EXAMPLES. 203 



14. A bucket of mass M Ibs. is raised from the bottom of a shaft of depth 

 h feet by means of a cord which is wound on a wheel of mass m Ibs. The 

 wheel is driven by a constant force, which is applied tangentially to its rim 

 for a certain time and then ceases. Prove that if the bucket just comes 

 to rest at the top of the shaft, t seconds after the beginning of the motion, the 

 greatest rate of working in foot-pounds per second is 



2kM*gt/{Mgf- - 2h (M+ m}}, 

 the mass of the wheel being regarded as condensed uniformly on its rim. 



15. An engine is pulling a train and works at a constant horse power 

 doing If units of work per second. If M is the mass of the whole train and 

 F the resistance (supposed constant), prove that the time of generating velocity 

 v from rest is 



/MH H Mv 



16. A two- wheeled vehicle is being drawn along a level road with velocity 

 v ; the wheels (radius c) are connected by an axle (radius r) fixed to them, the 

 weight of the vehicle exclusive of the wheels and axle is W, and its centre of 

 inertia is vertically over the middle point of the axle. Show that, if the 

 shafts are in a horizontal plane with the tops of the wheels, the horse is 

 working at a rate W r vrsinX/ v /(c 2 -r 2 sin 2 X), where X is the angle of friction 

 between the axle and its bearings. 



17. Two bodies hang by a cord over a fixed pulley : show that, neglecting 

 the inertia of the pulley, the spaces described by the bodies in successive equal 

 intervals of time are in arithmetic progression. 



If instead of one of the bodies a pulley of negligible mass is substituted, 

 and bodies of masses m and m slung over it, find the mass of the single body 

 in order that m may remain at rest if initially so, and prove that the 



1 m m 

 acceleration of the pulley is - - q. 



2 m 



18. For one of the moving bodies in an Atwood s machine a pulley is 

 substituted, round which passes a cord connecting two masses P, Q, which 

 hang freely. Show that, if the ratio P : Q lies between 3 and ^, certain values 

 of the other moving body may be found which will keep either P or Q 

 stationary, and that these values are in the ratio 3P- Q : 3Q-P. 



19. If in an Atwood s machine the chain can only support a tension 

 equal to one quarter of the sum of the weights at its two ends, show that the 

 greater weight cannot be much less than six times the smaller, and that the 

 least acceleration possible is 



20. In an Atwood s machine the groove in the pulley in which the chain 

 runs is cut to that depth at which it is found that the inertia of the pulley 

 may be divided equally between the moving bodies, and Q is the weight 



