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&quot; 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 



tf = 4rrypf a (a- b)/{b (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 



b 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 /x/(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 



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 , 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 27rf 



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



L. 14 



