EXAMPLES 57 



6. Two smooth planes, whose inclinations are a and /3, meet in a hori- 

 zontal line AB. At a point in A B is a small smooth ring through which 

 passes a string with a weight at either end, resting one on each of the 

 given planes, and in the same vertical plane with the ring. If the weights 

 are in the equilibrium, find the tension of the string and the ratio of the 

 weights. 



7. Two smooth rings of weights W and W 2 are connected by a string 

 and rest in equilibrium on the convex side of a circular wire in the vertical 

 plane. Show that, if the string subtends the angle a at the center of the 

 circle, the angle of inclination 6 of the string to the vertical is given by 



W l + W t a 



8. Two weights rest on a rough inclined plane and are connected by a 

 string which passes over a smooth peg in the plane ; if the angle of inclina- 

 tion a is greater than the angle of friction c, show that the least ratio of 

 the less to the greater is sin (a: e)/sin(a + e). 



9. Two weights support one another on a rough double inclined plane, 

 by means of a fine string passing over the vertex, and both weights are 

 about to move. Show that if the plane be tilted un-til both weights are 

 again on the point of motion, the angle through which the plane will be 

 turned is twice the angle of friction. 



10. Two weights P, Q of similar material, resting on a double inclined 

 plane, are connected by a fine string passing over the common vertex, and 

 Q is on the point of motion down the plane. Prove that the greatest weight 

 which can be added to P without disturbing the equilibrium is 



P sin 2 e sin (a + /3) 

 sin (a e) sin (/3 e) 



<r, /3 being the angles of inclination of the planes, and e the angle of 

 friction. 



11. A body is supported on a rough inclined plane by a force acting 

 along it. If the least magnitude of the force, when the plane is inclined 

 at an angle a to the horizon, be equal to the greatest magnitude, when the 

 plane is inclined at an angle /3, show that the angle of friction is \(a. /3). 



12. Two equal rings of weight W are movable along a curtain pole, the 

 coefficient of friction being p. The rings are connected by a loose string 

 of length I, which supports by means of a smooth ring a weight W v How 

 far apart must the rings be so that they will not come together ? 



13. Two weights P, Q of different material are laid on a rough plane, 

 whose inclination is 0, and connected by a taut string inclined at 45 to the 

 intersection of the plane with the horizon. Both weights are on the point 



