318 MOTION OF RIGID BODIES 



change in the length of a day is about -^ of a day, where a is the radius 



of the earth in feet and D, p are the density of the earth and the meteoric 

 dust respectively. 



6. Two masses M and m suspended from a wheel and axle of radii 

 a, b do not balance. Show that the acceleration of M is 



Ma mb 



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



7. A uniform cylinder has coiled round its central section a light, per- 

 fectly flexible, inextensible string. One end of the string is attached to a 

 fixed point, and the cylinder is allowed to fall. Show that it will fall with 

 acceleration f g. 



8. Two equal uniform rods of length 2 a, loosely joined at one extremity, 



are placed symmetrically upon a fixed sphere of radius - and raised 



3 



into a horizontal position so that the hinge is touching the sphere. They 

 are then allowed to descend. Show that when they are first at rest they 

 are inclined at an angle cos- 1 ^ to the horizontal, that the pressure on the 

 sphere at each point of contact is one quarter of the weight of a rod, and 

 that there is no strain at the joint. 



9. A rod rests with one extremity on a smooth horizontal plane and the 

 .other on a smooth vertical wall, the rod being inclined at an angle a to 



the horizon. If it is allowed to slip down, show that it will separate from 

 the wall when its inclination to the horizontal is sin- 1 (| sin a). 



10. If the sun gradually contracts in such a way as always to remain 

 similar to itself in constitution and form, show that when every radius has 

 contracted an nth part of its length, where n is large, the angular velocity 



will have increased to 1 1 + J times its former value. Examine the change 

 in the kinetic energy of rotation. 



11. An elastic band of natural length 2?ra, mass m, modulus X, 'rests 

 against a rough wheel of radius a in a horizontal plane. The string is held 

 against the circumference of the wheel, which is made to rotate with angu- 

 lar velocity O. If the string is left to itself, show that it will expand, and 



that when its radius is r its angular velocity will be , and that its radial 

 velocity will be 



. 2X(,-art 



Tfld J 



