EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA. 51 



inertia for a given weight, that is to say those which have the 

 greatest radius of gyration. 



Coulomb used a solid cylinder movable about its axis; Sir W. 

 Thomson recommends the use of a hollow cylinder resting on a thin 

 square or circular plate, so that it can be accurately centred in re- 

 ference to the axis of rotation. These two shapes have the advantage 

 that they can be readily turned, and made true, and have only a very 

 slight friction against the external medium. The hollow cylinder is 

 less easy to regulate, but any defects of homogenity in the substance 

 are of less importance, and do not perceptibly alter the radius of 

 gyration. 



A rectangular parallelopipedon, in the shape of a very elongated thin 

 plate, may also be advantageously employed. The dimensions of the 

 plate may be more easily measured than those of the cylinder, and the 

 thickness only necessitates a very small term of correction ; other 

 things being equal, a plate gives a greater radius of gyration than the 

 annular cylinder employed by Sir W. Thomson ; for if we suppose the 

 cylinder slit, and stretched out straight, its moment of inertia in the 



o 



shape of plate is times as great as in the form of ring. 



O 



703. EXPERIMENTAL DETERMINATION OF MOMENTS OF INERTIA, 

 AND OF COUPLES. When the shape of the moving system is 

 complicated, or its structure is not homogeneous, its moment of 

 inertia can only in general be determined by experiment, and by 

 comparison with that of a homogeneous body of simple form. 



Two distinct cases present themselves, according as the directing 

 couple, which tends to restore the system to its position of rest, is, or 

 is not, independent of the weight of the system ; if the damping 

 cannot be neglected, the value T is deduced from the true time T of 

 infinitely small oscillations, correcting it for the effect of damping. 

 An auxiliary body of known moment of inertia K' is added to the 

 system, and the new time T' of oscillations is determined in the 

 same way. We have then 



C K K + K' 



7T 2 T 2 T' 2 

 from which is deduced 



-wr _ _ -rrt 



X' 2 - T 2 



E2 



