Question

Derive an expression for variation in gravitational acceleration of the Earth at with latitude.

Answer 1

Variation of g with latitude

1. Latitude is an angle made by the radius vector of any point from the center of the Earth with the equatorial plane.
2. The Earth rotates about its polar axis from west to east with uniform angular velocity ω as shown in the figure.
   Hence, every point on the surface of the Earth (except the poles) moves in a circle parallel to the equator.
3. The motion of a mass m at point P on the Earth is shown by the dotted circle with the center at O′.
4. Let the latitude of P be θ and the radius of the circle be r.
   ∴ PO' = r
   ∠EOP = θ, E being a point on the equator
   ∴ ∠OPO' = θ
   In Δ OPO', cos θ = `"PO'"/"PO" = "r"/"R"`
   ∴ r = R cos θ
5. The centripetal acceleration for the mass m, directed along PO' is,
   a = rω2
   ∴ a = rω2 cos θ
   The component of this centripetal acceleration along PO, i.e., towards the centre of the Earth is,
   `"a"_"r" = "a" cos theta`
   ∴ `"a"_"r" = "R"omega^2 cos theta xx cos theta`
   `"a"_"r" = "R"omega^2cos^2theta`
6. Part of the gravitational force of attraction on P acting towards PO is utilized in providing this component of centripetal acceleration. Thus, the effective force of gravitational attraction on m at P can be written as,
   mg' = mg - mRω2cos2θ
   Thus, the effective acceleration due to gravity at P is given as,
   g' = g - Rω2cos2θ