KINEMATICS. [249. 



249. Exercises. 



(1) Two points A, A 1 of a plane figure move on two fixed circles 

 described with radii a, a' about O, O' ; show that the angular velocities 

 w, <i>' of OA, O'A' about O, O' are inversely proportional to OM, O'M, 

 M being the point of intersection of OO' with A A 1 . 



(2) Given the magnitudes v, v' of the velocities of two points A, A ( 

 of an invariable plane figure and the angle (v, v') formed by their 

 directions ; find the instantaneous centre C and the angular velocity 

 CD about C. 



(3) Show that in the " elliptic motion " of a plane figure (Arts. 

 25-27) the velocity of any point (x', y') is 



v = [a 2 + x 12 +/ 2 - 2a(x' cos 2 <j> +/ sin 2 



dt 



(4) In the same motion find the velocities of B and O' (Fig. 6, 

 Art. 26) when ^4 moves uniformly along the axis of x. 



250. The continuous motion of a rigid body is called a trans- 

 lation when the velocities of all its points are equal and parallel 

 at every moment (Art. 9). All points describe therefore equal 

 and similar curves, and every line of the body remains par- 

 allel to itself. The velocity v = ds/dt of any point is called the 

 velocity of translation of the body. 



251. A rigid body can be imagined to be subjected to several 

 velocities of translation simultaneously ; the resulting motion is 

 a translation whose velocity is found by geometrically adding 

 the component velocities. 



Conversely, the velocity of translation of a rigid body can be 

 resolved into components in given directions. 



252. The continuous motion of a rigid body is called rotation 

 when two points of the body are fixed ; the line joining these 

 points is the axis of rotation. All points excepting those on the' 

 axis describe arcs of circles whose centres lie on the axis. 



The velocity of any point P of the body at the distance 

 OPr from the axis is v = a>r=r dO/dt, if w = d0/dt is the 



