330 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



122. Two particles of masses M, m are attached to a thread which hangs 

 vertically from a fixed point, m being above M ; (1) m is held slightly pulled 

 aside a distance h from the position of equilibrium, and, being let go, the 

 system performs small oscillations; (2) M is held slightly pulled aside a 

 distance k, without disturbing m, and, being let go, the system performs 

 small oscillations. Prove that the angular motion of the lower thread in the 

 first case will be the same as that of the upper thread in the second case if 



123. A particle is placed on one of the plane faces of a uniform gravitat- 

 ing circular cylinder at a very small distance from the centre of the face ; 

 prove that it will make small oscillations in a period 



where a, h, p are the radius of the cylinder, its height, and the density of its 

 material. 



124. The lower end of a uniform rod of length a slides on an inextensible 

 thread of length 2a whose ends are fixed to two points distant 2 x /(a 2 -6 2 ) 

 apart in a horizontal line, and the upper end of the rod slides on a fixed 

 vertical rod which bisects the line joining the two fixed points. Prove that 

 the time of a small oscillation about the vertical position of equilibrium is 



125. The extremities of a uniform rod of length 4 slide without friction 

 on the circumference of a three-cusped hypocycloid whose plane is vertical, 

 one of the cusps being at the highest point of the circumscribing circle 

 (radius 3a). Prove that the length of the simple equivalent pendulum 

 iafo. 



126. A number n of uniform isosceles triangular discs are smoothly 

 jointed at a common vertex so as to form a pyramid whose base is a regular 

 polygon inscribed in a circle of radius a, and whose edges lie on a cone of 

 vertical angle 2cot~ 1 ^/(S + sin 2 ^/^). Prove that, if the system is placed so 

 as to rest on a smooth sphere with each of its planes inclined to the vertical 

 at an angle a [>sin~ 1 (^cos7r/^)], the length of the simple equivalent pendu- 

 lum for its small oscillations is 



$ a cos a(l+8 cos 2 a)/( l-j-2 cos 2 a). 



127. In a heavy plane lamina, whose centre of gravity is 6r, are two 

 narrow straight slits BA, AC, such that AGr bisects the angle BAG. Through 

 each slit passes a fixed peg, the pegs, P, Q, being in the same horizontal 

 line. Prove that the time of a small oscillation of the lamina in its own 

 plane, about a position of equilibrium in which the vertex A of the triangle 

 APQ is upwards, is 



/ JLPq (P&+& BinM)_ 

 V 9 sin A (4PQ* -AG* sin 2 A) ' 



where k is the radius of gyration of the lamina about a line through G per- 

 pendicular to its plane. 



