152 V. OSCILLATIONS AND CYCLIC MOTIONS. 



so that if the damping is small, as is usually the case, it affects the 

 period only by small quantities of the second order. 



As has been shown in 38 we have here an instance of the 

 use of a dissipation function 



and the energy is dissipated at a rate proportional to the exponen- 

 tial e~ xt . 



44. Forced Vibrations. Resonance. The motion considered 

 in the last section being that of a system left to itself is called a 

 free oscillation or vibration. We shall now consider a problem of a 

 different sort from. any yet treated and involving a force depending 

 upon the time, and thus introducing or withdrawing energy from 

 the system. Let us suppose a particle to be subject to the same 

 conditions as above, but in addition to be acted upon by an 

 extraneous force varying according to a harmonic function of the time, 



40) F=Eco$pt, 



so that the differential equation of motion is 



We may find a particular solution by putting 



s = acos(pt a), 



42) ds . f , d*s o / j \ 

 -^= -apsin(pt-a), -^ = - aj 8 cos (pt a). 



Substituting in the differential equation, we have 



43) a (h 2 p*) cos (pt a) axp sin (pt a) = Ecospt 



= E {GOS a cos (pt a) sin a sin (pt a)}. 



This can be identically true for all values of t only if the coefficients 

 of the sine and cosine of the variable angle (pt cc) are respectively 

 equal on both sides of the equation, accordingly we must have 



axp = 

 a(h 2 - 



from which eliminating first E and then a, 



xp 



46) tan = ^, 



from which we obtain the amplitude 



46) a = - = 



