2.1 Variation of acceleration due to gravity $(g)$ due to shape of the earth

  Gravitation

    1.0 Newton's law of gravitation
    2.0 Variation of acceleration due to gravity
    3.0 Gravitational field
    4.0 Gravitational potential
    5.0 Gravitational potential energy
    6.0 Satellites
    7.0 Kepler's law of planetary motion
    8.0 Problem solving technique

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2.1 Variation of acceleration due to gravity $(g)$ due to shape of the earth

• If the radius of the earth decreases by $n\% $ keeping its mass unchanged, then the acceleration due to gravity on its surfaces increases by $2n\% $
  
  As we know, $$g = \frac{{GM}}{{{R^2}}}$$$$\frac{{\Delta g}}{g} = - 2\frac{{\Delta R}}{R}$$$$\frac{{\Delta g}}{g} = - 2\left( { - n\% } \right)$$$$\frac{{\Delta g}}{g} = 2n\% $$

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• If the mass of the planet increases by $n\%$ keeping its radius unchanged, the acceleration due to gravity on its surface increases by $n\%$.
  $$\frac{{\Delta g}}{g} = \frac{{\Delta m}}{m}$$ $$\frac{{\Delta g}}{g} = n\% $$

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• If the density of a planet decreases by $n\%$ keeping its radius unchanged, the acceleration due to gravity decreases by $n\%$.
  $$g = \frac{4}{3}\pi G{R_e}\rho $$ $$\frac{{\Delta g}}{g} = \frac{{\Delta \rho }}{\rho }$$ $$\frac{{\Delta g}}{g} = - n\% $$