24. F. K fflipher — Surfaces of the Compound Pendulum. 



linear acceleration in falling as a simple pendulum would be g 

 sin (6+a) and the moment of the force required to produce this 

 acceleration would be 



F'V = dm rg sin (6+ a) (2) 



Subtracting (1) from (2), 



r(F"— F) = dm g r sin (6 + a) — dm — g sin (3) 



The factor F" — ¥' = d F is a force which must be impressed 

 upon dm in excess of its tangential weight-component, in order 

 to impart to the element its real acceleration at the given in- 

 stant. This force may be either positive or negative, the sign 

 depending upon the position of d m, and the direction of swing. 



The integral of (3) for the entire system is necessarily zero, or, 



/ dmgr sin (6 + a) — -y- sin 6 I dm ? ,a =0. (4) 



The first term is the moment of the weight of the system, 

 referred to the plane VS. . The second integral is the moment 

 of inertia I, referred to the axis S. Hence, 



M^K sin 6 - -| sin 6 I = 0. 



Where M represents the entire mass of the pendulum. This 

 gives the well known value of I. 



l= W7K 



The loci of the isodynamic lines in the disc pendulum are 

 determined from (3), which may be put into the following- 

 form : 



rdF • ,c ^ r* . . 



= r sin (o + a) — — sin o (5) 



g dm I 



This expression represents the moment of the impressed force 

 d F per unit of weight at any point determined by the values 

 r, d and a. Making this value constant = a, it gives the condi- 

 tion for an isodynamic line, which is therefore — 



a = r sin (d+a) — — sin 6 (6) 



Let S be the origin of a system of rectangular coordinates, x 

 being the horizontal and y the vertical coordinate of dm. Then 



as ^—x^ + y^ and sin (d+a) = — , equation (6) becomes, 



^+2/" — - x + -r— 7, a = 0. (7) 



° sin 6 sin o 



For a fixed value of 6 and a varying value of a, this is the 

 equation of a series of concentric circles, the common center 



being on the horizontal through S at a distance ±— - — ^ from 

 & s 2 sin 



