EXAMPLES. 207 



41. A smooth cycloid has its axis AB inclined to the vertical and its 

 convexity upwards ; a particle begins to slide down the arc from A, and leaves 

 the curve at P ; the perpendicular from P on AB cuts at Q the circle on AB 

 as diameter, and QR is a diameter of this circle. Prove that PR is horizontal. 



42. A particle moves on a smooth curve in a vertical plane, the form 

 of the curve being such that the pressure on the curve is always m times 

 the weight of the particle. Prove that the time of a complete revolution is 



27T /- , and that the length of the vertical axis of the curve is 

 ( OT 2_l)t V 9 



2ma , , , , ,, Pjl , . 2m 2 +1 

 -. s rr-o i the whole length of the curve being ira . 



/ iyY)& 1 \& / O -i \-lJ- 



\ m L ) (m 2 -!)^ 



43. Prove that, if a particle moves in a smooth tube under the action 

 of forces tending to centres, the pressure on the tube at any point will be 

 proportional to 



r \. f T J 



where -=- is the acceleration towards any one of the centres, and p is the 

 radius of curvature. 



44. A smooth circular tube of radius a is fixed in a vertical plane, and 

 contains a particle, which is attached to the highest point of the tube by an 

 elastic thread inside the tube ; the modulus of elasticity is ^\/3 of the weight 

 of the particle, and the natural length of the thread subtends an angle ^rr at 

 the centre. Prove that, if when the particle is in equilibrium it receives by an 

 impulse a downwards velocity x /{(27rV3 3)a^}, it will just reach the lowest 

 point. 



45. Two equal smooth circular tubes are fixed so as to touch at their 

 lowest points the same horizontal plane, their planes being at different 

 inclinations ; two small heavy beads are projected at the same instant along 

 these circles from their lowest points, the velocity of each bead being due to 

 falling from the highest point of the other circle. Show that throughout the 

 motion the two beads will always be at the same height. 



46. Assuming that the mass of the Moon is ^ of that of the Earth, and 

 that the Moon s distance is 60 times the Earth s radius, prove that owing 

 to the Moon s attraction a seconds pendulum will be losing at a rate 

 T &amp;lt;j(3sm 2 a-l) seconds per day, where a is the altitude of the Moon at 

 the place of observation. 



47. A bead moves on a smooth circular wire in a vertical plane its velocity 

 being that due to falling from a horizontal line HK above the circle. Prove 

 that, if / is the internal limiting point of the co-axal system of which the 

 circle and the line HK are members, then any chord through / divides the 

 wire into two parts which are described in equal times. 



