EXAMPLES. 221 



1 34. Prove that if the path of a particle moving on a right circular cone 

 cuts the generators at an angle x the acceleration in the tangent plane to the 

 surface and normal to the path is 



where v is the velocity, and r the distance from the vertex. 



If the axis of the cone is vertical, and the vertex upwards, and if the 

 velocity is that due to falling from the vertex, prove that, when the particle 

 leaves the cone, 



2 sin 2 x tan 2 a, 



2a being the vertical angle of the cone. What happens when tan 2 a>2 ? 



135. A particle moves on a smooth surface of revolution. The velocity is 

 v at a point where the normal terminated by the axis of revolution is of length 

 v, and this normal makes an angle 6 with the axis ; prove that, if ds is the 

 element of arc of the path, and x the angle at which it cuts the meridian, the 

 acceleration in the tangent plane to the surface and normal to the path is 



2 f^X s * n X c t ff\ 

 \ds v ) ' 



136. A particle describes a rhumb line on a sphere in such a way that 

 the longitude increases uniformly; prove that the resultant acceleration 

 varies as the cosine of the latitude, and that its direction makes with the 

 normal an angle equal to the latitude. 



137. A particle describes a rhumb line on a smooth sphere under a force 

 parallel to its axis. Show that the force varies inversely as the fourth power 

 of the distance from the axis and directly as the distance from the diametral 

 plane perpendicular to the axis. 



138. A particle of unit mass moves on a smooth sphere under two central 

 attractive forces /i/^Vg 2 , and p/r/r^ in the distances r lt r 2 of the point from 

 the two poles. Prove that, if the velocity at starting is that due to falling 

 from an infinite distance, the path on the sphere is a rhumb line. 



139. A particle is placed at rest on the smooth inner surface of a vertical 

 circular cylinder, which rotates with uniform angular velocity o> about the 

 generator furthest from the particle. Prove that the particle leaves the 

 surface when it has descended a distance 



140. A particle is attached by a thread of length a to a point of a rough 

 fixed plane inclined to the horizon at an angle equal to the angle of friction 

 between the particle and the plane. The particle is projected down the plane 

 at right angles to the thread, which is initially straight and horizontal. Prove 

 that it comes to rest at the lowest point of its path if the square of the initial 

 velocity is (n - 2) pga/J(l +/* 2 ), where p. is the coefficient of friction. 



