Propulsive Effects of a Rotating Mass 



where 1/^2 mv^ is the kinetic energy of the mass m, 1/2 ]u>^ is the kinetic energy 

 of the remaining masses that rotate around the y axis and in respect to which 

 the moment of inertia equals j, and pz is the energy of the vertical motion of 

 the weight p . 



Equations (5), (4), and the third equation of Eqs. (1) yield 



E =: -m [R2(1 + sin2 61) + r^ + 2Rr cos 5] oj^ + - J co^ + p (R cos 6 + r) sin 9 . 



K we put h = j/mR2, we obtain 



E - P(Rcos i9+ r) sin5 



- m [R2(1 + sin2 0) + r^ + 2Rr cos 6 + hR^; 



(6) 



Differentiating Eq. (5) with respect to the time t, we obtain e. To determine 

 the value of E necessary for the calculation of w, we can resort to the mean 

 value of the number of revolutions N. In fact, from dd = codt, using Eq. (6), 

 we obtain the period 



-~f^^r'^-^r 



1 V '2 



- m [R2(1 + sin2 0) + r^ + 2Rr cos ^ + hR2] 



P(Rcos (9+ r) sine* 



N (7) 



To deduce E from this expression, we can proceed graphically, choosing arbi- 

 trary values Ej, Ej, E3, . . . , calculating the integral and determining the cor- 

 responding values Tj, T,, T3, . . . . Entering in graphs having E as a function 

 of T with the value of i/n, we can obtain the value of e. 



By applying the procedure used for the device indicated in Fig. 3 to other 

 devices, the corresponding expressions can be obtained. 



It is particularly useful for what will be said to consider the device indi- 

 cated in Fig. 5. 



The coordinates of points P^ and Pj in which the masses are concentrated 



are: 



R sin^ (9 - r cos 



-R sin^ 9 + T cos 



-R cos 



y = -R cos 



(R cos 9 + r) sin i 



-(R cos 9 + r ) sin 



With R = r and putting mi = m^ = m/2, the coordinates of the center of gravity 

 G of the two masses are 



Xq = , y^ = -R cos 9 , Zq = . 



1379 



