SIMPLE HARMONIC MOTION 263 



5. The length of a pendulum which makes 2 n beats per day is changed 

 from I to I + L. Show that the pendulum will lose beats per day approxi- 

 mately. 



6. A balloon ascends with constant acceleration, and reaches a height of 

 3600 feet in two minutes. Show that during the ascent a pendulum clock will 

 have gained about one second. 



7. A pendulum of length I is adjusted by moving a small part only of the 

 bob of the pendulum, this being of mass equal to one nth of the complete bob. 

 How far must this be moved to correct an error of p seconds a day ? 



SIMPLE HARMONIC MOTION 



208. We have seen that throughout the motion of a pendulum 

 which moves so that its maximum inclination to the vertical 

 is small, the acceleration is proportional to the distance from 

 the middle point of its path, and is directed towards that point. 

 A point which moves in this way is said to move with simple 

 harmonic motion. Thus if s is the distance from a fixed point, 

 of a point which moves with simple harmonic motion, we have 

 an equation of the form 



d?s 



d?=-^ s ' 



where k is a constant. 



Integrating, we obtain, as before in the case of the pendulum 

 (cf. equation (96)), 



4 ,2 _ Z>2/ C 2 2\ 



v fa ( s o s )y 



and from this again 



s=s Q cosk(t ). (97) 



The constant k is known as the frequency of the motion. Thus 



the frequency of a simple pendulum is \\ 



^ff 



209. A simple geometrical interpretation can be given of simple 



harmonic motion, and this enables us to obtain a complete knowl- 

 edge of the motion without any use of the integral calculus or of 

 the theory of differential equations. In fig. 131 let the arm OP 

 rotate about with uniform angular velocity Jc, so that P describes 

 a circle of radius a with uniform velocity ka. Let a perpendicular 



