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In the reported figure of earth , the value of acceleration due to gravity is same at point A and C but it is smaller than that of its value at point B (surface of the earth ) . The value of OA : AB will be x : y. The value of x is __________

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answer is 4.

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Detailed Solution

$\begin{array}{l} At point C, g_{h} = g \frac{R^{2}}{(R + h)^{2}} = \frac{g \cdot R^{2}}{{(\frac{3 R}{2})}^{2}} = \frac{4 g}{9} \\ At point A, g_{d} = g [1 - \frac{d}{R}] \end{array}$

$G i v e n : g_{h} = g_{d}$

$\begin{array}{l} \Rightarrow \frac{4 g}{9} = g [1 - \frac{d}{R}] \\ \Rightarrow \frac{5}{9} = \frac{d}{R} \\ \Rightarrow d = \frac{5 R}{9} \\ \Rightarrow A B = \frac{5 R}{9} \\ and O A = O B - A B = \frac{4 R}{9} \\ Thus, O A : A B = 4 : 5 \\ So, x = 4 \end{array}$

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