228 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



the first line. Then a is invariable, for if it were to change the 

 body would be deformed. Now the second line of particles makes 

 an angle 6 + a with the fixed line, and this angle also increases at 

 a rate 6. We thus see that every line of particles parallel to the 

 plane turns with the same angular velocity, and this is the angular 

 velocity of the rigid body. 



215. Moment of Inertia. Consider a rigid body turning 

 about an axis with angular velocity o&amp;gt;. Let m be the mass of a 

 particle of a body at a distance r from the axis. Then this 

 particle is describing a circle of radius r with velocity ro&amp;gt;. Hence 

 its moment of momentum about the axis is rar 2 o&amp;gt;, and its kinetic 

 energy is ^mr 2 &&amp;gt; 2 . 



It follows that the moment of momentum of the rigid body 

 about the axis is 



the summation referring to all the particles, and the kinetic 

 energy is 



These expressions become 



a) 1 1 Ip (a? + 2/ 2 ) dxdydz, 



and a 



for a body of density p at a point (#, y, z) the axis of rotation 

 being the axis of z. 



The multiplier of &&amp;gt; and ^o&amp;gt; 2 in these expressions is called the 

 moment of inertia of the body about the axis. We shall see 

 presently that it enters into the expressions for the kinetic energy 

 and moment of momentum of a rotating body, whether the axis of 

 rotation is fixed or not. 



The moment of inertia of a body about an axis depends only 

 on the shape of the body, its situation with reference to the axis, 

 and the distribution of density within it. 



216. Theorems concerning Moments of Inertia. I. The moment 

 of inertia of a system about any axis is equal to the moment of inertia about 

 a parallel axis through the centre of inertia together with the moment of 



