282-284] ROTATION OF THE EARTH. 359 



equilibrium position ; it must be projected at right angles to 

 the vertical plane containing it with velocity all sin I. When 

 thus set going it moves like a simple pendulum of the same 

 length in a plane which turns about the vertical from East to 

 West with angular velocity fl sin I. 



This result accords well with observed facts. 



*284. Examples. 



[In these examples the Earth is regarded as a homogeneous sphere.] 



1. If the Earth were to rotate so fast that bodies at the equator had no 

 weight, prove that in any latitude the plumb-line would be parallel to the 

 polar axis. 



2. If the acceleration due to gravity at the poles is g Q and at the equator 

 g e , prove that in (geocentric) latitude X the value of g is 



and that the deviation .of the plumb-line from the (geometrical) vertical is 

 tan- 1 {(g -g e ) sin X cos X/(# sin 2 X +g 6 cos 2 X)}. 



3. Prove that a pendulum which beats seconds at the poles will lose 

 approximately 30m cos 2 1 beats per minute in latitude , where 1-fwi : 1 is 

 the ratio of the weight of a body at the poles to its weight at the equator. 



4. A train of mass m is travelling with uniform speed v along a parallel 

 of latitude in latitude I. Prove that the difference between the pressures on 

 the rails when the train travels due East and when it travels due West is 

 kmvQ. cos I approximately. 



5. A projectile is projected from a point on the Earth's surface with 

 velocity V at an elevation a in a vertical plane making an angle /3 with 

 the meridian (East of South). Prove that after an interval t it will have 

 moved southwards through #, eastwards through y, and upwards through 2, 

 where 



x = Vt cos a (cos /3 -f- &t sin I sin /3}, 



y = Vt (cos a sin /3 - Qt (sin I cos ft cos a -f cos I sin a)} 4- $ &gt z cos I, > 



z=Vt (sin a + Qt cos I sin ft cos a} - \gfi-, 



approximately, G 2 y being neglected. 



6. Prove that, if the bob of a pendulum of length L is let go from a 

 position of rest relative to the Earth when its displacement from its equi- 

 librium position is a, and the vertical plane through it makes an angle ft with 

 the meridian (East of South), its path is given approximately by the equation 



higher powers of LQ?lg being neglected. 



