MOMENT OF MOMENTUM 



EXAMPLES 



1. Show that the acceleration of a hoop rolling down a hill of inclination a 

 is \ g sin a. 



2. Find the acceleration of a pair of locomotive wheels running down a gra- 

 dient of 1 in 50, each wheel consisting of a rim and of spokes of uniform thick- 

 ness, the weight of the rim being twice that of the spokes and the weight of the 

 axle being half of that of a wheel. (Neglect the thickness of the axle.) 



3. Two bicyclists, riding exactly similar machines, coast down a hill, start- 

 ing with equal velocities at the top. Neglecting the forces of friction and the 

 resistance of the air, show that the heavier rider will reach the bottom first. 



4. The pulley of an Atwood's machine is a uniform disk of mass M. When 

 masses mi , w 2 are attached to the ends of the string, show that the acceleration 



of mi is im - m 2 



gr. 



mi + m 2 + M 



5. Two spheres, one a hollow shell and the other a homogeneous solid, roll 

 down hill together, starting simultaneously from rest at the top. Show that 

 their times over any part of the path are in the ratio 5 : V21. 



6. If the masses of the wheels of a carriage are supposed to be all collected 

 at the rim, show that the energy of the carriage when moving with velocity V 

 is \ M F 2 , where M is the weight of the complete carriage plus the weights of 

 the wheels. 



7. A straight piece of uniform wire is stood vertically on end and allowed to 

 fall over. With what velocity does it strike the ground ? 



8. A homogeneous solid cigar-shaped spheroid, semi-axes a and 6, is stood 

 on its point on a horizontal plane and is allowed to roll over. Find its angular 

 velocity when the end of its minor axis is in contact with the plane, and find 

 the pressure on the plane at this instant. 



MOMENT OF MOMENTUM 



242. Let x, y, z be the coordinates of any particle of mass m. 

 Let the components of the total resultant force acting on the par- 

 ticle be X, Y, Z. Then the equations of motion are 



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