EXAMPLES. 215 



coming to rest at its lowest point. Prove that at the starting point the 

 tangent makes with the horizon an angle 2tan~ 1 /ij and that the velocity 

 is greatest when the angle &amp;lt;f&amp;gt; which the direction of motion makes with the 

 horizon is given by the equation 



2u 2 - 1 sin &amp;lt; + 3* cos = 2n. 



94. Two particles, each of unit mass, attracting each other with a force 

 H (distance), are placed in two rough straight intersecting tubes at right angles 

 to each other and the friction is equal to the pressure on each tube : prove 

 that, if they are initially at unequal distances from the point of intersection, 

 one moves for a time | TT/ V //A before the other starts, and that, while they are 

 approaching the point of intersection of the tubes, they move in the same 

 manner as the projections of the two extremities of the diameter of a circle 

 upon a straight line on which the circle rolls uniformly. 



95. A ring moves on a rough elliptic wire, of semi-axes a, b, under the 

 attraction of a thin uniform gravitating rod of mass M in the line of foci. 

 Prove that, if it is projected from an end of the minor axis and comes to rest 

 at the end of the major axis through which it first passes, the velocity v 

 of projection is given by the equation 



2 __4yMfM [ e~^d6 

 ~(a + 6) 2 Jo l-2acos0+a 2 



where /-t is the coefficient of friction, and a = (a b)/(a -f 6). 



96. A particle on a plane is moving with constant velocity V relative to 

 it, the plane at the same time turning round a fixed axis perpendicular to it 

 with angular velocity a&amp;gt;. Prove that the path of the particle is given by the (^^ 

 equation 



V n 



a 2 ) + cos-i 



r and 6 being referred to fixed axes, and a being the least distance of the 

 particle from the axis of rotation. 



97. A point P moves along a plane curve which rotates in its plane about 

 a point with uniform angular velocity to. Prove that the curvature of its 



path is 



V(&amp;lt;r V+ 2a&amp;gt;) ( F+ r&amp;lt;B sin i/r) + rco ( Va&amp;gt; sin ^ -/cos ^ + r&amp;lt;a 2 ) 



( F 2 + r 2 o&amp;gt; 2 + 2 Frco sin ^)* 



where r is the length OP, a- is the curvature of the curve at P, \//- the angle 

 between OP and the tangent, V the velocity of P relative to the curve, and / 

 the rate of increase of F. 



98. A particle P is free to move on a smooth circular wire whose centre 

 C rotates with constant angular velocity in the plane of the wire about a fixed 

 point 0. Show that, if CP = 30C and the particle just makes complete 

 revolutions, the pressure between the particle and the wire vanishes when 

 CP makes with OC an angle sec* 1 3. 



