186 WORK 



9. A spider hangs from the ceiling by a thread of modulus of elasticity 

 equal to its weight. Show that it can climb to the ceiling with an expend- 

 iture of work equal to only three quarters of what would be required if 

 the thread were inelastic. 



10. A fine thread having two masses each equal to P suspended at its 

 ends is hung over two smooth pegs in the same horizontal line, distant 

 2 a apart. A mass Q is then attached to the middle point of the portion of 

 the string between the pegs and allowed to descend under gravity. Show 

 that its velocity after falling a depth x will be 



l{2g(x 2 + a 2 )(Qx + 2 Pa - 2Pa: 2 + a 2 ) > [ 

 \t Q(x 2 + a*) + 2Px* JT 



11. Assuming that the attraction of the earth on a body outside the 

 earth falls off inversely as the square of the distance of the body from the 

 earth's center, find with what velocity a shot would have to be fired ver- 

 tically upwards from the earth's surface so as never to return to the earth 

 at all. 



12. A steam hammer of weight 30 tons is pressed down partly by its 

 weight and partly by the pressure of steam in a vertical cylinder acting on 

 a piston which moves with the hammer. The area of the piston is 4 square 

 feet, and the steam pressure is 225 pounds to the inch. If the hammer is 

 raised a height of 2 feet above its block, and set free, find the velocity with 

 which it will strike the block. 



13. The ends of a uniform rod of length / are connected by a string of 

 length a which is placed over a smooth peg. Show that the rod can only 

 hang in a horizontal or vertical position, and examine the stability or 

 instability of these positions. 



14. Two equal uniform rods are rigidly jointed in the shape of the let- 

 ter L, and placed astride a smooth circular cylinder of radius a. Find 

 the smallest length of the rods consistent with stability of equilibrium, the 

 rods being constrained to remain in a vertical plane perpendicular to the 

 axis of the cylinder. 



15. A cube of stone of edge a rests symmetrically and with its base 

 horizontal on a rough circular log of diameter b. Show that the equilib- 

 rium is stable or unstable according as b > or < a. 



16. A rocking stone rests on a fixed stone, the contact being rough, and 

 the common normal at the point of contact being vertical. If p, p' be 

 the radii of curvature of the surfaces of the two stones at the point of 

 contact, and if h be the height of the center of gravity of the movable stone, 

 show that the equilibrium of the rocking stone will be stable or unstable 

 according as 111 



- > or < - + 

 h p p 



