EXAMPLES. 217 



106. If while one particle oscillates in a smooth circular tube of radius a 

 in a vertical plane through an arc of height k, another particle circulates in a 

 smooth helical tube described on the cylinder of diameter h whose axis is 

 horizontal, touching the circular tube at the lowest point, with velocity due to 

 a height 2a above the lowest point, the two particles can move so as always to 

 be at the same level provided the length of one turn of the helix is equal 

 to the circumference of the circular tube. 



107. A particle slides on a smooth helix of angle a and radius a under a 

 force to a fixed point on the axis equal to p. (distance). Show that the pressure 

 cannot vanish unless the greatest velocity of the particle is ajp sec a. 



108. A particle moves on a helical wire whose axis is vertical. Prove 

 that the velocity v after describing an arc s is given by the equations 



ds , sec 2 a cosh <f> 



where a is the radius of the cylinder in which the helix lies, a the inclination 

 of the helix to the horizon, and p the coefficient of friction. 



109. A small smooth groove is cut on the surface of a right circular cone 

 whose axis is vertical and vertex upwards in such a manner as that the 

 tangent is always inclined to the vertical at the same angle /3. A particle 

 slides down the groove from rest at the vertex ; show that the time of 

 descending through a vertical height h is equal to the time of falling freely 

 through a height h sec 2 . Show also that the pressure is constant and makes 

 with the principal normal to the path a constant angle 



tan - 1 { \ sin a/V(cos 2 a - cos 2 )}, 

 where 2a is the angle of the cone. 



110. A smooth helical tube of pitch a has its axis inclined at an angle 

 /3 (>a) to the vertical, and a particle rests in the tube. The tube is made to 

 turn about its axis with uniform angular velocity w. Prove that the particle 

 makes complete revolutions round the axis if 



ao> 2 /<7>[(7r - 2y) sin y 2 cos y] sin /3 cot 2 a cosec 2 a, 

 where sin y = tan a cot , and a is the radius of the helix. 



111. A smooth tube is bent so as to lie on a cone of vertical angle 2a and 

 to cut the generators at a constant angle /3, the axis of the cone being vertical 

 and the vertex uppermost. The tube is made to rotate uniformly about the 

 axis of the cone with angular velocity Q. Prove that if a particle starts from 

 rest at the vertex it will in time t describe along the tube a distance 



[ cosh ( Qt sin cos ft) ~ *] 

 L 



2 2 

 G 2 sm 2 a cos /3 



112. A particle moves in a smooth tube in the form of a loxodrome on a 

 sphere of radius a, while the tube turns uniformly about the polar axis with 

 angular velocity >. The particle is projected from a point in the equatoreal 



