36 H. PARTICULAR MOTIONS OF A POINT. 



of tlie displacement. Suppose such a displacement depends upon a 

 single variable x, then as the force F developed will usually be a 

 uniform, continuous function of x, we may develop by Taylor's 

 Theorem, 



dF 



If now x be small enough, the term in the first power of x is more 

 important than those that follow, which may therefore be neglected. 

 If we measure x from the configuration of equilibrium, when x = 0, 

 j^=0 so that we have 



If the coefficient of x is negative, the force tends to restore the 

 system to the configuration of equilibrium, and being proportional 

 to the displacement, the system will execute harmonic vibrations 

 about this configuration Thus small vibrations are harmonic, which 

 explains the extreme frequency of such motions in nature. A common 

 method of realizing such vibrations is by the use of a tuning-fork. 

 If a point moves so as to describe the resultant of two simple 

 harmonic motions of the same frequency in lines intersecting at right 

 angles, its equations of motion are 



12) + '*-0, 



The resultant acceleration is directed toward the origin and is 

 directly proportional to the radius vector. The path is obtained by 

 the elimination of t between the integrals 



x = a sin (nt cc) = a (sin nt cos a cos nt sin a) 



13) 



y = 1} sin (nt /3) & (sin nt cos /3 cos nt sin /?), 



where a, &, a, /3 are constants of integration. Solving for sinnt and 



sinnt = 



X . n II . 



sin p f- sin a 

 a b 



_ j 



sin ( a) 



x 11 



cos 3 f- cos a 

 a b 



cos n t = 



sin (jfl a) 



Squaring and adding we have the equation of the path 



. 



sin 2 (a-/?) 



which represents an ellipse. The motion is called elliptic harmonic 

 motion. If cc = /3, that is if both components vanish together, the 



