EXAMPLES. 205 



of weights P' and W are substituted, P' will descend with acceleration /, 

 such that 



W- W], 



all the pulleys being of equal weight. 



28. In any machine without friction and inertia a body of weight P 

 supports a body of weight W t both hanging by vertical cords. These bodies 

 are replaced by bodies of weights P' and JP, which, in the subsequent motion 

 move vertically. Prove that the centre of inertia of P' and W will descend 

 with acceleration 



g ( WP' - W'PfK W 2 P' + W'P 2 ) ( W' + P). 



29. Two particles of masses P and Q lie near to each other on a smooth 

 horizontal table, being connected by a thread on which is a ring of mass R 

 hanging just over the edge of the table. Prove that it falls with acceleration 



30. Two particles of masses m, m' are attached to the ends of a thread 

 passing over a pulley and are held on two inclined planes each of angle a 

 placed back to back with their highest points beneath the centre of the pulley. 

 Prove that if each portion of the thread makes an angle /3 with the corresponding 

 plane the particle of greater mass m will at once pull the other off the plane if 



a tan /3 - 1. 



31. Two equal bodies, each of mass M t are attached to the chain of an 

 Atwood's machine, and oscillate up and down through two fixed horizontal 

 rings so that each time one of them passes up through a ring it lifts a bar of 

 mass m, while at the same instant the other passes down through its ring 

 and deposits on it a bar of equal mass. Prove that, neglecting friction, the 

 period of an excursion of amplitude a is 



.a \ //' 



and that the successive amplitudes form a diminishing geometric progression 

 of which the ratio is 



where p is a mass which distributed over the circumference of the pulley will 

 produce the same effect on the motion as the inertia of the actual mechanism. 



32. A series of vertical circles touch at their highest points, and smooth 

 particles slide down the arcs starting from rest at the highest point ; prove 

 that the foci of the free paths lie on a straight line whose inclination to the 

 vertical is tan" 1 (1^/5). 



33. A particle is projected along the circumference of a smooth vertical 

 circle of radius a. It starts from the lowest point and leaves the circle before 



