258-260] 



ENERGY AND MOMENTUM. 



295 



point by an inextensible thread, and the shell rests on a horizontal plane. 

 Suddenly the lower thread breaks, the particle jumps up to the highest point 

 of the shell and adheres there, and it is observed that the shell jumps up 

 through a height h. Prove that the modulus of elasticity of the upper 

 thread is 



Explain what external forces produce momentum in the system as 

 a whole. 



7. Three equal particles are connected by an inextensible thread of length 

 a 4- b so that the middle one is at distances a and b from the other two. The 

 middle one is held fixed and the other two describe circles about it with the 

 same uniform angular velocity so that the two portions of the thread are 

 always in a straight line. Prove that, if the middle particle is set free, the 

 tensions in the two parts of the thread are altered in the ratios 2a -f b : 3a 

 and 26 + a : 36, there being no external forces. 



8. Two equal particles are connected by an inextensible thread of length 

 I ; one of them A is on a smooth table and the other is just over the edge, 

 the thread being straight and at right angles to the edge. Find the velocities 

 of the particles immediately after they have become free of the table, and 

 prove (i) that in the subsequent motion the tension of the thread is always 

 half the weight of either particle, and (ii) that the initial radius of curvature 

 of the path of A immediately after it leaves the table is ^ J51. 



*260. Stability of steady motions. The principles we are 

 now illustrating may frequently be applied to problems concerning 

 the stability of steady motions of which we had an example in 

 Article 65. We shall illustrate the method by considering the 

 steady motion of a spherical pendulum. 



Let be the angle the radius vector from the centre of the 

 sphere to the particle makes with the downwards vertical at time 

 t, a the radius of the sphere, </> 

 the angle contained between the 

 plane through the particle and the 

 vertical diameter and a fixed plane 

 through the same diameter. 



The energy equation is 



ma 2 (fc + sin 2 0< 2 ) + mga (1 - cos 6) 



= const. 



and the equation of momentum 

 about the vertical diameter is 



ma? sin 2 0d> = const. Fig. 78. 



