THE VIBRATION OF A SPRING 857 



x m d*x 



T " ~ T ~^2 

 h g at* 



d?x g 



and -T5- - - -^r x 



dt 2 mh 



a 



- n 2 x, where n 2 = -^r 

 mh 



Then a; = A sin nt + B cos nt 



Let the initial displacement of the body from its equilibrium 

 position be d ft. 



The initial conditions are x = d and v = when t = 0. 



Then x = d when < =0, hence B = d, 



dx 



also v = -7- = nA cos ?rf nB sin nt 



at 



but u = when t = 0, hence A = 0. 



Finally, a; = d cos n/. 



The body vibrates with simple harmonic motion the amplitude 

 of which is d ft. 



The periodic time = 

 n 



irnh 



= 27TA/ sees. 

 V g 



/n 



The frequency or the number of vibrations per sec. = 



2ic 



181. Theory of Struts. Euler's Formula. In this work we have 

 to find the buckling load that is, the least load which can be 

 applied at the ends of the strut to just cause the strut to bend. 

 There are three distinct cases to consider : 



(1) When the strut is free at both ends. 



(2) When the strut is fixed at both ends. 



(3) When the strut is free at one end and fixed at the other. 



Case I. When the strut is free at both ends. 



Let W be the load on the strut and y the deflection at a point 

 A situated at a distance x from the end O. 



