53-J DETERMINATION OF CENTROIDS. 29 



53, Exercises. 



(1) Three beads of masses 3, 5, 12, are strung on a straight wire 

 ivhose mass is neglected, the bead of mass 5 being midway between 

 he other two. Find the centroid. (Take moments about the middle 

 Doint.) 



(2) Show that the centroid of three equal particles placed at the 

 Trtices of a triangle is at the intersection of the medians of the triangle. 



(3) Show that the centroid of three masses m 1} m 2 , m s situated at 

 he vertices of a triangle and proportional to the opposite sides, is at 

 ;he centre of the inscribed circle. 



(4) Equal particles are placed at five of the six vertices of a regu- 

 ar hexagon. Find the distance of the centroid from the centre of 

 igure. 



(5) Find the centroid of a homogeneous triangular frame. 



(6) Show that the centroid of a homogeneous semicircular wire lies 



2 



it the distance - r from the centre, r being the radius. 



7T 



(7) Find the co-ordinates of the centroid of the arc of a quadrant 

 of a circle by using the first proposition of Pappus (Art. 30). 



(8) Find the centroid of a circular arc AB of angle A OB = a, 

 whose density varies as the length of the arc measured from A. 



Find the centroids of the following homogeneous arcs of curves : 



(9) Parabola f=$ax from the vertex to the end of the latus 

 rectum. 



(10) Cycloid x= a (0 sm$),y = a (i cos0), from cusp to cusp. 



( 1 1 ) Half the cardioid r = a ( i + cos 0) . 



(12) Catenary y=-(e^ + e~~c) between two points equally distant 

 from the axis of x. 



(13) Common helix : x = r cos0, y = r sin 9, z = krQ, from = o to 

 = 9. 



(14) The sides of a right-angled triangle are a and b. Find the dis- 

 tances of the centroid of the triangular area from the vertices. 



(15) From a square A BCD one corner EAF is cut off so that 

 AE = %a, AF\a, a being the side of the square. Find the centroid 

 of the remaining area. 



