EXAMPLES. 71 



23. Prove that the component accelerations of a moving particle are X 

 parallel to the axis of x, and R perpendicular to the radius vector, where 



TT (r 2 a?) (rx xr} 2 

 ~lc(j*~-a?) 



an d R- r ^ ~ ^ ^ ~ ^ ~ (^ ~ ^ 



~~ &quot; 



24. The position of a point is given by x, y, r, where x, y, z, r have their 

 usual signification relative to rectangular axes ; show that the component 

 accelerations are 



uw vw . . 

 u + , v+~ , (w-uwx+vwy}lr*, 



u, v, w being component velocities in the directions of x, y, r. 



25. If x, y are the coordinates of a point referred to rectangular axes 

 turning with angular velocity co, prove that the accelerations in the directions 

 of the axes are 



x yo&amp;gt; 2j/o&amp;gt; - co% and y+xd&amp;gt;+2d;a&amp;gt; o&amp;gt; 2 y. 



26. Prove that, if rectangular axes Ox, Oy revolve with uniform angular 

 velocity co, and the component velocities of a point (x, y) parallel to the axes 

 are Ajx and Bjy, then the square of the distance of the point from the origin 

 increases uniformly with the time. 



27. The sides CA, CB of a triangle are fixed in position and the side AB 

 is of constant length. The velocities of A and B along CA and CB are u and 

 v, the corresponding accelerations are U, V, and to is the angular velocity of 

 AB ; prove that 



u cos A + v cos B = 0, u sin A ~ v sin B = co&amp;gt;, 



U cos A + Fcos B = ecu 2 , 7 sin A ~ Fsin jB = cd&amp;gt;. 



28. Two axes &r, Oy are inclined at an angle a and rotate with angular 

 velocity o&amp;gt; about 0. Show that the component velocities are 



x - &amp;lt;*x cot a - a&amp;gt;y cosec a, y + vy cot a + ax cosec a. 



If the position of a point is defined by the perpendiculars , 77 drawn to 

 the instantaneous positions of Ox, Oy, prove that the component velocities 

 u, v in these directions are given by 



u = ( -f fj cos a) /sin 2 a + o)//sin a\ 

 -y = (?) + 1 cos a) /sin 2 a co/sin aj 

 and the component accelerations are 



u - a&amp;gt;u cot a-\-o&amp;gt;v cosec a, 

 v + &amp;lt;&amp;gt;# cot a cow cosec a. 



29. Two fixed points are taken on a circle and any point on the circle is 

 at distances r lt r% from them, the radii vectores r x , r 2 containing an angle a ; 

 prove that the component velocities in the directions of r^ and r 2 are u lt u. 2 

 where 



and that the component accelerations in the same directions are 



and u 2 -u l r l fr 2 . 



