EXAMPLES. 329 



116. Two equal particles are connected by a string of length 2, which 

 passes over a fixed pulley, and they rest on a smooth inclined plane ; /3 is the 

 inclination of the two portions of the string to the plane when the particles 

 are together, and a that of the plane to the horizon. Prove that when they 

 are slightly displaced the length of the simple equivalent pendulum is 



I cot $ cosec /3 cosec a. 



117. A triangle ABC is formed of equal smooth rods each of length 2a, 

 and small equal rings rest on the rods at the middle points of AS, AC, being 

 attached to A by equal elastic threads of natural length I, and connected 

 together by an inextensible thread passing through a fixed smooth ring at 

 the middle point of BC. Prove that, if there are no external forces, and if 

 one of the rings is slightly displaced, the period of the small oscillations is 



2irJ{2alm/E (5a - 3Z)}, 

 where m is the mass of each ring and E is the modulus of elasticity. 



118. A particle is attached to the middle point of an elastic thread 

 whose ends are attached to two points in the same horizontal plane. The 

 distance between the points and the unstretched length of the thread are 

 each equal to 2a, and in the position of equilibrium the two parts of the 

 thread contain a right angle. Prove that the time of a small oscillation is 

 the same as for a simple pendulum of length 



119. A uniform elastic ring of mass m modulus X and natural length 2?rc 

 in the form of a circle is under the action of a force p (distance) per unit 

 mass directed from its centre. Prove that its radius will vary harmonically 

 about a mean length 2wXc/(27rX - w/zc), provided 2ir\&amp;gt;mpc. What happens if 

 this condition is not satisfied ? 



120. Three small equal rings are fitted on three smooth rods, which are 

 parallel and in the same plane, one being midway between the other two, and 

 the distance between neighbouring rods being a. Prove that, if the rings 

 attract each other according to the law of gravitation and are placed so that 

 the line joining any two of them is nearly perpendicular to the rods, the 

 middle ring and the centre of inertia of the other two will oscillate in a period 

 2ir// s /(3/i), and the other two relatively to each other in period 47r/V(5/i), pa 

 being the attraction at distance . 



121. A circular hoop of negligible mass and of radius b carries a particle 

 rigidly attached to it at a point distant c from its centre, and its inner 

 surface is constrained to roll on the outer surface of a fixed circle of radius 

 a, (b&amp;gt;a\ under the action of a repulsive force from the centre of the fixed 

 circle equal to p times the distance. Prove that the period of small oscilla 

 tions of the hoop will be 



2:r 



b + c /b-a 

 a V cp 



