242 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



One of the quantities o&amp;gt; and ii can be eliminated by means of equation 

 (2), and there then remain two unknown quantities in terms of which the 

 motion can be completely expressed by solving the equations (3) and (4). 



Two first integrals of these equations can be obtained, one of them being 

 the equation of energy. 



Fig. 61. 

 Examples. 



1. Prove that, in the problem just considered, there is an integral equa 

 tion of the form 



maQ. (1 +& 2 /a 2 ) + m (aQ. -(a + b) & cos 6 - o&amp;gt;# 2 /&} = const., 

 and that & and 6 are connected by an equation of the form 



J (a + 6) 2 [(1 + 2 /& 2 ) - m (cos 6 - F/6 2 ) 2 / {M(l+ 2 /a 2 ) + m (1 + #/6 8 )}] +g cos 6 



const. 



2. A uniform rod of length I rests on a fixed horizontal cylinder of radius 

 a with its middle point at the top : prove that, if it is displaced in a vertical 

 plane, so as to remain in contact with the cylinder, and if it rocks without 

 slipping, the angle 6 it makes with the horizontal at time t is given by the 

 equation J (^ I 2 + a 2 2 ) # 2 +ffa (cos + 6 sin 0) = const., 



and the length of the simple equivalent pendulum for small oscillations is 



3. A homogeneous sphere rolls down a rough plane of inclination a 

 Prove that the acceleration of its centre is f gsiu a, and that the ratio of the 

 friction to the pressure is f tan a. 



4. A thread unwinds from a reel of radius a, the uppermost point of the 

 thread being held fixed, the unwound part of the thread being vertical, and 

 the axis of the reel being horizontal. Prove that the acceleration of the 



