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



3. A compound pendulum consists of a rod which can turn about a 

 fixed horizontal axis and a spherical bob which can slide on the rod. Prove 

 that the period of oscillation will be prolonged by sliding the bob up or down 

 according as the length of the equivalent simple pendulum is &amp;gt; or &amp;lt; twice 

 the distance of the centre of gravity of the bob from the axis of rotation. 



4. Two rigid pendulums of masses m and m turn about the same hori 

 zontal axis. The distances of the centres of gravity and of oscillation from 

 the axis are A, h! and I, I respectively. Prove that, if the pendulums are 

 fastened together in the position of equilibrium, the length of the simple 

 equivalent pendulum for the compound body will be (mhl + m h l }l(mh-\-m h }. 



225. Illustrative Problems. It will be most convenient to exemplify 

 the application of the principles that have been laid down in this chapter 

 and in Chapters VII. and VIII. by partially working out some problems. 

 The most important matters to be illustrated are actions between two rigid 

 bodies whether smooth or rough, and the expression of the effects of the 

 inertia of a rigid body by means of the moment of inertia. Other matters 

 of subsidiary interest are the kinematical expression of velocities and 

 accelerations in terms of a small number of independent geometrical quanti 

 ties, the expression of kinematical conditions, and the calculation of resultant 

 stresses. Some of the problems selected for discussion are of a complicated 

 character, and have been chosen in order to illustrate a number of points. 



I. Inertia of machines. We shall consider Atwood s machine. To 

 avoid having to take account of the motion of the pulley in our preliminary 

 notice of Atwood s machine (Article 186) we assumed the pulley to be 

 perfectly smooth, or that the rope slides over it 

 without frictional resistance and without setting it 

 in motion. It will now be most convenient in 

 order to get some idea of the way the motion of 

 the pulley affects the result to suppose the pulley 

 to be so rough that the particles of the rope and 

 the pulley in contact move with the same velocity 

 along the tangents to the pulley. 



Now let M be the mass of the pulley, a its 

 radius, k its radius of gyration about its axis, 6 the 

 angle through which it has turned up to time t. 



Let m and m be the masses of the bodies at- m [ 

 tached to the rope, and x the distance through 

 which m has fallen up to time t. Then x=a6. 



Neglecting the mass of the rope, the kinetic energy is 



and the work done is 



so that the energy equation is 



Fig. 48 (bis). 



(m - m } gx, 



Jf^ 2 +(m + m )* 2 



(m - m ) gx + const. 



