182-185] PRESSURE OF CURVE. 173 



When the constraint is applied by means of a smooth tube 

 within which the particle moves, or when the particle is the centre 

 of inertia of a small ring sliding on a smooth wire in the form of 

 the curve, R can act either inwards or outwards. When the con- 

 straint is applied by means of a smooth surface the particle is 

 either on the concave side or on the convex side of its path on the 

 surface ; in the former case R acts inwards, in the latter outwards. 



When the pressure, if there is any, must act inwards or must 

 act outwards the particle leaves the curve at the instant when the 

 pressure changes sign. Putting R = 0, we find as the condition for 



this 



v 2 = gp cos 0, 



showing that when the particle leaves the curve the velocity is 

 that due to falling through one quarter of the vertical chord of 

 curvature. 



If we wish to find the acceleration along the tangent to the 

 curve we have only to resolve along the tangent, and the required 

 acceleration is the resolved part of g along the tangent. 



184. Motion down an inclined tube. The case just discussed includes 

 in particular the case of a particle sliding down a line of greatest slope on a 

 smooth inclined plane, or sliding down a smooth straight tube held in an 

 inclined position. 



If a is the inclination to the horizon, the acceleration down the tube (or 

 down the plane) is g sin a, and the pressure on the tube (or on the plane) is 

 mg cos a, where ra is the mass of the particle. 



The motion of the particle is uniformly accelerated motion, and the dis- 

 cussion of Article 42 applies to it. The first experimental determination of 

 the value of g was made by observing the velocity acquired in sliding down 

 an inclined plane. 



185. Examples. 



1. Prove that the time of descent down all chords of a vertical circle 

 starting from the highest (or ending at the lowest point) is the same. 



2. Prove that the line of quickest descent from a point A to a curve, 

 which is in a vertical plane containing A, is the line from A to the point of 

 contact with the curve of a circle described to have A as its highest point and 

 to touch the curve. Prove also that the line of quickest descent from a 

 curve to a point A is the line from A to the point of contact with the curve 

 of a circle described to have A as its lowest point and to touch the curve. 



