374 PRACTICAL MATHEMATICS 



(17) A force of 10 Ib. extends a spring by 4 inches. A mass of 

 8 Ib. is suspended from such a spring in a medium whose resistance 

 to the motion is 2v Ib. where v ft. per sec. is the velocity of the 

 body at any instant. The body is displaced 3 inches from its 

 equilibrium position and is then allowed to oscillate in the medium. 

 If x ft. is the distance of the body from the equilibrium position 

 at any subsequent time t sees., find the equation of motion for the 

 body, and then find an expression giving x in terms of t. What 

 is the periodic time of the motion ? 



(18) The two simple harmonic motions x = 3-2 sin (nt + 0-732) 

 and x% = 5-6 sin (nt + 1-346) can be expressed as one simple har- 

 monic motion x = A sin (nt + E). Find the values of A and E. 



/72/w (Lx 



(19) Solve -3-5 + 2/-J-+ n 2 x = Q. Take n* = 200, /= 7-485. 



(ii at 



dx 



Let x = and also -,- = 10 when t = 0. (B. of E., 1912.) 

 dt 



(20) A body capable of damped vibration is acted upon by 

 simply varying force which has a frequency /. If x is the dis- 

 placement of the body at any instant t, and if the motion is 

 defined by 



d?x , dx 



-^ + 0-:- + n?x = a sm 27r 



at* at 



we wish to study the forced vibration. 



Take a = 1, 6=1-5, n 2 = 4 find x, first when /= 0-2547 and 

 second when/= 0-3820. (B. of E., 1910.) 



//*'V> fj'V 



(21) -TY + 2/-T- + n 2 x = a sin qt expresses the forced vibration 



( (t (.1C 



of a system. Imagine the natural vibrations to have been damped 

 out. Take n 2 = 49, /= 3, q = 5 ; find as. (B. of E., 1913.) 



(22) A weight W Ib. hangs from a spiral spring whose stiffness 

 is such that a force of 1 Ib. weight elongates it h feet. A down- 

 ward force F Ib., in addition to the force of gravity, acts upon 

 the weight. At any instant the weight is x feet below the mean 

 position it would have if F were zero. Neglecting friction and 



d?x 

 the mass of the spring, prove that -^ + n z x = n 2 hF where n? =g/*Wk 



M 



If the natural frequency, /or , is 10 and if hF = a sin qt, neglecting 



Ztt 



the natural vibration, find the forced vibration ; first when the 

 forced frequency / x or q/2n is 2 ; second when it is 5 ; third when 

 it is 15. (B. of E., 1914.) 



(23) A condenser of capacity k farads is charged so that the 

 potential difference of the plates is V Q volts. The plates are con- 



