292 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



equivalent pendulum for oscillations in the vertical plane through the equi- 

 librium position of the rod is 



^pa cos a (1 -f 2 cos 2 a)/(a p sin 3 a), 

 provided this expression is positive. 



2. A uniform rod of length 2a passes through a smooth ring fixed at a 

 height b above the vertex of a smooth bowl in the form of a surface of revolu- 

 tion whose axis is vertical, and rests in a vertical position. Prove that, if c is 

 the radius of curvature of the meridian curve at the vertex, the length of the 

 simple equivalent pendulum for small oscillations is 



provided this expression is positive. 



3. A uniform rod of length 2a is supported symmetrically by two cords 

 each of length I attached to its ends and to two fixed points at the same level 

 distant 2 (a + 1 cos a) apart. Prove that the length of the simple equivalent 

 pendulum for small oscillations in the vertical plane through the cords is 



^al sin a (1 +3 sin 2 a)/(a + Z cos 3 a). 



4. A number of equal uniform rods each of length 2a are freely jointed 

 at a common end and arranged at equal intervals like the ribs of an umbrella, 

 and this cone of rods is placed in equilibrium over a smooth sphere so that 

 the angle of the cone is 2a. Prove that for small vertical oscillations of the 

 joint the length of the simple equivalent pendulum is 



^ a cos a (1 + 3cos 2 a)/(l+2cos 2 a). 



5. Prove that the length of the simple equivalent pendulum for small 

 oscillations of the handle of a garden roller rolling on a horizontal walk is 



where a is the radius of the roller, M the mass of the roller alone, k its radius 

 of gyration about its axis, h the distance of the centre of inertia of the 

 handle from the axis of the roller, and I the length of the simple equivalent 

 pendulum for the oscillations of the handle when the roller is held fixed. 



6. Four equal uniform rods are freely jointed so as to have a common 

 extremity, and four other like rods are similarly jointed; the free ends of the 

 rods are then jointed in pairs so as to form eight edges of an octahedron. 

 One of the joints where four rods meet is fixed and the other is attached to 

 it by an elastic thread, so that in equilibrium the octahedron is regular and 

 the thread vertical. Prove that the length of the simple equivalent pendulum 

 for small vertical oscillations of the lowest point is f (I Z ), where I and 1 

 are the equilibrium length and the natural length of the thread. 



