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



proportional to the square of the length of that diameter. This ellipse is 

 known as the ellipse of inertia. 



IV. If two plane systems in the same plane have the same mass, the 

 same centre of inertia, the same principal axes at the centre of inertia, and 

 the same moments of inertia about these principal axes, they have the same 

 moments of inertia about any axis in or perpendicular to the plane. 



For, in the first place, the two systems have by Theorem III. the same 

 moments of inertia about any axis lying in the plane and passing through the 

 common centre of inertia, by Theorem I. they have the same moments of 

 inertia about any other axis in the plane, and by Theorem II. they have the 

 same moment of inertia about any axis perpendicular to the plane. 



Such systems are described as momental equivalents. 



217. Calculations of moments of inertia. 



I. Uniform ring. Radius of gyration of a body. For a circular ring of 

 mass m, and of very small section, the mass between two sections made by 



planes through the axis containing an angle d6 is d6. If a is the radius 



. ^7T 



of the ring the moment of inertia about the axis is 



/" 27r ra 

 



Jo 2?r 



This might have been seen at once since every element is at distance a 

 from the axis. 



In the case of a body of any shape and of mass m we can always express 

 the moment of inertia about any axis in the form mk 2 , where k represents the 

 length of a line, and thus we see that k is the radius of a ring such that, if 

 the mass of the body were condensed uniformly upon the ring, the moment 

 of inertia of the ring about its axis would be the same as the moment of 

 inertia of the body about the axis in question. The quantity Tc for any body 

 and any axis is known as the radius of gyration of that body about that axis. 



II. Uniform rod. Let m be the mass of the rod, and 2a its length, r the 

 distance of any section from the middle point. The mass of the element 



between the sections r and r + dr is dr. Therefore, if the thickness of the 



rod is disregarded, the moment of inertia about an axis through the middle 

 point at right angles to the rod is 



1 



* m 7 



r 2 dr = 

 -2a 



The radius of gyration of the rod is a/^/3. 



III. Circular disc. For a uniform circular disc, of radius a and mass m, 

 the mass per unit area is m/7ra 2 , and thus the moment of inertia about an 

 axis through the centre perpendicular to the plane is 



{*"(" 

 Jo Jo net? 



or ma 2 . Hence the radius of gyration about the axis of the disc is a/ 



