i8 



INTRODUCTION TO DYNAMICS. 



[36. 



36. To find the centroid of the cross-section of a T-iron (Fig. 8) 

 it is only necessary to find its distance ~x from the lower side 

 AB ; for it must lie on the axis of symmetry CD. Taking 

 moments with respect to AB we obtain with the notation 

 indicated in the figure : 



hence 



If a, fi are nearly equal and very 

 small in comparison with a, b> we 

 have approximately 



TG 



_ 



X - 



a + b 



Fig. 8. 



/37. The area of a homogeneous cir- 

 cular sector (Fig. 4, -p. 13) of radius r 



and angle AOB = 2a can be resolved into triangular elements 

 POP^ ^dQ, the bisecting . radius OC being taken as polar 

 axis. The centroid of such an element lies, by Art. 32, at 

 the distance -|r from the centre O. Regarding the mass, 

 p-^r^dd, of each element as concentrated at its centroid, the 

 sector is replaced by a homogeneous circular arc of radius ^r 

 and density ^pr*d9. By Art. 29, the centroid of such an arc, 

 which is the required centroid of the sector, lies on the bisect- 

 ing radius OC at the distance |r5S? from the centre O. 



a 

 Hence 



- 9 sin a 



38. In general, for areas bounded by curves we must resort to 

 integration, using the general formulae of Art. 15. 



If the area 5 be plane, we have in rectangular co-ordinates 



