253-256] OSCILLATIONS OF RIGID BODIES. 289 



4. Prove that, if the fixed points in Example 3 are at a distance 2c apart 

 and the particle is displaced vertically, the length of the simple equivalent 

 pendulum is 



5. A pulley of negligible mass is hung from a fixed point by an elastic cord 

 of modulus X and natural length a, and an inextensible cord passing over the 

 pulley carries at its ends bodies of masses M and m. Prove that the time of a 

 small oscillation in which the pulley moves vertically is 4?r *J{Mmal(M+m) X}. 



*255. Rigid bodies and connected systems. In the 



application of the method of Article 253 to problems of oscillations 

 of rigid bodies and connected systems usually the most important 

 matter to attend to is that the potential energy V must be 

 expressed correctly to the second order of the small quantity 6. 



If we formed the equation of motion by a direct process it 

 would be necessary in it to retain only the first power of 6. 



Now there are cases in which the equations of motion of a 

 single rigid body can be readily obtained in a form free from 

 unknown reactions by taking moments about the instantaneous 

 centre, but in case this method is adopted there is again a matter 

 to attend to, in that the equation obtained becomes nugatory if 

 moments are taken about the instantaneous centre in the position 

 of equilibrium. This position is, of course, occupied by the instan- 

 taneous centre at a single instant during the period, viz. : at the 

 instant when 6 = 0, and at any other instant during the period 

 the instantaneous centre is in a slightly different position. The 

 method which is now effective is to form the equation of moments 

 about the instantaneous centre in a displaced position. In the 

 application of this method it is worth while to remark that the 

 moment of the kinetic reactions about the instantaneous centre 

 is expressed, correctly to the first order of small quantities, by the 

 product of the angular acceleration and the moment of inertia of 

 the body about an axis through the instantaneous centre perpen- 

 dicular to the plane of motion. [Example 2 of Article 220.] 



We shall illustrate the application of the method just described 

 and of that described in Article 253 by solving some problems. 



*256. Illustrative problems. 



I. A uniform rod can slide with its ends on two smooth straight wires 

 equally inclined to the horizontal and fixed in a vertical plane. To find the 

 oscillation about the horizontal position. 



L. 19 



