EXAMPLES. 339 



176. A plane disc of any form has in its surface a flat circular cavity of 

 radius a, and it is free to turn about a vertical axis through a point in the 

 circumference of the cavity. A circular disc of mass m and radius c(<a) 

 with a smooth face and a rough edge is in the cavity, and the whole system is 

 rotating steadily about the vertical axis with angular velocity . Prove that 

 the length of the simple equivalent pendulum for small oscillations about the 

 state of steady motion is 



g(a-c} 



" ac 2 o> 2 /+ m [ 2 + (2a - c) 2 } ' 



where I is the moment of inertia of the first disc about the axis, and k is the 

 radius of gyration of the circular disc about a vertical axis through its 

 centre. 



177. A plane tube the equation of which is y i =f(x] is turning freely 

 about the axis of symmetry which is vertical with angular velocity *J(glc\ 

 and a particle of mass m is in the tube close to the lowest point. Prove that, 

 if the radius of curvature of the tube at the lowest point is greater than c, 

 the particle will rise in the tube to a vertical height h which is the least 

 positive root of the equation 



where / is the moment of inertia of the tube about the axis of symmetry. 



178. One end of a rigid uniform rod of length 2a formed of gravitating 

 matter is constrained to move uniformly in a circle of radius c with angular 

 velocity &>, and the rod is attracted to a fixed particle of mass m at the centre 

 of the circle. Prove that the rod can move steadily projecting inwards 

 towards the centre, and that this steady motion is stable if 



c (c 2a) 2 . 



179. If an elastic thread whose length is the same as that of a uniform 

 rod is attached to the rod at both ends, and suspended by the middle point, 

 prove that the rod will sink until the parts of the thread are inclined to the 

 horizon at an angle 6 which satisfies the equation 



where n is the ratio of the modulus of elasticity of the thread to the weight 

 of the rod. 



180. A bead is free to slide on a rod whose ends slide without friction 

 on a fixed circle. Prove that it moves relatively to the rod as if repelled 

 from the middle point with a force varying inversely as the cube of the 

 distance. 



181. A smooth rigid uniform circular tube of mass M contains two 

 particles of masses m lt m 2 and being placed on a table is set in motion by a 

 blow in a line passing through the centre of inertia of the system. Prove 

 that, if 1} 2 are the angles which the radii to the particles make at time t 

 with a fixed line on the table, then throughout the motion 



+ m 2 2 ) + 2ro 1 i 2 (0\ + 2 ) sin 2 J (B l - 2 ) = 0. 



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