300 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



2. A uniform straight tube of length 2a contains a particle of equal 

 mass, and, the particle being at the middle point, the tube is started to 

 rotate about that point with angular velocity G>. Prove that, if there are no 

 external forces, the velocity of the particle relative to the tube when it leaves 

 it is aco^/f- 



3. Two horizontal threads are attached to a circular cylinder of negligible 

 mass whose axis is vertical, are coiled in opposite directions round it, and carry 

 equal particles which are initially at rest on two smooth horizontal planes. 

 One of the particles is struck at right angles to its thread so that it starts 

 off with velocity V and its thread begins to unwind from the cylinder. 

 Prove that, if the initial length of the straight portion of the thread 

 attached to the particle struck is c, its length r at time t is given by the 

 equation 



the cylinder being free to turn about its axis. 



4. A thread is attached to a rigid cylinder of radius a and moment of 

 inertia / about its axis, and carries a particle of mass m which is free to 

 move on a smooth plane perpendicular to the axis, while the cylinder is free 

 to rotate about the axis. The particle is projected on the plane at right 

 angles to the thread with velocity V so that the thread tends to wind up 

 round the cylinder. Prove that the length r of the straight portion at any 

 subsequent time is given by the equation 



where c is the initial value of r, and deduce that 



r 2 - c 2 = 2a Vt + V 2 t 2 m/(M+ m\ 

 where M=l/a 2 . 



5. A cone of vertical angle 2a is free to turn about its axis, and a smooth 

 groove is cut in its surface so as to make with the generators an angle /3. 

 A particle of mass m moves in the groove, and starts at a distance c from 

 the vertex. Prove that, if at any subsequent time the particle is at a 

 distance r from the vertex and the cone has turned through an angle 0, 

 r and 6 are connected by the equation 



(1+mc* sin 2 a) e 2dsinacot ^ =(/+mr 2 sin 2 a), 

 where / is the moment of inertia of the cone about its axis. 



6. An elliptic tube of latus rectum 2, eccentricity e, and moment of 

 inertia / about its major axis, is rotating freely about its major axis, which 

 is fixed, with angular velocity Q, and contains a particle of mass m which is 

 attracted to one focus by a force /im/(distance) 2 and is initially at rest at the 

 end of the major axis nearest the centre of force. Prove that, if the particle 

 is slightly displaced, and if /ie(l + e) 2 <W, it will come to rest relatively to 

 the tube at an end of the nearer latus rectum, provided 



