Question

There is a planet which is \(8\) times massive and \(27\) times denser than the earth. If \(g'\) and \(g\) are the accelerations due to gravity on the surfaces of the planet and the earth respectively, then:

• A. \(g'=8g\)
• B. \(g'=27g\)
• C. \(g'=18g\)
• D. \(g'=\dfrac{9}{4}g\)

Hint:

For a spherical planet, use \(\rho \propto \frac{M}{R^3}\) to first find the radius ratio, then apply \(g=\frac{GM}{R^2}\).

Answer 1

The Correct Option is C

Step 1: Find the planet's radius from its mass and density.
\( M' = 8M \), \( \rho' = 27\rho \). Since \( M = \frac{4}{3}\pi R^3\rho \), \( 8M = \frac{4}{3}\pi R'^3(27\rho) \Rightarrow R'^3 = \frac{8R^3}{27} \Rightarrow R' = \frac{2R}{3} \).

Step 2: Calculate surface gravity ratio.
\[ \frac{g'}{g} = \frac{M'}{M}\cdot\frac{R^2}{R'^2} = 8\cdot\frac{R^2}{(2R/3)^2} = 8\cdot\frac{9}{4} = 18 \] \[ \boxed{g' = 18g} \]