300 BOOK OF MATHEMATICAL PROBLEMS. 



1435. A circular tube containing a smooth particle revolves 

 about a vertical diameter with uniform angular velocity to, find 

 the position of relative equilibrium of the particle; and prove that 



2^ 



it will oscillate about this position in a time : , a being 



o> tun a 



the angle which the normal at the point makes with the vertical. 



1436. A heavy particle is placed in a tube in the form of a 

 plane curve, which revolves with uniform angular velocity to, 

 about a vertical axis in its plane, and the particle oscillates about 

 its position of relative equilibrium; prove that the time of oscil- 

 lation is 



27r // p sin a \ 



oj \/ \k p sin a cos" aj ' 



k being the distance from the axis of revolution, a the angle made 

 by the normal with the vertical, and p the radius of curvature 

 of the curve, at the point of equilibrium. 



1437. A straight tube, inclined to the vertical at an angle a, 

 revolves with uniform angular velocity to about a vertical axis 

 whose shortest distance from the tubs is a, and contains a smooth 

 particle placed initially at its shortest distance from the axis ; 

 prove that 



_ 2} + c 



* being the space described along the tube in a time t. 



1438. A heavy particle is attached to two points in the same 

 horizontal line, at a distance a, by two elastic strings, each of 

 natural length a, and is set free when the strings are at their 

 natural length ; prove that the initial radius of curvature of its 



O / Q 



path is ~ v ; the coefficients of elasticity being respectively m 

 and n times the weight of the particle. 



1439. A uniform heavy chain is placed on the arc of a smooth 

 vertical circle, its length being equal to a quadrant, and one ex- 



