# mass of planet expressed as multiple of earths mass

**1. The problem statement, all variables and given/known data**

On the surface of Planet X, the gravitational acceleration is 2g. If the diameter of Planet X is 1/3 that of Earth’s, what is the mass of Planet X, expressed as a multiple of Earth’s mass?

**2. Relevant equations**

[itex] F=G\frac{m1m2}{r^2} [/itex]

**3. The attempt at a solution**

let Mx = mass of planet x, Me = mass of earth, Rx = radius of planet x and Re = radius of planet earth. the diameter of planet x is 2Rx and the diameter of earth is 2Re.

[itex] 2Rx = \frac{2Re}{3} [/itex]

[itex] Rx = \frac{Re}{3} [/itex]

we have

[itex] a = \frac{GMx}{Rx^2} = \frac{GMx}{(\frac{Re}{3})^2} = \frac{9GMx}{Re^2} = 2g [/itex]

since 2g = [itex] \frac{2GMe}{Re^2} [/itex] we have

[itex] \frac{9GMx}{Re^2} = \frac{2GMe}{Re^2} [/itex]

[itex] GMx = \frac{2GMe}{9} [/itex]

[itex] Mx = \frac{2Me}{9} [/itex]

Is this correct?

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