303.] BODY WITH FIXED AXIS. ^3 



With these values the equation of motion (i) assumes the 

 simple form 



h . 



(9) 



As shown in Art. 175, the equation of the simple pendulum 

 of length / is 



d*0 g Q 



|. ^ = -5 sm * 



The two equations differ only in the constant factor of sin 0, 

 and it appears that the motion of a compound pendulum is the 

 same as that of a simple pendulum whose length is 



302. The problem of the compound pendulum has thus been 

 reduced to that of the simple pendulum. The length / is called 

 the length of the equivalent simple pendulum. The foot O 

 (Fig. 35) of the perpendicular let fall from the centroid on the 

 axis is called the centre of suspension. If on the line OG a 

 length OC=-l be laid off, the point C is called the centre of 

 oscillation. It appears, from (10), that G lies between O and C. 



The relation (10) can be written in the form 



h(l-K)=q\ or OG- 6Y7=const. 



As this relation is not altered by interchanging O and C, it 

 follows that the centres of oscillation and suspension are inter- 

 changeable ; i.e. the period of a compound pendulum remains 

 the same if it be made to swing about a parallel axis through 

 the centre of oscillation. 



303. Exercises, 



(1) A pendulum, formed of a cylindrical rod of radius a and length 

 Z, swings about a diameter of one of the bases. Find the time of a 

 small oscillation. 



(2) A cube, whose edge is a, swings as a pendulum about an edge. 

 Find the length of the equivalent simple pendulum. 



