23-25] Rotating Systems 27 



The position of the system may be supposed defined by >/r, a coordinate 

 fixing the position of the axes, such that ^ = o>, and n 1 Lagrangian 

 coordinates O l , 2 . . . Q n ^ fixing the configuration of the system relative to 

 the axes, so that the system has n degrees of freedom in all. 



The equations of motion are (cf. equations (5)), 



in which G is the generalised force corresponding to the coordinate >|r, and 

 so is the couple about the axis of z which acts upon the system. 



From the value of T given by equation (25), we clearly have 95 rT /9^r = 

 and dT/dco = M, so that equation (28) reduces to 



expressing simply that the rate of increase of the moment of momentum M 

 is equal to the couple G. 



If a mass is rotating freely in space, G = 0, so that M remains constant. 



If a mass is constrained to rotate at a constant angular velocity while M 

 changes, a couple G will be necessary to maintain the rotation, and the 

 amount of this couple will be determined by equation (30). 



Mass rotating with Constant Angular Velocity 



25. Let us first consider the problem' when (o is kept constant. To 

 transform equations (29) we notice that 



dx _ -, dx dO s 



so that = ^- . 



de s MS 



We accordingly have 



= ^ 



de s ~ 



so that 



d fdU\ ^ (. dy ,.dx\ ^ [" d ( dy\ d fdx\~\ 



& = ~ m ( x m - y wj +* m [ x dt bi) - y dt y J 



\~] 



) J 



dU ^ ffa . dy .\ ~ [ d fdy\ d /dx\~] 



Also w. = 2m (aT s y - d x ) + Sm r dt - 



d fdU\ dU ^ f.dy . dx 

 so that T- I - o/i = 2zm (x ^ y ^- 



dt \r)0J OV S \ dV s 00 



dx 



