THE MOMENT OF INERTIA 228 



where I is the moment of inertia of the body about the axis of 

 rotation. 



If the whole mass of the body was considered to be concen- 

 trated at a point situated at a distance k from the axis, the 

 circumferential velocity of this point = kw ft. per second. 



Hence the total kinetic energy = -Mk 2 w 2 ft. pdls. 



2 



Then I = MA: 2 



and k is defined as the " radius of gyration " of the body. 



(2) Considering the case of a lamina immersed in a liquid to 

 a certain depth. Then the pressure at any depth is px, where x 

 is the depth and p is the weight of unit volume of the liquid. 



Let the whole area be made up of a large number of small 

 elementary areas, a lf a z , a 3 . . . situated at depths x lt x 2 , x 3 . . . 

 respectively. 



Pressure at the depth # t = px l 



Thrust on the area a l = pa 1 x l 



The resultant thrust on the whole area will be the sum of all 

 of the thrusts on the elementary areas. 



Resultant thrust = p{a 1 x 1 + a 2 x 2 + a 3 # 3 + . . .} 



where x is the depth of the centroid of the area below the surface. 

 Moment of the thrust on the area a x = pa^x^. 



The moment of the resultant thrust on the whole area will be 

 the sum of the moments of all the thrusts on the elementary 

 areas 



Total moment = p{a 1 x j 2 + a 2 x 2 2 + a 3 x 3 2 + . . .} 



= PI 



where I is the moment of inertia of the area about the free surface. 

 If 2 is the depth of the " centre of pressure," that is the point 

 at which the resultant thrust acts 



Total moment 

 1 hen 



Resultant thrust 



pAx 

 l_ 



Ax 



where k is the radius of gyration of the area about the free surface. 



