336 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



158. Two equal particles are connected by a thread one point of which 

 is fixed, and the particles are describing circles of radii a and b about this 

 point with the same angular velocity so that the thread is always straight. 

 Prove that, if the thread is suddenly released, the tensions in the two portions 

 are altered in the ratios (a + 6) : 2a and (a + b) : 26. 



159. Three equal particles are attached at equal intervals to a thread. 

 One extreme particle A is held fixed, and the other two are describing circles 

 about it with the same angular velocity so that the thread is straight. Prove 

 that, if the particle A is let go, the tensions in the two portions of the thread 

 are diminished in the ratios 1 : 3 and 1 : 2. 



160. Two particles each of mass m are connected by a rod of negligible 

 mass and of length I, and lie on a rough horizontal plane (coefficient of 

 friction p). One of the particles is projected vertically upwards with velocity 

 F, prove that the other particle will begin to move when the rod makes with 

 the plane an angle a, where a is the least angle which satisfies the equation 



( F 2 - 3gl sin a) (cos a + p. sin a) = pgl, 



provided V 2 lgl is less than 3sina+cosec a, and find the radius of curvature 

 of the path immediately after. 



161. Two particles, each of mass m, are connected by an inextensible 

 thread of length I passing over a smooth pulley at the top of a smooth plane 

 of inclination a on which one of the particles rests at a distance a from the 

 top (a&amp;lt;). Prove that in the motion which ensues after the system is free of 

 the plane the tension of the thread is constant and equal to 



\ingal~ 1 cos 2 a (1 - sin a), 



and that the radius of curvature of the path of the upper particle immediately 

 after it leaves the plane is 



1 - sin a [cos 2 a + 1(1- sin q) 2 ]^ 



COSq 1+ ^a -1 COS 2 q(l shlq) * 



162. A spherical shell contains a particle of equal mass supported by 

 springs of equal length and strength, attached at opposite ends of a diameter, 

 and the system, all parts of which are moving in the line of the springs with 

 the same velocity, strikes directly a fixed plane. Show that, if the coefficient 

 of restitution between the shell and the plane is unity, the shell will strike 

 the plane again after an interval of time equal to half the period of free 

 oscillation. 



163. In Example 162 the spherical shell is of mass km and the particle 

 of mass m. Prove that the shell will or will not strike the plane again 

 according as k&amp;lt; or &amp;gt; 1 + 2 cos q, where q is the least positive root of the 

 equation 



164. In Example 162 the particle and the shell have equal masses but 

 there is imperfect restitution (coefficient e] between the shell and the plane. 





