221-224] RIGID PENDULUM. 237 



to distinguish it from the " simple pendulum " whose motion was 

 discussed in Articles 188 193. 



Let G be the centre of inertia of the 

 body, G8 the perpendicular from G to the 

 axis, the angle GS makes with the 

 vertical at time t. Then the whole motion 

 takes place in the vertical plane through 

 G at right angles to the axis, and depends 

 only on the quantity 6. Fi 8- 



Let GS = h. Let M be the mass of the body, k its radius of 

 gyration about an axis through G perpendicular to the plane of 

 motion. 



The velocity of the centre of inertia is hO, and the kinetic 

 energy is 



The potential energy of the body in the field of the earth's 

 gravitation is 



Mgh (1 - cos 0), 



the standard position being the equilibrium position. 

 Hence the equation of energy can be written 

 Pf (h 2 + k 2 ) 2 = Mgh cos 6 + const. 



Comparing this equation with that obtained in Article 188, we 

 see that the motion is the same as that of a simple pendulum of 

 length (k* + h?)lh. 



A point in the line SG at this distance from S is known as the 

 " centre of oscillation," 8 is called the " centre of suspension." 

 The distance between these centres is the " length of the simple 

 equivalent pendulum." 



224. Examples. 



1. Prove that if a rigid pendulum, for which S and are respectively a 

 centre of suspension and the corresponding centre of oscillation, is hung up 

 so that it can oscillate in the same vertical plane as before but with as 

 centre of suspension, then S will be the centre of oscillation. 



2. A uniform rod moves with its ends on a smooth circular wire fixed in 

 a vertical plane. Prove that if it subtends an angle of 120 at the centre 

 the length of the simple equivalent pendulum is equal to the radius of the 

 circle. 



