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, both hanging by vertical cords. These bodies 

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

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

 with acceleration 



g ( wp 1 - WP^K W 2 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 



m /m&amp;lt;2tanatan/3 1. 



31. Two equal bodies, each of mass M, 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 



vt7 



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 (| N / 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 



