THE SIMPLE PENDULUM 



259 



THE SIMPLE PENDULUM 



205. One of the most important cases of a variable force arises 

 in the motion of a simple pendulum. To obtain a first approxima- 

 tion, we can suppose that the whole weight of the pendulum is 

 concentrated in the bob, which may be 



treated as a particle, and that this is sus- 

 pended from a fixed point by a weight- 

 less string or rod so that it is constrained 

 to move in a vertical circle. 



Let s denote the distance along this 

 circle described by the particle, this dis- 

 tance being measured from the lowest 

 point 0. Let the angle PCO between 

 the string and the vertical be denoted 

 by 0, so that s = ad. The forces acting 

 on the particle consist of its weight and the tension of the 

 string. The latter has no component in the direction in which 

 the particle moves. The former has a component mg sin 6. 



Thus the equation of motion is 



^ 



A 



^ = #sin0, (94) 



where 6 = s/a. 



206. This equation cannot be solved by elementary mathe- 

 matics, except in the simple case in which the angle is small, 



i.e. the case in which the pendulum never swings through 

 more than a small angle from the vertical. Confining our atten- 

 tion to this case, we may replace sin 6 by 0, and 9 by s/a, and 

 write the equation of motion in the form 



Thus the acceleration of the bob of the pendulum is proportional 

 to its distance from 0, and is towafds O. 



