Question

The radius of the Earth is 3.7 times the radius of the Moon, and the acceleration due to gravity on the surface of the Moon is 1/6 g, where g is the acceleration on the surface of the Earth. What is the mass of the Moon?

Answer 1

The gravitational acceleration on the Moon can be written as follows:

$$g_M=\dfrac{GM_M}{(R_M)^2}=\dfrac{GM_M}{(\dfrac{R_E}{3.7})^2}=13.69\dfrac{GM_M}{(R_E)^2}.$$

From the other hand, $g_M=\dfrac{1}{6}g_E$, we can write:

$$g_M=\dfrac{GM_E}{6(R_E)^2}.$$

We can equate both equations:

$$13.69\dfrac{GM_M}{(R_E)^2}=\dfrac{GM_E}{6(R_E)^2},$$ $$13.69M_M=\dfrac{M_E}{6},$$ $$M_M=\dfrac{M_E}{13.69\cdot6}=\dfrac{M_E}{82},$$ $$M_M=\dfrac{5.97\cdot10^{24}\ kg}{82}=7.28\cdot10^{22}\ kg.$$