202 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



8. A cycloid is placed with its axis vertical and base upwards, and 

 particles starting from various points of the base run down chords of quickest 

 descent to the curve. Prove that if x is the length of such a chord, and 2A 

 the vertical height through which a particle would fall freely in the time of 

 describing it, then x 2 - x */(h?/a) - 2A 2 = 0. 



9. Two equal parabolas of latus rectum 2 are placed with their axes 

 vertical at a distance 2 from each other and with the vertex of the lower at 

 a depth I below the vertex of the higher, the convexities being opposed. The 

 line of quickest descent from the higher to the lower is of length h and makes 

 an angle (j> with the vertical. Prove that 



h/l = sec sec 2 20 = 2^/2 cosec sec 20 cos (^r + 20). 



10. A window is supported by two cords passing over pulleys in the 

 framework of the window (which it loosely fits) and is connected with counter- 

 poises each equal to half the weight of the window. One cord breaks, and the 

 window descends with acceleration /. Show that the coefficient of friction 

 between the window and the framework is 



where a is the height and b the breadth of the window. 



11. A train of mass m runs from rest at one station to stop at the next 

 at a distance I. The full speed is F, and the average speed is v. The 

 resistance of the rails when the brake is not applied is u V/lg of the weight of 

 the train, and when the brake is applied it is u' Vjlg of the weight of the train. 

 The pull of the engine has one constant value while the train is getting up 

 speed, and another constant value while it is running at full speed, prove that 

 the average rate at which the engine works in starting the train is 



US 1 1 



I \ U 2/V-2/F-1/VJ ' 



12. A train starts from rest at one station and stops at the next, the pull 

 of the engine having one constant value while the train is getting up speed, 

 and another constant value while it is running at full speed. Prove that the 

 work done by the engine in getting up speed exceeds that done by the brake 

 in stopping the train by (V/vl) times the work done by the resistance 

 during the whole journey, V and v being respectively the full speed and the 

 average speed of the train. 



13. It is required to find in horse-power the average rate of working of, 

 and in pounds weight the pull exerted by, each horse of a two-horse omnibus 

 which maintains an average speed of 6 miles an hour without exceeding 7J 

 miles an hour and slows down to 1 foot per second every hundred yards to 

 pick up or set down given the following data weight of 'bus = 25 cwt., weight 

 of two horses = 30 cwt., weight of driver, conductor and passengers = 35 cwt., 

 and brake power produces a friction equal to one-fifth of the pressure. The 

 brake is supposed to be applied to one wheel only, and no work is done by the 

 horses when going at full speed. 



