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



12. When a particle moves on a smooth 

 cycloid under gravity, the vertex of the 

 cycloid being at the lowest point, the equa- 

 tion of motion, by resolution along the 

 tangent in direction QP, may be written 



s = g sin #, 



s being the arc measured from the vertex 

 to P, and 6 the angle the normal OP makes 

 with the vertical. Now, by a known pro- 

 Fig. 47. perty of the cycloid, s 4a sin $, where a is 

 the radius of the generating circle, and thus the above equation becomes 



showing that the motion in s is simple harmonic with period 



Thus the time taken to fall to the vertex from any point on the curve is 



independent of the starting point, and in fact is TT *J(a/g). 



[This property is known as the " Isochronism of the cycloid."] 



13. Show that the time a train, if unresisted, takes to pass through a 

 tunnel under a river in the form of an arc of an inverted cycloid of length 

 2s and height h cut off by a horizontal line is 



'-2gh\ 

 cos" 1 1 ^ i 



where v is the velocity with which the train enters and leaves the tunnel. 



186. Motion of two bodies connected by an inexten- 

 sible thread. We shall consider the case 

 of two bodies, of masses m and m', attached 

 to the ends of an inextensible thread which 

 passes over a smooth pulley. We neglect 

 the mass of the thread. 



The bodies will move like particles at 

 their centres of inertia. 



The particles in question move in the 

 line of the thread. 



Suppose that the particle of mass m has 

 descended through a distance x in the inter- 

 val t. Then m' has risen through a distance 

 x. Hence at any instant the velocities of the particles are equal 

 in magnitude. 



The kinetic energy is therefore J (m 4- m') # 2 . 

 Also the potential energy is Const. + m'gx mgx. 



Fig. 48. 



