130 



CENTER OF GRAVITY 



replaced by a wire of uniform density. The center of gravity of 

 this wire has already been determined. If 2 a is the angle of the 

 wire, the center of gravity lies on the radius to the middle point 



of the wire at a distance a - from the center. 



o a 



Thus the center of gravity of the original sector of a circle of 

 radius a and angle 2 a is found to lie on the central radius of the 



sector at a distance - a from the center. 



3 a 



101. Center of gravity of a spherical cap. The piece cut off 



from a spherical shell by a plane is called a spherical cap. 



The center of gravity of a spherical cap 

 cut from a uniform shell can easily be 

 found by the methods already explained. 

 Let PQ be the spherical cap, being 

 the center of the sphere from which it 

 is cut. Let OE be the radius perpen- 

 dicular to the plane PQ by which the 

 cap is bounded, and let a denote the 

 radius of the sphere. 

 Any plane parallel to PQ will cut the sphere in a circle of 



which the center will lie on OE. Hence by taking a great num- 

 ber of planes parallel to PQ, we can 



divide' the spherical cap into a number 



of narrow circular rings, each having its 



center on the line OE. Let us consider a 



single circular ring cut off by the planes 



A a A', BIB 1 . Let the angles AOE, BOE 



be equal to 6 and + dO respectively, so 



that the ring itself subtends an angle dd 



at the center. The width AB of the ring is 



a d6. Its circumference may, in the limit, 



be supposed equal to the circumference 



of the circle AaA'. Since Aa = a sin 6, 



this circumference is 2 TTO, sin 6. Thus the ring under consideration 



FIG. 74 



