EXAMPLES. 319 



56. A small smooth ring of mass m slides on the side AB of a square 

 A BCD formed of four rigidly connected rods. An impulse R is applied to C 

 in direction DC. Prove that the initial velocity of the ring is 



where 2a is the length of a side, c is the distance of the ring from the middle 

 point of AB, M is the mass of the square and k its radius of gyration about 

 its centre. 



57. A uniform rod of length 2a moving in a vertical plane falls on a 

 horizontal smooth plane so as to make with it an angle 6 at the instant of 

 impact, and there is perfect restitution. Prove that, if at the instant of 

 impact, the rod is turning about any point in the vertical line through that 

 point of the rod which is distant a(l+^sec 2 #) from the lower end, the 

 angular velocity to and the vertical component of the velocity of the centre 

 of inertia will be immediately reversed, and further that if '36 cos 6 = ao^fg 

 the subsequent impacts on the plane will take place at equal intervals of time 

 20/o>. 



58. A smooth uniform cube of side 2a and radius of gyration k about an 

 axis through its centre is free to turn about an axis which is horizontal and 

 passes through the centres of two opposite faces, and the cube is at rest with 

 two faces horizontal. An equal cube falls without rotation and with velocity 

 V, and strikes the upper face of the first cube along a line parallel to the 

 fixed axis and at a distance c from the vertical plane through it. Prove that, 

 if e is the coefficient of restitution and a the angle which the lower face of 

 the falling cube makes with the horizontal, the angular velocity imparted to 

 the first cube is c V ( 1 + e)/(c 2 + k 2 + a 2 - a 2 sin 2a). 



59. Two uniform rods AB, BC of masses m, m' lie on a smooth table 

 inclined to each other at an angle a; they are jointed at B, and the end A 

 turns on a pivot fixed to the table. If AB is struck at the middle by a blow 

 P perpendicular to A B the kinetic energy of the resulting motion is 



%P 2 /(%m + 4m' - 3m' cos 2 a). 



If there is a smooth peg touching BC at its middle point on the proper 

 side to give constraint the kinetic energy is 



P 2 /(|m + 4m' - 1 m' cos 2 a). 



60. Two uniform rods AB, BC hinged together at B are moving about 

 the middle point of AC as instantaneous centre of rotation, with no motion 

 relative to each other, when a point in one of the rods is suddenly fixed, 

 ABC being at the moment a right angle. Prove that, if after impact the 

 relative motion of the rods is initially zero, the point must be the hinge. 



61. Two lengths 2a and 26 are cut from the same uniform rod of mass M 

 and freely jointed at one end of each. The rods being at rest in a straight 

 line, an impulse MV is applied at the free end of a. Prove that the kinetic 

 energy when 6 is free is to that when the further end of b is fixed in 

 the ratio (4a + 36) (3a + 46)/12 (a + 6) 2 . 



