EXAMPLES 319 



v 12. A uniform triangular disk ABC is so supported that it can oscillate 

 in its own plane, which is vertical, about A. Show that the length of the 

 simple equivalent pendulum is 



1 3 6 2 + c 2 - a 2 



13. A spherical hollow of radius a is made in a cube of glass of mass M, 

 and a particle of mass m is placed inside. The cube is then projected with 

 velocity V on a smooth horizontal plane. If the particle just gets round 

 the sphere, remaining in contact with it all the way, show that 



14. Three equal particles are attached to the ends and middle point of a 

 rod of negligible mass, and one of the end particles is struck by a blow at 

 right angles to the rod. Show that the velocities of the particles at starting 

 are in the ratio KOI 



O . < . J.. 



15. A rough horizontal cylinder of mass M and radius a is free to turn 

 about its axis. Round it is coiled a string, to the free extremity of which 

 is attached a chain of mass m and length /. The chain is gathered close 

 up and then let go. Show that if is the angle through which the cylinder 

 has turned after a time t before the chain is fully stretched, then 



16. A uniform flat circular disk is projected on a rough horizontal table, 

 the friction on any element moving with velocity V being cF 3 x (mass of 

 element), in a direction opposite to that of V. Find the path of the center 

 of the disk. 



