EXAMPLES. 209 



will descend through a vertical height which is a third proportional to the 

 natural length of the thread, and the increase of its length when in the lowest 

 position, the thread being stretched throughout the motion. 



55. A particle hangs in equilibrium under gravity being suspended by 

 an elastic thread whose modulus of elasticity is 3 times the weight of the 

 particle. The particle is slightly displaced in a direction making an angle 

 cot" 1 4 with the horizontal, and is then released. Show that it will oscillate 

 in an arc of a small parabola terminated by the ends of the latus rectum. 



56. A particle placed at an end of the major axis of a normal section of 

 a uniform gravitating elliptic cylinder is slightly disturbed in the plane of the 

 section. Prove that it can move round in contact with the cylinder, and that 

 its velocity v when at a distance y from the major axis of the section is given 

 by the equation 



v* = 4-nypy* a (a- &)/{& (a + &)}, 



where p is the density of the cylinder, and 2a, 26 are the principal axes of a 

 normal section. 



57. A particle moves in a smooth tube in the form of a catenary being 

 attracted to the directrix with a force proportional to the distance from the 

 directrix. Prove that the period of oscillation is independent of the amplitude. 



58. Prove that a hypocycloid, generated by the rolling of a circle of radius 

 6 on a circle of radius a, is isochronous for a force varying as the distance 

 from the centre of the fixed circle, and that the time of an oscillation is 



where the force per unit of mass at unit distance is p. (distance). 



59. A particle, of unit mass, is at rest in a smooth tube in the form of an 

 equiangular spiral of angle a at a distance 2d from the pole. Prove that, 

 under the action of a force ^/(distance) 2 towards the pole, it will reach the pole 



in time TT sec a 



60. A cycloidal wire in a vertical plane, with its axis vertical and vertex 

 upwards is completely occupied by equal small smooth rings. Prove that, if 

 the constraint at the cusps is removed, then in time t the length of the arc 

 cleared of rings will be 



2ZsinhV(^ 2 W, 



where I is the length of the cycloid. 



61. A particle slides down a smooth cycloidal tube with its axis vertical 

 and vertex downwards, starting from rest at an arc-distance s 1 from the vertex. 

 After a time t, and before the first particle has reached the vertex, a second 

 particle slides down the tube starting from rest at an arc-distance s 2 from the 

 vertex. Prove that the arc-distance from the vertex of the point where the 

 particles meet is 



1 2 27rA 



where T is the time of a complete oscillation in the tube. 



L. 14 



