EXAMPLES. 321 



that the impulsive force exerted at the hinge at the further end of the n th 

 rod is 



. 



12 



69. A set of (2/& + 1) equal rods OA, OB, OC, ... each of mass m and 

 length 2a are freely jointed at 0, and lie in one plane so that any two neigh- 

 bouring rods are inclined at an angle a, = 27r/(2w + l); an impulse P acts 

 along OA and o^, o> 2 , ... are the initial angular velocities communicated to the 

 rods on each side of OA in order. Prove that 



<>! cosec a = a> 2 cosec 2a = . . . = f w/a, 

 where M, =f P/'{(2w- 1) i}, is the initial velocity of OA. 



70. Four small smooth rings of equal mass are attached at equal 

 intervals to a thread and rest on a circular wire in a vertical plane. The 

 radius of the wire is one-third of the length of the thread, and the rings are 

 at the four upper corners of a regular hexagon inscribed in the circle, the two 

 lower rings being at the ends of the horizontal diameter. Prove that, if the 

 thread is cut between one of the extreme particles and one of the middle 

 ones the tension in the horizontal part is suddenly diminished in the 

 ratio 5 : 9. 



71. Particles of masses m and m' are fastened to the ends of a thread 

 which rests in a vertical plane on the surface of a smooth horizontal circular 

 cylinder of mass M which can slide on a horizontal plane. The system is 

 initially held at rest so that the radii of the circular section through the 

 particles make with the vertical angles a and /3. Prove that when the system 

 is released the tension of the thread immediately becomes 



M(sm a + sin /3) 4- (m sin a + m' sin /3) {1 - cos (a +/3)} 



y 



(m 4- m') (M+ m sin' 2 a + m' sin 2 /3) - mm' (cos a - cos 



2 ' 



72. A particle P, of mass M, rests in equilibrium on a smooth horizontal 

 table being attached to three particles of masses m, m', m" by cords which 

 pass over smooth pulleys at points A, B, C at the edge of the table. Prove 

 that, if the cord supporting m" is cut, M will begin to move in a direction 

 making with CP an angle 



H (m ~ m'} {(m + mj - m"* 

 '"* ' 



where = 2mV* + 2m" 2 m 2 + 2w 2 m' 2 - m* - w' 4 - m" 4 . 



73. Two particles A, B of masses m, m' are connected by a thread which 

 passes through a smooth ring C at the top of a smooth plane of inclination a 

 to the horizontal. Initially AC(=-a) is along a line of greatest slope, and 

 EC is vertical. Prove that, if A is projected at right angles to AC with 

 velocity v, B will begin to ascend or descend according as 



m'jm< or >sina + v 2 /#a. 

 L. 21 



