340 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



182. Two equal uniform rods AB, EC each of mass m and length 2a are 

 freely jointed at B and have their middle points joined by an elastic string, 

 and the system moves in one plane under no forces. Prove that, if B is the 

 angle between the string and either rod at any time, tp the angle the string 

 makes with a fixed line, and V the potential energy of the stretched string, 

 then throughout the motion 



(^ + cos 2 0)0 = const., 

 ma 2 {(I + cos 2 ff) 2 + ( J + sin 2 6} 2 } + F= const. 



183. Two equal uniform rods AC, CB, hinged at C, and having their 

 extremities A, B connected by a thread so that ACB is a right angle, are 

 revolving in their own plane with uniform angular velocity about the angle 

 A which is fixed. Prove that if the thread is severed the reaction at the 

 hinge is instantaneously changed in the ratio </5 : 4. 



184. A smooth uniform tube contains a smooth uniform rod and the 

 system moves under no external forces being set in motion by an impulse at 

 right angles to the tube when the distance between the middle points of the 

 rod and tube is a. Prove that the distance r between the middle points when 

 the system has turned through an angle 6 is given by the equation 



where b is a certain constant depending on the masses and moments of inertia 

 of the rod and tube. 



185. A smooth circular tube lying in a horizontal plane contains a 

 particle at a point C and can turn about a point A of its circumference. 

 Prove that, if the tube is struck by a horizontal blow, the particle can 

 oscillate about the point B furthest from A, and that, if CB subtends at the 

 centre an angle a and the line joining the particle to the centre at time t 

 makes with the radius to B an angle /3, the limits of oscillation are given by 



cos 3= 



186. One end of an inextensible thread of length a is attached to a 

 smooth circular wire of radius a whose plane is vertical at one end of a 

 horizontal diameter, and the other end is attached to one end of a rigid 

 uniform rod of length a whose other end can slide on the wire. The system 

 starts from rest with the thread and rod in a horizontal position ; find the 

 velocity of the rod when its middle point has fallen through any distance. 



187. A uniform rod of mass m and length 2a moves at right angles to 

 itself on a smooth table, and impinges symmetrically on a uniform circular 

 disc of mass m' and radius a spinning freely about its centre. Prove that, 

 if there is no restitution, and the edge of the disc is rough enough to prevent 

 slipping, the bodies will separate after an interval in which the unmolested 

 disc would have turned through an angle whose circular measure is 



