of determining the velocity of any point of a body.

Fig. 21

One of these methods is given by the theorem: The projections of the velocities of two points of a rigid body on the straight line joining those points are equal.

Consider any two points A and B of a body. Taking point A as the pole (Fig. 21) we have from Eq. (50) . Projecting both members of the equation on AB and taking into account that vector is perpendicular to AB, we obtain:

, (52)

and the theorem is proved. This result offers a simple method of determining the velocity of any point of a body if the direction of motion of that point and the velocity of any other point of the same body are known.

Comprehension check.

Ex.3. Answer the following questions.

1. What can we evolve from the equation ?

2. What is the formulation of the theorem about the projections of the velocities of two points of a body?

3. How can we prove this theorem?

Ex.4. Say whether the following statements are True or False.

1. In translation all points of the body move with different velocities.

2. The velocity of any point M of the body is the geometrical sum of its velocities for translational and rotational components of motion.

3. is the radius vector which specifies the position of point M with reference to the axes Ax'y'.

4. The axes Ax'y' perform translational motion together with the pole A.

Ex. 5. Substitute the verbs in bold in sentences for the following verbs in the correct form (these verbs are used in the text). In some cases more than one verb is possible.