Centrifugal Correction to g

Any particle of mass m on the surface of earth (regarded as a uniform sphere of mass M_e and Radius R_e) is subjected to an acceleration due to gravity given by



which we refer to as the ‘true’ value of g. The gravitational force of interaction between earth and particle is the cause for g : F_g = m g, where g is radially towards the center of earth.

However, when observed from the rotating frame attached to earth, the particle is also subjected to inertial forces. Suppose the particle is at rest at point B at latitude λ on earth’s surface. Note that it is located at a radial distance (R_e cos λ) from origin O’ of the frame with axes (OX’, OY’) rotating about the axis O’ N. Hence there acts a centrifugal force on particle along the direction O’ B given by the above equation.

F_cen = m ω² (R_e cos λ)                                                             (i)

where, ω is angular velocity of earth’s rotation. Since, particle is at rest relative to earth, there is no coriolis force acting on it.

Thus, for an observer on earth, the particle is subjected to two forces: F_g and F_cen. The resultant of these two forces would not be radially towards O, the center of earth. We usually resolve F_cen into two components: the radial component F_cen cos λ along OB and the component F_cen sin λ tangential to surface of earth. Thus,

F_r’ = F_g – F_cen cos λ

= m (g – ω² R_e cos² λ)                                                             (ii)

and, F_t’ = F_cen sin λ = m ω² R_e cos λ sin λ                          (iii)

The resultant radial force is sometimes written as mg’, where g’ denotes the ‘effective’ acceleration due to gravity, i.e. g modified by centrifugal acceleration. We find that g’ depends on λ as,

g’ = g – ω² R_e cos² λ                                                                (iv)

The plumb-line at latitude λ is deflected from radial direction OB by an angle θ given by,


Note that local plumb-line at position B defines the vertical direction there. It is along the direction of plumb-line that the resultant force F’ acts.
 

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