DYNAMICS OF A RIGID BODY. 313 



If m = 120, the suW.jurnt initial velocity is less than if the 

 whole plane hail been rough, in the ratio 67 : 77. 



1498. A straight tube AB, of small bore, containing a smooth 

 rod of the same length, is closed at the end ./?, and is in motion 

 about the fixed end -1 with angular velocity a>; E being opened, 



that the initial tension of the rod at any point P is equal 



A P PR 

 to J/d> 2 ' p ; J/ being the mass. 



L . 1 I > 



1499. The ends of a uniform heavy rod are fixed by smooth 

 rings to the arc of a circle, which is made to revolve uniformly 

 about a fixed vertical diameter ; find the positions of relative equi- 

 librium, and prove that any such position in which the rod is not 

 horizontal is stable. 



0. A smooth semicircular disc rests with its plane vertical 

 and vertex upwanls on a smooth horizontal table, and on it rest 

 two equal uniform rods, each of which passes through two fixrd 

 rings in a vertical line. The disc is slightly displaced, and in the 

 ensuing motion one rod leaves the disc when the other is at the 

 vertex ; prove that 



m _ 2 (2 sin a - 1 - Bin ft) - sin ft cos* g 

 p ~ sin 3 /i ' 



1 the masses of the disc and of either rod, a the angle 

 wlm-h the radius to either point of contact initially makes \vith 

 iiorizon, and ft = cos~ l (2 cos a). 



A uniform rod moves with Trinity on a 



smooth horizontal plane, and the other attached to a .^ 

 whi.-h is fixed to a point above iho plane; whn the rod 



an: in one straight line the rod in let go; prove that 



string is .lar Telocity is */ , 



a + /-/ 



i 



and thr, angular accelorut :< n of the rod is . 



the lengths of the string and rod, and h tho height of the fixed 



above thr plain-. 



