198-200] 



ROUGH CURVE. 



187 



and moment of velocity h may be described, relatively to a different frame 

 with the same origin, as uniform motion in a straight line, provided A 2 > p. 



3. A particle moves in a smooth plane tube, and is under a central force 

 to a fixed point about which the tube rotates uniformly. Prove that, if the 

 pressure is always zero, the central force is 



m [r 2 + 2ro> (h - r 2 o>)/j 2 + (h - r 2 co) 2 p ~ 3 dp/dr] , 



where m is the mass of the particle, mh is its moment of momentum about 

 the fixed point, CD is the angular velocity of the tube, r is the radius vector, 

 and p the perpendicular from the fixed point on the tangent to the tube at 

 the position of the particle. 



*200. Motion on a rough plane curve under gravity. 



When a particle is constrained to describe a plane curve in a 



vertical plane under gravity but there is 



frictional resistance to the motion as well 



as pressure on the curve we assume that 



the friction is fj, times the pressure, where 



fi is the coefficient of friction. The friction 



acts in the tangent to the curve in the 



sense opposite to that of the velocity. 



The equations of motion take different 

 forms in different circumstances. We shall 

 choose for investigation the case where the 

 particle is on the outside of the curve, and 

 is descending. 



Let the arc s of the curve be measured from some point of the 

 curve so that it increases in the sense of the velocity, and let (f> be 

 the angle contained between the inwards normal and the down- 

 wards vertical. ' Then < increases with s, and ds/dcfr (= p) is the 

 length of the radius of curvature. 



Let v be the velocity of the particle, m its mass, R the pressure 

 of the curve on the particle. The equations of motion are 



dv , 



mv -T- = mg sm </> fi 



Fig. 51. 



v 2 



mg cos (f> R 



Eliminating R we obtain the equation 

 dv 



or 



v -^ fj, = g (sin < /JL cos c/>), 

 = gp (sin (f> /JL cos <). 



f/7J 



v -7 , 



