58 FORCES ACTING ON A SINGLE PAETICLE 



of motion. Determine the coefficients of friction of P and Q, it being 

 known that that of the upper weight is twice that of the lower. 



14. A heavy ring is free to slide on a smooth elliptic wire of eccen- 

 tricity e in a vertical plane, the major axis of the ellipse making an angle a 

 with the horizontal, and a string fastened to the ring passes over a smooth 

 peg at the center of the ellipse and supports a body of equal weight. Show 

 that the angle < which the tangent to the wire at the ring makes with the 

 major axis is given by the equation 



tan (0 + a) (sec 2 - e 2 ) = e z tan 0. 



15. Two small smooth rings of weights W, W are connected by a string, 

 and slide on two fixed wires, the former of which is vertical and the latter 

 inclined at an angle a to the horizontal. A weight P is tied to the string, 

 and the two portions of it make angles 6, with the vertical. Prove that 



cot 6 : cot : cot a = W : P + W : P + W + W. 



16. Two particles of unequal mass are tied by fine inextensible strings 

 to a third particle. They lie on a rough inclined plane, with the strings 

 stretched and making angles a, /3 with the horizontal line in the plane. 

 Find the magnitude and direction of the least horizontal force which, on 

 being applied to the* third particle, will move all three. 



17. A heavy particle is placed on a rough inclined plane of which the 

 inclination a is equal to the angle of friction. A thread is attached to 

 the particle, and passed through a hole in the plane which is lower than 

 the particle, but is not in the line of greatest slope through it. Show that 

 if the thread be gradually drawn through the hole, the particle will describe 

 a straight line and a semicircle in succession. 



