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Graph of Variation of G

Consider two ...

Question

Consider two solid spheres of radii $R_{1} = 1 m, R_{2} = 2 m$ and masses $m_{1}$ and $m_{2}$ , respectively. The gravitational field due to sphere $1$ and $2$ are shown. The value of $\frac{m_{1}}{m_{2}}$ is:

A

\frac{2}{3}

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B

\frac{1}{6}

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C

\frac{1}{2}

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D

\frac{1}{3}

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Solution

The correct option is B $\frac{1}{6}$
Gravitational field at the surface is given by,

$E = \frac{G m}{R^{2}}$
$∴ E_{1} = \frac{G m_{1}}{R_{1}^{2}} and E_{2} = \frac{G m_{2}}{R_{2}^{2}}$

The peak value in the graphs given represents the gravitational field at the surface.

$\Rightarrow \frac{E_{1}}{E_{2}} = \frac{2}{3}$
$∴ \frac{E_{1}}{E_{2}} = {(\frac{R_{2}}{R_{1}})}^{2} (\frac{m_{1}}{m_{2}})$

$\Rightarrow \frac{2}{3} = {(\frac{2}{1})}^{2} (\frac{m_{1}}{m_{2}})$
$\Rightarrow (\frac{m_{1}}{m_{2}}) = \frac{1}{6}$

Hence, option $(B)$ is correct.

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