218 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



plane with velocity aa> relative to the tube. Prove that the particle will be at 

 an angular distance 6 from the equatoreal plane after a time 



{sec a - log (sec 6 + tan #)}/>, 

 and that the pressure on the tube in this position is 



2mcta) 2 ( 1 + sin a) cos 6, 

 m being the mass of the particle and a the angle of the loxodrome. 



113. A train starts from rest on a level uniform curve, and moves round 

 the curve so that its speed increases at a constant rate /. The outer rail is 

 raised so that the floor of a carriage is inclined at an angle a to the horizon. 

 Show that a body cannot rest on the floor of the carriage unless the coefficient 

 of friction between the body and the floor exceeds 



/cosa/V(# 2 +/ 2 sin 2 a). 



114. A locomotive, starting with a constant acceleration / from a point A 

 of a railway, comes to a curve PQ in the line. Prove that, if, in passing along 

 PQ, the pressure of the flanges of the wheels on the rails is constant, PQ must 

 be a portion of an equiangular spiral whose pole may be any point on a circle 

 touching AP at P and having its diameter equal to A P. Prove also that if 

 the track is tilted up at an angle 6 so that the constant pressure vanishes the 

 angle of the spiral must be tan" 1 (%gf~ l tan 6}. 



115. Two particles of masses m and m' are attached to the ends of a rigid rod 

 of negligible mass and of length 2, which is freely moveable about its middle 

 point. Show that the inclination a of the rod to the vertical when the particles 

 are moving with uniform angular velocity o> is given by the equation 



cos a = (m m'} gl{(m + m') a> 2 l}. 



116. A particle is fastened to one end of a thread of length I, the other 

 end being fixed to the top of a smooth sphere of radius a ; the particle 

 describes a horizontal circle with angular velocity <>, and the length of the 

 thread in contact with the sphere is aa. Prove that 



o> 2 g cot a/ '{a sin a + (I - aa) cos a}. 



117. A bead can slide on a rough straight wire which is rotating with 

 uniform angular velocity o> about a fixed vertical axis intersecting it, and a is 

 the inclination of the wire to the horizontal. Prove that for the ring to be in 

 relative equilibrium it must lie between two points in the wire whose distance 

 apart is 



ga> ~ 2 sec a (tan (a + X) - tan (a - X)}, 



where X is the angle of friction. 



118. A small ring can slide on a smooth plane curved wire which rotates 

 with angular velocity co about a vertical axis in its plane. Find the form of 

 the curve in order that the ring may be in relative equilibrium at any point. 



Prove that if the angular velocity is increased to ' the ring will still be in 

 relative equilibrium if the wire is rough and the coefficient of friction between 

 it and the ring not less than ^ ('/ - </<') 



