185] GRAVITY. 175 



direction of the arrow marked F', and this friction is rolling friction. The 

 resultant of all such frictions as F' is the resistance to the motion of the 

 train. 



It appears that the resistance of the rails arises from rolling friction, and 

 the pull of the engine from sliding friction, and, since the ratio of rolling 

 friction to pressure at a point of contact is always less (for the same materials) 

 than that of sliding friction to pressure, there is no difficulty in seeing how a 

 train can be set in motion by a locomotive of smaller mass than the train. 



The resistance of the air is not sufficiently great to affect the argument. 



6. A carriage is slipped from an express train, going at full speed, at a 

 distance I from a station, and comes to rest at the station. Prove that the 

 rest of the train will then be at a distance Mlj(Mm) beyond the station, 

 M and m being the masses of the whole train and of the carriage slipped, and 

 the pull of the engine being constant. 



7. Prove that the extra work required to take a train from one station to 

 stop at the next at a distance I in an interval t is 



times the work required to run through without stopping, where the incline 

 of the road is 1 in m, and the resistance of the road and the brake power per 

 unit mass are equal to the components of gravity down uniform inclines of 

 1 in n and 1 in k respectively. 



8. A cylinder whose section is a parabola is placed with its generators 

 horizontal, the axis of a normal section vertical, and the vertex upwards, and 

 a particle is projected along it in a vertical plane. Prove that if it leaves the 

 parabola anywhere it does so at the point of projection. 



9. A particle is projected from the lowest point of a vertical section of a 

 smooth hollow circular cylinder whose axis is horizontal so as to move round 

 inside the cylinder. Prove that, if the velocity is that due to falling from 

 the highest point, the particle leaves the circle when the radius through it 

 makes with the vertical an angle cos" 1 ^. 



Find the least velocity of projection in order that the particle may describe 

 the complete circle. 



10. A particle is constrained to describe a circle by means of an inex- 

 tensible thread, and leaves the circle when the thread makes an angle /3 with 

 the vertical drawn upwards. Prove that when it strikes the circle again the 

 thread makes an angle 3/3 with the same vertical. 



11. Prove that a particle projected in any manner on a smooth plane of 

 inclination a to the horizon describes a parabola as if under gravity diminished 

 in the ratio sin a : 1. 



