NIPHER SURFACES OF THE COMPOUND PENDULU\I. 645 



a line parallel to those axes. The pendulum then becomes a thin 

 plate of varying density, lying in the plane of the paper as in 

 Fig. I. 



This supposed condensation is really unnecessary in a rigid 

 system, as the center of gravity G and the element dm may lie in 

 different planes, at right angles to the axis S, v^ithout in any way 

 changing the result. 



At any instant the linear acceleration of O is g sin d^ and its 



or 



angular acceleration is - sin d. This is also the angular accel- 

 eration of every other particle in the system. The linear accel- 

 eration of dm is therefore -j- g sin d. The force required to 

 produce this acceleration on dm. is 



F' = dm -J g sin d. 

 The moment of this force about S is 



Y'r = dm J- g sin d. (i) 



If the element dm w^ere disconnected from the system, its 

 linear acceleration in falling as a simple pendulum w^ould be 

 ^ sin (w -|- a), and the moment of the force required to produce 

 this acceleration w^ould be 



Y"r= dm rg ?>m {n-\- a). (2) 



Subtracting (i) from (2), 



r{Y" — F') = dm gr sin {6 -{- a) — dm ~g sin 0. (3) 



The factor F' — Y'= dY is a force vs^hich 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 instant. 

 This force' may be either positive or negative, the sign depending 

 upon the position of dtn and the direction of swing. 



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



fdm g r sin C^-f-a) — ^ sin « f dm r'^ = 0. (4) 



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

 ferred to the axis S in the plane VS ; the second integral is the 

 moment of inertia I, referred to the axis S : hence. 



