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Infinite number of spheres, each of mass $m$ are placed on the $X$-axis at distances $1,2,4,8$, 16 ...... metres from origin. The magnitude of the gravitational field at the origin is
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The correct answer is:
$\frac{4}{3} \mathrm{Gm}$
Gravitational field at the origin due to mass at 1,2 , $4,8,16 \ldots$
$$
\begin{aligned}
& =\frac{G m}{(1)^2}+\frac{G m}{(2)^2}+\frac{G m}{(4)^2}+\frac{G m}{(8)^2} \ldots . . \\
& =G m\left[1+\frac{1}{4}+\frac{1}{16}+\ldots \ldots\right]=\frac{4 G m}{3}
\end{aligned}
$$
$$
\begin{aligned}
& =\frac{G m}{(1)^2}+\frac{G m}{(2)^2}+\frac{G m}{(4)^2}+\frac{G m}{(8)^2} \ldots . . \\
& =G m\left[1+\frac{1}{4}+\frac{1}{16}+\ldots \ldots\right]=\frac{4 G m}{3}
\end{aligned}
$$
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