320 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



62. An equilateral triangle formed of three equal uniform rods hinged at 

 their ends is held in a vertical plane with one side horizontal and the opposite 

 corner downwards. Prove that, if after falling through any height the 

 middle point of the highest rod is suddenly stopped, the impulsive stresses at 

 the upper and lower hinges will be in the ratio ^13 : 1. 



63. A rectangle, sides 2a and 26, formed of four uniform rods of the 

 same material and section, smoothly hinged at the ends, is moving without 

 rotation on a smooth horizontal plane, when a side of length 2a impinges on 

 a small rough peg (zero restitution). Prove that for that side to acquire the 

 greatest possible angular velocity, the point of impact must be at a distance 



a{(36+a)/(36 + 3a)}* from its centre. Prove also that the rectangle cannot 

 begin to rotate as a rigid body unless the direction of motion before impact 

 makes with the impinging side an angle greater than 



6(26>3a) 



64. A rhombus formed of four similar uniform rods freely hinged at the 

 angular points is rotating on a smooth horizontal plane about its centre when 

 one corner is suddenly fixed. Prove that the relative angular velocities of 

 the rods just before and just after the impulse are in the ratio 5-3 cos a : 2, 

 where a is the angle of the rhombus at the corner fixed. 



65. A rhombus formed of four equal uniform rods each of length 2a 

 freely jointed at common extremities is moving with velocity v in the direction 

 of one of its diagonals of length 4a cos a, when the middle point of one of 

 the front sides is suddenly fixed. Prove that the initial angular velocity of 

 that side is zero, and that of the adjacent sides is f (vja) sin a. 



66. Eight equal uniform rods AB, BC, ... HK freely jointed at their 

 extremities are placed on a table in the form of a square with two rods in 

 each side, the ends A, K being in contact but free. Prove that if the end A 

 is set in motion with a given velocity in a direction making an angle 6 with 

 A B, then K will start in a direction making an angle $ with AB, where 



3409 tan = 433 tan 6. 



67. Twelve equal rods each of length 2a are so jointed together that 

 they can be the edges of a cube, and the framework moves symmetrically 

 through a configuration in which each rod makes an angle 6 with the vertical ; 

 prove that, if u is velocity of the centre of inertia, the kinetic energy is 

 ^M(^a?d 2 + ii 2 \ where M is the mass of the framework, and that, if the frame 

 strikes the ground when u= V and = 0, then u is reduced to 



F/(l+-/ 7 cosec 2 0). 



68. An indefinite number of equal uniform rods are loosely jointed 

 together and are in a straight line and at rest when a blow P is struck at the 

 free end of the extreme rod in a direction perpendicular to its length. Prove 



