ILLUSTRATIVE EXAMPLES 



49 



end a weight W resting on the plane. If the coefficient of friction /* is so large 

 that motion does not take place, find through what angle the plane must be 

 tilted before motion begins. 



7. A tourist of mass M is roped to a guide of mass TO on the side of a moun- 

 tain, the side of which may be taken to be an inverted hemisphere. The length 

 of the rope subtends an angle a at the center of the mountain, and the rope is 

 not supposed to touch the mountain at any point. If the coefficient of friction 

 between either man and the mountain is yw, how far can the tourist venture 

 down the side of the mountain before both he and the guide fall to the bottom ? 



ILLUSTRATIVE EXAMPLES 



1. A heavy particle C rests on a smooth inclined plane, being supported by 

 two strings of lengths Zi, 1%, which are attached to two points A, B in the plane, 

 these points being in the same horizontal line and at a distance h apart. Find the 

 tensions of the strings and the reaction with the plane. 



Let W be the weight of the particle and let a be the inclination of the plane to 

 the horizon. The particle is in equilibrium, being acted on by the following forces : 



(a) Its weight W, which acts vertically downwards. 



(6) The reaction between the particle and the plane. Since the plane is 

 smooth, this reaction acts at right angles to the plane. Let the amount of the 

 reaction be E. 



(c) The two tensions of which 

 the amount is required. Let 

 the amounts of these be denoted 

 by Ti, T 2 . 



Since these four forces pro- 

 duce equilibrium, the sum of 

 their resolved parts in any di- 

 rection must vanish. The two 

 tensions have no resolved parts 

 at right angles to the plane ; 

 hence, by resolving at right an- 

 gles to the plane, we shall get 



FIG. 22 



an equation in which only two of the forces are involved. 



The resolved part of the weight at right angles to the plane is TFcos a. The 

 reaction is wholly at right angles to the plane ; hence the equation for which 



we are in search is 



E - Wcosa = 0. 



This gives us the amount of the reaction at once. 



Let us now consider the resolved parts of the forces in the inclined plane. 

 The only forces which have components in this plane are the following : 



(a) The weight, of which the component isTFsina, down the line of greatest slope. 



(6) The tensions of the strings, which are entirely in the plane and which 

 act along the strings CA, CB. 



