322 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



74. A sphere of mass m hangs by a chain of length b and negligible mass 

 to one end of a rigid horizontal arm of length c, which is free to rotate about 

 a fixed vertical axis passing through its other end. The arm is seized and 

 made to rotate with angular velocity Q. Prove that the tension of the chain 

 immediately becomes m(<7 + Q 2 c 2 /6), and that the plane through the chain and 

 the radius from the centre of the sphere to the point of attachment starts to 

 rotate with angular velocity |ii about the radius. 



75. A thread ABC is fixed at A and has particles of masses m, m' 

 attached to it at B and C, and the system is held in a vertical plane so that 

 AB and BC make acute angles a and a+/3 with the vertical. Prove that, 

 when B and C are let go, the initial tension of A B is 



m(m+ m'} g cos a/(m + m' sin 2 /3). 



76. A circular wire of mass M is held at rest in a vertical plane, on a 

 smooth table, and a particle of mass m rests against it being supported by 

 an inextensible thread which passes over the wire and is secured to a fixed 

 point in the plane of the wire at the same level as the highest point of the 

 wire. Prove that if the wire is set free the pressure of the particle upon it 

 is immediately diminished by an amount ra 2 #sm 2 a/(J/-|-4msin 2 |a), where a 

 is the angular distance of the ring from the highest point of the wire. 



77. Four particles A, B, C, D of equal mass connected by equal threads 

 are placed on a smooth plane of inclination a(<|7r) to the horizontal, so 

 that AC is a line of greatest slope and AB, AD make angles a with AC on 

 opposite sides of it. If the uppermost particle A is held, and the particles 

 B and D are released, prove that the tension in each of the lower threads is 

 instantly diminished in the ratio 



(l-2sin 2 a)/(l+2sin 2 a). 



78. A bead of mass m' can slide on a thread one end of which is fixed 

 while the other end carries a particle of mass m. Initially m is held at the 

 level of the fixed end, and the two parts of the thread make equal angles a 

 with the vertical. Prove that, if the particle m is released, the initial tension 

 in the thread is mm'gcos a/(m' + 4m cos 2 a), and the initial acceleration of the 

 bead m' is 



g (m' + 2m cos 2 a)/(m' + 4m cos 2 a). 



79. One end of a thread PQ is fixed to a point P on a smooth horizontal 

 plane, and the other end Q is attached to a small smooth ring of mass m 

 which rests on the plane ; another thread passes through the ring and is 

 fixed at one end to a point R of the plane while its other end 8 carries a 

 particle of mass M. Initially the angle PQR is obtuse and equal to /3, and 

 the angle RQS is right; the particle M is projected parallel to QR with 

 velocity V. Prove that the initial tension in PQ is 



Mm F 2 (sin - cos )/ (m + M+ M sin 2/3), 

 where a is the length of QS. 



