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



87. A particle slides down the arc of a rough circle (/* = ) fixed in a 

 vertical plane, and the particle starts from rest at an end of the horizontal 

 diameter. Prove that, if 6 is the angle the radius vector through the particle 

 makes with the horizontal when the velocity is a maximum, then 



88. A particle of unit mass moves in a rough straight tube AB under the 

 action of a central repulsive force from a point C of magnitude X/r at a 

 distance r from C. The point A is the foot of the perpendicular from C on 

 the tube, and the particle is projected from A along the tube with velocity v. 

 Prove that it comes to rest when the radius vector from C makes with CA an 

 angle 6 satisfying the equation 



where p is the coefficient of friction. 



89. A particle is started with indefinitely small velocity from that point 

 of a rough cycloidal arc (vertex uppermost) at which it could rest in limiting 

 equilibrium. Show that the velocity at a point at which the tangent makes 

 an angle < with the horizon is 2 >J(ag) sin ($ - e), and that the particle leaves 

 the cycloid when the velocity is *J(2ag] (sin ^6 + cos ^e), where e is the angle of 

 friction. 



90. A particle slides down a rough cycloid whose axis is vertical and 

 vertex downwards. Prove that the time of reaching a certain point oil the 

 cycloid is independent of the starting point. 



Prove also that, if X is the angle of friction, and if the tangent at the 

 starting point makes with the horizontal an angle greater than a, where a is 

 the least positive angle which satisfies the equation 



sin(a-X) = e (a+x)tanA sin2X, 

 the particle will oscillate. 



91. A ring moves on a rough cycloidal wire with its axis vertical and 

 vertex downwards. Prove that, if it starts from the lowest point with velocity 

 UQ, its velocity u when its direction of motion is inclined at an angle to the 

 horizontal is given by 



where a is the radius of the generating circle and e is the angle of friction. 



Prove also that if it starts from a cusp with velocity v 0t its velocity v 

 during its descent is given by 



tf = ( Vo 2 + 4 a g cos 2 e) e ( * ~ 2ir) ten e - 4a# sin 2 (0 - e). 



92. A particle is projected from a point on the lowest generator of a rough 

 horizontal cylinder of radius a with velocity V at right angles to the generator. 

 Prove that it returns to the point of projection after a time a(e 2 M*-- !)/(/* F), 

 where p is the coefficient of friction. 



93. A rough wire in the form of an equiangular spiral whose angle is 

 cot~ 1 2/* is placed in a vertical plane, and a heavy particle slides down it, 



