Answer to Question #157634 in Physics for Lm

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:

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