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 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 e 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 I, where 1-fm : 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 

 4m^Q 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 z, 

 where 



x= Fcosa{cosj3 + Q8inZsin}, &quot;j 



y = Vt (cos a sin fi-Qt (sin I cos j3 cos a + cos I sin a)} + J Qgt 3 cos , &amp;gt; 

 z = Vt (sin a + Q.t cos I sin 3 cos a} - Jgtf 2 , J 



approximately, Q?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 /3 with 

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



- *) = Q V(%) sin l W (* 8 - r ^l r -cos 1 (Mi 

 higher powers of LQ?jg being neglected. 



