223.] PLANE MOTION. 



I2 3 



If H be the height of the initial point N (0 = ) above the 

 lowest point A of the circle, we have by (44) 



A i cos n H 



- - ,, 



2 2 2/ 



so that (46) can be written in the form 



7 



222. Exercises. 



( i ) Show that 4 = 7r\/-(i +-i + -5- + 225 .f ... ) if the ampli- 



VV l6 I02 4 147456 / 

 tude 20 of the oscillation is 120. 



(2) Show that as a second approximation to the time of a small 

 oscillation we have ti=ir^t/g(i-t-0 z /i6). 



(3) Find the time of 'oscillation of a pendulum whose length is i 

 metre at a place where -=980.8, to four decimal places. 



(4) A pendulum hanging at rest is given an initial velocity v^. Find 

 to what height h\ it will rise. 



(5) Discuss the pendulum problem in the particular case when MN 

 (Fig. 52) touches the circle at B, that is when the initial velocity is due 

 to falling from the highest point of the circle. 



223. Central Motion. The motion of a point P is called 

 central if the following two conditions are fulfilled : (i) the 

 direction of the acceleration must pass constantly through a 

 fixed point O ; (2) the magnitude of the acceleration must be a 

 function of the distance OP =r only, say 



/=/. (48) 



The fixed point O is in this case usually regarded as the seat 

 of an attractive or repulsive force producing the acceleration, 

 and is therefore called the centre of force. 



Harmonic motion as discussed in Arts. 172-214 is a special 

 case of central motion, viz. the case in which the acceleration/ is 



