Ex.1. Look at Appendix 1 and read the following mathematical symbols.

     ,       ,         

 

In kinematics, as in statics, we shall regard all solids as rigid bodies, i.e., we shall assume that the distance between any two points of a body remains the same during the whole period of motion.

                          Fig. 8                              Fig. 9

 

Problems of kinematics of rigid bodies are basically of two types: (1) definition of the motion and analysis of the kinematics characteristics of the motion of a body as a whole; (2) analysis of the motion of every point of the body in particular. We shall begin with the consideration of the motion of translation of a rigid body.

  Translation of a rigid body is such a motion in which any straight line through the body remains continually parallel to itself.

Translation should not be confused with rectilinear motion. In translation the particles of a body may move on any curved paths.

The properties of translational motion are defined by the following theorem: In translational motion, all the particles of a body move along similar paths (which will coincide if superimposed) and have at any instant the same velocity and acceleration. To prove the theorem, consider a rigid body translated with reference to a system of axes Oxyz. Take two arbitrary points A and B on the body whose positions at time t are specified by radius vectors  and  (Fig. 9). Draw a vector   joining the two points. It is evident that

                                                                         (35)

The length of  is constant, being the distance between two points of a rigid body, and the direction of  is constant by virtue of the translational motion of the body. Thus, the vector  is constant throughout the motion of the body (  = const.). It follows then from Eq.(35) (and the diagram) that the path of particle B can be obtained by a parallel displacement of all the points of the path of particle A through a constant vector . Hence, the paths of particles A and B are identical curves, which will coincide if superimposed.

To determine the velocities of points A and B, we differentiate both parts of Eq. (35) with respect to time. We have

.

But the derivative of the constant vector  is zero while the derivatives of vectors  and  with respect to time give the velocities of points A and B. Thus we find that

,

i.e., at any instant the velocities of points A and B are equal in magnitude and direction.

Again, differentiating both sides of the equation with respect to time, we obtain

 or .

Hence, at any instant the accelerations of A and B are equal in magnitude and direction.

As points A and B are arbitrary, it follows that the paths and the velocities and accelerations of all the points of a body at any instant are the same, which proves the theorem.

It follows from the theorem that the translational motion of a rigid body is fully described by the motion of any point belonging to it. Thus, the analysis of translational motion of a rigid body is reduced to the methods of particle kinematics examined before.

The common velocity  of all the points of a body in translational motion is called the velocity of translation, and the common acceleration w is called the acceleration of translation. Vectors  and  can, obviously, be shown as applied at any point of the body.

 

Comprehension  check.