203, 204] CONICAL PENDULUM. 191 



determine the two components of the velocity (<j and y$) in two 

 directions at right angles which lie in the tangent plane to the 

 surface. 



204. Examples. 



1. Show that a particle projected along a circular section of a smooth 

 surface of revolution with its axis vertical can describe the circle under the 

 action of gravity and the reaction of the surface, provided the velocity V of 

 projection and the radius y of the circle are connected by the equation 



V*=gy tan 0, 



where /3 is the angle which the normal to the surface at any point on the 

 circle makes with the vertical. 



Prove also that the reaction of the surface is mg sec /3, where m is the 

 mass of the particle. 



[In the particular case where the surface is spherical this motion is 

 frequently referred to as motion of a " spherical pendulum," and the constraint 

 can be provided by an inextensible string with one end fixed at the centre of 

 the sphere. Since the string describes a right circular cone the name " conical 

 pendulum " is sometimes used.] 



2. A train rounds a curve whose radius of curvature is p with velocity v. 

 Prove that, to prevent the train from leaving the metals the outer rail 

 ought to be raised a height bv 2 /pg above the inner, b being the distance 

 between the rails. 



3. A railway carriage is travelling on a curve of radius r with velocity 

 v, 2a is the distance between the rails and h is the height of the centre of 

 inertia of the carriage above the rails. Show that the weight of the carriage 

 is divided between the rails in the ratio gra v 2 k : gra + v 2 h, and hence that 

 the carriage will upset if v>*J(gra/h). 



4. The point of suspension of a simple pendulum of length I is carried 

 round in a horizontal circle of radius c with angular velocity o> \ show that 

 when the motion is steady the inclination a of its suspending thread to the 

 vertical is given by the equation 



w 2 (c + 1 sin a) = g tan a. 



Show that, if (gl^Y < ^- c 7 , the inclination can be inwards towards the 

 axis. 



5. A particle moves on a smooth surface of revolution whose axis 

 is vertical. Prove that the polar equation of the projection of the path on 

 a horizontal plane is given by the equation 



where z=f(u) is the equation of the meridian curve, u~ l being the distance 

 from the axis, and h is a constant. 



