or 



ILLUSTEATIVE EXAMPLES 



After replacing a6 by (a + b) <j> from equation (6), we have 



- ma (a + 6) 2 = mga (a + b) sin 0, 

 5 dt 2 



(a + 6)-^ = - 



337 



showing that the center of the moving sphere moves with five sevenths of the 

 acceleration of a smooth particle sliding down a sphere of radius a -\- b. 



The same result could have been obtained by eliminating from equations 

 (a) and (6), and then regarding as a single Lagrangian coordinate. 



2, A flywheel is connected by a crank and rod to a piston moving in a hori- 

 zontal cylinder. When there is no steam in the engine the flywheel rests in its 

 position of equilibrium. Find its motion if displaced. 



Let a, 6 denote the length of the crank and rod, and let 0, be the angles 

 they make with the horizontal in any position of the flywheel. Then the posi- 

 tion of the engine is known fully when 6 and are known. Not only do the 



FIG. 155 



values of 6 and suffice to determine the position of the engine, but if we 

 assign arbitrary values to 6 and 0, we do not necessarily obtain a possible 

 position for the engine. 



The velocity of rotation of flywheel, axle, and crank is 0, so that the kinetic 

 energy of this motion is \ Id 2 , where J is the moment of inertia of this part of 

 the engine about the axis of the flywheel. The coordinates of the center of 

 gravity of the rod, which we shall assume to be its middle point, measured 

 from the axis of the flywheel, are : 



horizontal : a cos + 6 cos 0, 

 vertical : ^ 6 sin 0. 



Thus the velocity of its center of gravity has components 

 (a sin - + i 6 sin . 0) 



