If ' $R$ ' is the radius of earth \& ' $g$ ' is acceleration due to gravity on earth's surface, then mean…

If ' $R$ ' is the radius of earth \& ' $g$ ' is acceleration due to gravity on earth's surface, then mean density of earth is
  1. $\frac{4 \pi \mathrm{G}}{3 g R}$
  2. $\frac{3 \pi R}{4 g G}$
  3. $\frac{3 \mathrm{~g}}{4 \pi \mathrm{RG}}$
  4. $\frac{\pi \mathrm{RG}}{12 \mathrm{~g}}$

Solution

Mass of the Earth is given by, $\mathrm{M}=\mathrm{V} \rho=\frac{4}{3} \pi \mathrm{R}^3 \rho...(i)$ Acceleration due to gravity is given by, $\mathrm{g}=\frac{\mathrm{GM}}{\mathrm{R}^2}...(ii)$
Substituting equation (i) in equation (ii), $g=\frac{G}{R^2} \times \frac{4}{3} \pi R^3 \rho \Rightarrow \rho=\frac{3 g}{4 \pi R G}$ :

Asked in: MHT CET 2024 (04 May Shift 1)

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