I74-] MOTION ON A FIXED CURVE. 93 



(4) In Ex. (3), let m = 2 Ibs. ; /= 3 ft. ; find the tension in pounds : 

 (a) when the mass makes one revolution per second ; (t>) when it 

 makes 10 revolutions per second. (<:) If the "string cannot stand a ten- 

 sion of more than 300 pounds, what is the greatest allowable velocity? 



(5) A locomotive weighing 32 tons moves in a curve of 800 ft. 

 radius with a velocity of 30 miles an hour ; find the horizontal pressure 

 on the rails. 



(6) To prevent the lateral pressure on the rails in a curve, the track 

 is inclined inwards. Determine the required elevation e of the outer 

 above the inner rail for a given velocity v and radius R if the gauge 

 (i.e. the distance between the rails) is 4 ft. 8 in. 



(7) A plummet is suspended from the roof of a railroad car; how 

 much will it be deflected from the vertical when the train is running 

 45 miles an hour in a curve of 300 yards' radius? 



(8) A body on the surface of the earth partakes of the earth's daily 

 rotation on its axis. The constraint holding it in its circular path is due 

 to the attractive force of the earth. Taking the earth's equatorial radius 

 as 3963 miles, show that the centripetal acceleration of a particle at the 

 equator is about -^ ft. per second, or about ^^ of the actually observed 

 acceleration g = 32-09 of a body falling in vacua. 



(9) If the earth were at rest, what would be the acceleration of a 

 body falling in vacuo at the equator ? 



(10) Show that if the velocity of the earth's rotation were over 

 17 times as large as it actually is, the force of gravity would not be 

 sufficient to detain a body near the surface at the equator. 



(n) Show that in latitude < the acceleration of a falling body, if 

 the earth were at rest, would be gi ==g -\-j cos 2 <, where g is the observed 

 acceleration of a falling body on the rotating earth and j the centripetal 

 acceleration at the equator. Thus, in latitude < '= 45, g= 980-6 cm. ; 

 hence # = 982. 3. 



(12) A chandelier weighing 75 Ibs. is suspended from the ceiling 

 of a hall by means of a chain 12^- ft. long whose weight is neglected. 

 By how much is the tension of the chain increased if it be set swinging 

 so that the velocity at the lowest point is 5 ft. per second ? 



(13) A cord of 2 ft. length passes at its middle point through a hole 

 in a smooth horizontal table. It carries at its lower end a mass of 

 2 Ibs., at its other end a mass of i Ib. The latter is set to revolve in a 



