EXAMPLES 317 



The rim will touch the ground when the range of wobbling is given by 



tan = - , and after this the top will roll on the ground. If we take 



a 

 a = 6 inches and h = 2 inches, as before, this gives tan = |, so that 



o o 2 _ O2 1 Q 



cos = = and ^<L_^_ = .106, approximately. Thus O = - fl , roughly. 



Thus such a top will " sleep " until its rim velocity is reduced to 4.7 feet 

 a second. After this it will wobble, and as soon as its rim velocity is 

 reduced to about 4.5 feet a second the top will begin to roll on the ground. 

 For an ordinary small, pear-shaped peg top we may take roughly h = 1^ 

 inches, and the radii of gyration about axes through the point of contact 

 as | inch and 2 inches. Thus in inches 



A = ^ M, B = 4;M, 



9 _ 4 Mgh B _ 2048 

 ~~A*~ ''^T g ' 



Taking g 386 inches per second per second, this gives O = 170 revolu- 

 tions per second. If thrown from a string of which the end is coiled round 

 the top in circles of radius 1 inch, the string must be withdrawn with a 

 velocity of about 60 miles an hour relative to the top to set up the required 

 angular velocity. 



GENERAL EXAMPLES 



1. A fly wheel whose moment of inertia is / has a string wound round 

 its axle of radius b. A tension equal to a weight w is applied to the string 

 for 1 second. What is the angular velocity of the fly wheel at the end of 

 1 second? I^^\J*2^ 



2. A fleet of total displacement 200,000 tons steams from east to west 

 along the equator, covering 20 minutes of longitude per hour. Regarding 

 the earth as a homogeneous sphere of mass 6 x 10 21 tons, find the change 

 in the earth's angular velocity produced by the motion of the fleet. Show 

 that the day is lengthened by, roughly, 16 x 10~ 14 seconds. 



3. The earth's mass is 6 x 10 21 tons, and icebergs and melted snow 

 weighing 10 10 tons move from the north pole to latitude 45 degrees. Find 

 the change in the length of the day. 



4. A train of mass m runs due north at 60 miles an hour. Show that 

 there must be a pressure between the eastern rail and the flanges of the 

 wheel in consequence of the earth's rotation, and find the amount of 

 the pressure. 



5. A thin layer of dust, thickness h feet, is formed on the earth by 

 'the fall of meteors, reaching the earth from all directions. Show that the 



