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Two planets have the same average density, but their radii are \(R\) and \(2R\). If the acceleration due to gravity on these planets is \(g_1\) and \(g_2\) respectively, then

1. \(\frac{g_1}{g_2}=\frac{R_1}{R_2}\)
2. \(\frac{g_1}{g_2}=\frac{R_2}{R_1}\)
3. \(\frac{g_1}{g_2}=\left(\frac{R_2}{R_1}\right)^2\)
4. \(\frac{g_1}{g_2}=\left(\frac{R_2}{R_1}\right)^3\)

Correct answer: \(\frac{g_1}{g_2}=\frac{R_1}{R_2}\)

Solution

For a planet of density \(\rho\), mass \(M=\frac{4}{3}\pi R^3\rho\). Hence \(g=\frac{GM}{R^2}=\frac{4}{3}\pi G\rho R\), so for the same density, \(g\propto R\). Therefore \(\frac{g_1}{g_2}=\frac{R_1}{R_2}\).

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