182.] 



PLANE MOTION. 



97 



(4) Show that the period of a simple harmonic oscillation can be 

 expressed in the form T= 2 TT V x/j x where j x is the acceleration of 

 the oscillating point at the time when its distance from the centre, or 

 its displacement, is x. 



(5) P x , P' x being the positions of the oscillating point at the times 

 /, /', respectively, and 8 the angle POP, i.e. the difference of phase, 

 .show that f /=8/w. 



(6) Show that v x = o> V# 2 x?. 



181. Compound Harmonic Motion. We have seen (Art. 176) 

 that the motion of a point, whose acceleration is directly propor- 

 tional to its distance from a fixed centre, and directed towards 

 this centre, is simply harmonic, provided the centre lies in the 

 line of the initial velocity. Removing this last restriction, we 

 have the more generalise of compound harmonic motion. 



Let O (Fig. 46) be the centre, P the position of the moving 

 point at the time t, OP s its distance from the centre, v its 

 velocity, j= /j?s its accelera- 

 tion, at that time. Referring 

 the motion to two rectangular 

 axes Ox, Oy in the plane deter- 

 mined by v and O, we can 

 resolve v and j into their com- 

 ponents along these axes : 



v x = v cos , v y = v sin a, 

 and j x = fix, jy= ^ 2 y, where 

 a is the angle made by v with the axis Ox, and x, y are the 

 co-ordinates of P. 



The projections P x , P y of P on the axes have therefore each 

 a simple harmonic motion, and the motion of P may be regarded 

 as the resultant of these component motions. 



182. In general, the motion of P will be curvilinear. We 

 proceed to examine somewhat more in detail the most important 

 cases of the composition of two or more simple harmonic motions, 



PART I 7 



