18, 19] SIMPLE HARMONIC MOTION. 35 



These formulae are of little practical value in gunnery, because 

 beside the attraction of the earth a projectile is acted on by retard- 

 ing forces due to its motion relative to the air, and depending on 

 the velocity of the projectile. 



19. Harmonic Motions. Next in simplicity to motions under 

 constant forces are those in which the force is directed toward a 

 fixed point, and depends upon the distance of the particle from it. 

 The simplest way in which it can depend upon the distance is by 

 being proportional to it. If the particle moves in a straight line 

 with an acceleration toward the origin proportional to its distance x 

 from it, we have 



8) -j-g- = n 2 x, where n is a constant. 

 The integral of this differential equation is 



9) x = Acosnt + Bsin.nt, 



where A and B are arbitrary constants. If we put A = a cos a, 

 IB = a sin a, this may be written 



10) x = acos(nt a), 



which as before contains two arbitrary constants, a and a. 



Obviously by giving a a value differing by we may use the 



sine instead of cosine. If we increase nt by 2jc the value of the 

 sine and cosine is unchanged, consequently the motion is periodic, 

 or the point is found in a given position at times separated by an 

 interval T, called the period, given by nT 2%, so that we may write 



11) x 



The maximum excursion of the point on either side of the 

 origin is called the amplitude a, and it is to be noticed that it does 

 not occur in the differential equation 8). Since x takes on positive 

 and negative values in symmetrical succession, the motion is an 

 oscillation with period T, and frequency, that is the number of 

 oscillations in unit time, 



A - JL 



~T ~~ 2V 



An oscillation expressed as above, 10), by a single sine or cosine function 

 of a linear function of the time is called a simple harmonic motion, 

 the name arising from the occurrence of such motions in musical 

 sounds. The frequency of harmonic motions in nature is due to the 

 fact that in any system which is disturbed from a position of rest 

 forces are called into play which depend in general on the magnitude 



