114] 



FREEDOM OF THE FIRST ORDER. 



105 



Then A, must be that screw on the cylindroid which is reciprocal to a, for a 

 wrench on X, and the given wrench on 77, must compound into a wrench on p, 

 whence the three screws must be co-cylindroidal * ; also X must be reciprocal 

 to a, so that its position on the cylindroid is known ( 20). Finally, as the 

 impulsive intensity 17 &quot; is given, and as the three screws 77, X, /* are all 

 known, the impulsive intensity X &quot; becomes determined ( 14). 



114. Small Oscillations. 



We shall now suppose that a rigid body which has freedom of the first 

 order occupies a position of stable equilibrium under the influence of a 

 system of forces. If the body be displaced by a small twist about the screw 

 a which prescribes the freedom, and if it further receive a small initial twist 

 velocity about the same screw, the body will continue to perform small 

 twist oscillations about the screw a. We propose to determine the time 

 of an oscillation. 



The kinetic energy of the body, when animated by a twist velocity 



7 / 



, 



CLt&amp;gt; 



is Mu a ~ (~ J ( 89). The potential energy due to the position attained by 



giving the body a twist of amplitude a away from its position of equili 

 brium, is Fv a 2 af 2 ( 102). But the sum of the potential and kinetic energies 

 must be constant, whence 



Mu a 



-?-) 

 dt I 



Fv a -a- = const. 



Differentiating we have 



Integrating this equation we have 



a = A sin A/ 





- 

 Mu a 



t + B cos 



/ Fv a 

 V Mu 



where A and B are arbitrary constants. The time of one oscillation is 

 therefore 



IM 

 F&quot; 



Regarding the rigid body and the forces as given, and comparing 

 inter se the periods about different screws a, on which the body might have 

 been constrained to twist, we see from the result just arrived at that the 



time for each screw a is proportional to -- . 



We shall often for convenience speak of three screws on the same cylindroid as co-cylindroidal. 



