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



This equation can be integrated after multiplication by the 

 factor e~^, in fact it becomes 



gper*+ (sin - ^ cos </>), 



so that v*er** = 2g ipf** (sin < - /^ cos </>) d<j> -f const., 



an equation which determines v as a function of <, and therefore 

 gives the velocity at any point of the curve. The velocity being 

 determined, the second of the equations of motion gives the 

 pressure, and, as in the case of a smooth curve, if R vanishes the 

 particle leaves the curve. 



The equations of motion take different forms according as the 

 particle is inside or outside the curve, and according as it is 

 ascending or descending. But in each case the equations can be 

 integrated by the above method. There is accordingly no definite 

 expression for the velocity at any point of the curve in terms of 

 the position, but the expressions obtained are different in the 

 different cases. 



201. Examples. 



1. Write down the equations of motion in the three cases not investigated 

 in Article 200 and the integrating factor in each case. 



2. A particle is projected horizontally from the lowest point of a rough 

 sphere of radius a, and returns to this point after describing an arc aa, 

 (a < TT\ coming to rest at the lowest point. Prove that the initial velocity 

 is sinav/{2#a(l+/A 2 )/(l-2/z 2 )}, where p is the coefficient of friction. 



3. A particle slides down a rough cycloid, whose base is horizontal and 

 vertex downwards, starting from rest at a cusp and coming to rest at the 

 vertex. Prove that, if p is the coefficient of friction, /* 2 e M7r = 1. 



4. A ring moves on a rough cycloidal wire whose base is horizontal and 

 vertex downwards ; prove that during the ascent the direction of motion at 

 time t makes with the horizontal an angle <, given by the equation 



where e is the angle of friction. 



*202. Motion on a curve in general. When a particle 

 moves on a given curve under any forces, we take m for the mass 

 of the particle, $ for the tangential component of the resultant 

 force of the field, N for the component along the principal normal, 



