EXAMPLES. 337 



Prove that, if ZTr/n is the period of the free oscillations of the system, the 

 time t until the shell again strikes the plane is the smallest positive root of 

 the equation 



(1 + e] sin nt = (l e) nt. 



165. In Example 163 prove that the period of oscillation when the shell 

 is free to move is less than it would be if the shell were held fixed in the ratio 



166. In a smooth table are two small holes A, B at a distance 2a apart ; 

 a particle of mass M rests on the table at the middle point of AB, being 

 connected with a particle of mass m hanging beneath the table by two 

 inextensible threads, each of length a (1 + sec a), passing through the holes. 

 A blow J is applied to M at right angles to AB. Prove that, if 



J 2 &amp;gt;2Mmag tan a, 

 M will oscillate to and fro through a distance 2a tan a, but if 



J 2 = 2Mmag (tan a tan /3) 



where tan /3 is positive, the distance through which M oscillates will be 

 2a N /{(sec a sec /3) (sec a - sec /3 + 2)}. 



167. A particle can move in a smooth plane tube which rotates uniformly 

 with angular velocity to about a vertical axis. Prove that the time of a small 

 oscillation about a position of relative equilibrium is 



2?r // psina \ 

 co V \a p sin a cos 2 a/ 



where p is the radius of curvature at the point of equilibrium, a the angle 

 the normal at this point makes with the vertical, and a the distance of the 

 point from the axis. 



168. A smooth circular wire is made to rotate uniformly about a vertical 

 diameter. A bead of mass m can move on the wire, and is attached to a 

 thread, which passes through a fixed smooth ring at the lowest point of the 

 circle and supports a body of mass m. Prove that, if a is the angle which the 

 radius through the bead in steady motion makes with the vertical, the steady 

 motion is stable or unstable according as 



is negative or positive. 



169. A particle describes a horizontal circle in steady motion at a depth 

 d below the centre of a smooth oblate spheroid of axes 2a, 26, the axis of 

 revolution being vertical. Prove that, if the tangent plane at any point of 

 the circle makes an angle -^ with the vertical, the velocity is acot^-^f(gd}/b, 

 and the period of the small oscillations when the steady motion is slightly 

 disturbed is the same as that of a simple pendulum of length 



2 d/(a 2 cos 2 ^ + 46 2 sin 2 ^-). 



170. A particle is describing a circle of radius r in a smooth bowl in the 

 form of a surface of revolution whose axis is vertical. Prove that, if the 



L. 22 



