30 INTRODUCTION TO DYNAMICS. [53. 



(16) In a trapezoid the parallel sides are a, b, the height is h, and 

 one of the non-parallel sides is perpendicular to the parallel sides ; 

 show that the co-ordinates of the centroid with a as axis of x and the 



perpendicular side as axis of y are * = <* + *> + 



(17) Find the centroid of the cross-section of a bar formed by 

 placing four angle-irons with their edges together, two of the irons 

 having the dimensions a, b, a, ft, as in Fig. 8, Art. 36, while the other 

 two have the dimension a different, say a'. 



(18) Find the centroid of the cross-section of a U- iron, the length 

 of the flanges being a= 12 in., that of the web 2^ = 8 in., and the 

 thickness 8 = i in. Deduce the general formula for x, and an approxi- 

 mate formula for a small 8, and compare the numerical results. 



(19) In the cross-section of an unsymmetrical double T, the flanges 

 are 2^= 12 in., 2 V =8 in.; the web is 0= 10 in.; and the thickness 

 of each of the two channel-irons forming the bar is 8 = i in. throughout ; 

 find the centroid. 



(20) In a T-iron the width of the flange is b, its thickness a ; the 

 depth of the web is a, its thickness ft. Find the distance of the centroid 

 from the outer side of the flange ; give an approximate expression and 

 investigate it for a = b, a = ft = a. 



(21) If one-fourth be cut away from a triangle by a parallel to the 

 base, show that in the remaining area the centroid divides the median 

 in the ratio 4:5. 



(22) Prove that the centroid of any plane quadrilateral ABCD 

 coincides with that of the triangle ACF, if the point F be constructed 

 by laying vftBF=DE on the diagonal BD, E being the intersection 

 of the diagonals. 



(23) The centroid of a homogeneous semicircular area of radius r 



lies at the distance x = r from the centre. 

 3 71 " 



(24) The centroid of the area of a homogeneous circular segment 

 of radius r subtending at the centre an angle 2 a is at the distance 

 - sin 3 a 3 



> if ' is the Ch rd > h itS dis " 



tance from the centre, and s the arc 



