204 MOTION UNDEK CONSTANT FORCES 



4. Two masses ra, m' are tied to the two ends of a string which is slung over 

 a peg, as in an Atwood's machine. The peg is not smooth, the angle of friction 

 between it and the string being e. Find the motion. 



5. In question 3, let the coefficient of friction between the table and the 

 weight M be /A, that between the table and string being /*'. Find the motion. 



,, 6. A rope hangs over a smooth pulley. Find the uniform acceleration 

 with which a man weighing 10 stone must pull himself up one end of the 

 rope, for the rope to be. kept at rest by a weight of 12 stone hanging from 

 the other end. ' Of" 



7. A monkey is tied to one end of the string of an Atwood's machine, the 

 other end having attached to it a weight exactly equal to that of the monkey, 

 and at just the same depth below the pulley. The monkey suddenly starts to 

 climb up his string. Which will rise the faster, the monkey or the weight on 

 the other string ? 



8. Weights of 10 pounds and 2 pounds, hanging by vertical strings, balance 

 on a wheel and axle. If a mass of 1 pound be added to the smaller weight, find 

 the acceleration with which it will begin to descend, and the tension of each 

 rope. (The inertia of the wheel and axle is to be neglected.) 



9. A mass of 5 pounds resting on a smooth plane inclined at 30 degrees to 

 the horizon is connected by a fine thread, which passes over a pulley at the 

 summit of the plane, with a mass of 3 pounds hanging vertically. Compare the 

 pull in the thread when the mass on the plane is "held fixed and when it is let 

 go. If the thread is severed or burnt 8 seconds after this mass has been let go, 

 find how far it will rise on the plane before falling back. 



10. A light thread passes over two fixed pulleys A and B, and carries 

 between them a movable pulley block O, under which it passes. A mass M is 

 attached to each end of the thread, and a mass m to the movable block. The 

 masses of the pullers- are negligible, and the pulleys are so arranged that all 

 the segments of the thread are vertical. Show that, when the system is let go, 

 the tension in the thread is mM/(M + ^ m) pounds, and find the acceleration 

 with which the mass m falls. 



11. An elastic band of mass m, natural length a, and modulus X is placed 

 round a rough horizontal wheel of circumference b (> a). How fast must the 

 wheel be made to rotate for the band to leave the wheel ? 



12. The elastic band of question 11 is placed on a smooth sphere of circum- 

 ference b rotating with angular velocity w. Find the position of rest. 



13. If the earth rotates faster and faster until ultimately bodies fly off from 

 its equator, show that by the time this stage is reached the plumb line at any 

 point will be parallel to the earth's axis. 



14. A body is placed on a spring balance when in a ship which is sailing 

 along the equator with velocity v. Show that if the balance weighs accurately 

 when the ship is at rest, its reading when the ship is in motion will show an 



error of times the weight of the body (approximately), where w is the 



9 

 angular velocity of the earth. 



