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 



siii 6=\ cos 6 + e~ . 



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

 action of a central repulsive force from a point G 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 (p with the horizon is 2 *J(ag) sin (&amp;lt;j&amp;gt; - e), and that the particle leaves 

 the cycloid when the velocity is *J(&ag) (sin|e + 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 on 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 



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 &amp;lt; 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 , its velocity v 

 during its descent is given by 



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*n*- !)/(/* P), 

 where /n is the coefficient of friction. 



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

 cot-^ is placed in a vertical plane, and a heavy particle slides down it, 



