EXAMPLES. 219 



119. A rod of length 2 rotates in a horizontal plane about one of its 

 ends with uniform angular velocity o&amp;gt;. The ends of a thread of length Zl are 

 attached to the ends of the rod, and a bead can slide on the thread. Prove 

 that when the motion is steady and the bead is at a distance a + x from 

 the axis, the acceleration towards the axis is 



120. A smooth cycloidal tube rotates with uniform angular velocity about 

 its base which is vertical. Prove that a particle cannot rest in the tube 

 anywhere except at the lowest point unless the angular velocity o&amp;gt; of the tube 

 exceeds J(gla\ where a is the radius of the generating circle, and that, when 

 a&amp;gt; exceeds this value, there are two positions of relative equilibrium, the 

 arc-distances of which from the vertex of the cycloid are 



2o&amp;gt; - 1 / 



121. A particle moves in a smooth circular tube of radius a which rotates 

 about a fixed vertical diameter with angular velocity o&amp;gt;. Prove that, if 6 is 

 the angular distance of the particle from the lowest point, and if initially it is 

 at rest relative to the tube with the value a for 6 where a&amp;gt;cos^a=*J(gla), 

 then at any subsequent time t 



cot ^6 = cot -|a cosh (att sin ^a). 



122. A particle of mass m is constrained to remain on the surface of 

 a sphere of radius a, and is attached to a fixed point of the sphere by a 

 slightly extensible thread of natural length aa and modulus X. Prove that, if 

 the particle is projected at right angles to the unstretched thread with velocity 

 v the greatest subsequent elongation is 2aX &quot; l mv 2 cot a. 



123. A particle is projected horizontally on the interior surface of a 

 smooth cone whose axis is vertical and vertex downwards. Prove that its 

 path when the cone is developed into a plane is the same as the path of a 

 particle under the action of a constant force to a fixed point. 



124. A particle moves on a smooth cone under a force to the vertex 

 varying inversely as the square of the distance. Prove that, if the cone is 

 developed into a plane, the path will be a conic having one focus at the vertex 

 of the cone. 



125. A particle moves under gravity on a right circular cone with a 

 vertical axis. Show that, if the equations of motion can be integrated without 

 elliptic functions, the particle must be below the vertex, and that its distance 

 r from the vertex at time t is given by an equation of the form 



(rr) 2 = 2g cos a (r - r ) (r + 2r )2, 

 where 2a is the vertical angle of the cone. 



126. A particle moves on the inside of a smooth circular cone of vertical 

 angle 2a under a force to the vertex varying inversely as the square of the 

 distance. It is projected from an apse at a distance c from the axis with 



