[satterly] new vibration EXPERIMENT 87 



ad cos 6 ^1-^1 



t.e., 1 must be > 1 



b sin 6 



a , ^ tan 6 



or — must be > 



b d 



When 6 is very small = 1 very nearly. As d increases 



6 



tan 6 increases at a greater rate than 6. For an angle as large as 5° 



tan ^ .0875 



.0873 



1.002. 



For 10° tan^ ^ ,1763^ ^ ^^ 



d . 1745 



so that for all angles that are likelv to occur in the experiment - 



b 



must be greater than unity or the arrangement will be unstable. 



Calculation of the Period of Vibration by the "Force and 



Moment" Method 



The moment of the weight of the balancing cylinder about B 

 = mgXDE = mg {CE-CD)=mg (ad cosd-b sin 6). 

 Also Ii = I-j-m GiB^ = I+m (b^+a- 6-). 

 By the ordinary formula in mechanics 



/^ ^ + mg (ad cosd-b sin ^) = • 



dt- 



whence if the angle 6 is so small that squares and higher powers of 6 



may be neglected we get 



{I-\-mb^) — -^ mg{a-b)d = 

 dt" 



whence we see that the motion is simple harmonic and of period 



T = 2^y'^J±!!^. (1) 



mg (a — b) 



Calculation of the Period of Vibration by the "Energy" 



Method 



The height of G above = a-\-b. 



Let a = angle of extreme tilt of balancing cylinder. The height 

 of the centre of gravity of balancing cylinder abo\-e is now OF 

 -hBD-\-EGu 



