298 



MOTION OF RIGID BODIES 



To see the truth of this, it is only necessary to notice that a par- 

 ticle of mass m at distance p from the axis has momentum mpw, 

 so that the moment of momentum of the whole system will be 



and since M and k z do not vary with the time, the rate of change 

 of angular momentum will be M J<? - 



Oscillation of a Pendulum 



245. An important application of the last theorem enables us 

 to find the time of oscillation of a pendulum of any description. 

 Let be the pivot about which the pendulum turns, let G be 

 its center of gravity, let OG = h, and let the line OG 

 make an angle with the vertical at any instant, so 



= is the angular velocity of the pendulum 



(Jut 



FIG. 146 



that 



about its axis. 



Let M be the mass, and Jc the radius of gyration 

 about its axis, of the whole pendulum. Then the 

 equation of motion is 



dt 



w hi cn < = The value of L is equal to the 



moment of the weight about the axis through 0, and is therefore 

 Mgh sin 0. 



Thus the equation becomes 



dt 



or, 



