Question

The moon’s radius is \[1/4\] that of the earth and its mass \[1/80\] times that of the earth. If \[g\] represents the acceleration due to gravity on the surface of the earth, then on the surface of the moon its value is:
(A) \[\dfrac{g}{4}\]
(B) \[\dfrac{g}{5}\]
(C) \[\dfrac{g}{6}\]
(D) \[\dfrac{g}{8}\]

Answer 1

Hint: We have been provided with the information about the ratio of the masses of the moon and the earth and the value of the ratio of the radius of the moon to that of the earth. Since no actual data has been provided to us, we can say that we have to solve this question by comparison of the two scenarios. Let’s see how it will be done.

Formula Used:
\[g=\dfrac{GM}{{{r}^{2}}}\]

Complete step by step answer:
The acceleration due to gravity on the surface of the earth is given as
\[{{g}_{earth}}=\dfrac{G\times mas{{s}_{earth}}}{radiu{{s}_{earth}}^{2}}\] where \[G\] is the universal gravitational constant
Similarly, the acceleration due to gravity on the surface of the moon will be given as
\[{{g}_{moon}}=\dfrac{G\times mas{{s}_{moon}}}{radiu{{s}_{moon}}^{2}}\] where \[G\] is the universal gravitational constant
Taking the ratio of the acceleration due to the gravity of the moon and the earth, we get
$\implies$\[\dfrac{{{g}_{moon}}}{{{g}_{earth}}}=\dfrac{\dfrac{G\times mas{{s}_{moon}}}{radiu{{s}_{moon}}^{2}}}{\dfrac{G\times mas{{s}_{earth}}}{radiu{{s}_{earth}}^{2}}}\]
The universal gravitational constant will get cancelled out, leaving us with
$\implies$\[\dfrac{{{g}_{moon}}}{{{g}_{earth}}}=\dfrac{mas{{s}_{moon}}}{mas{{s}_{earth}}}\times {{\left( \dfrac{radiu{{s}_{earth}}}{radiu{{s}_{moon}}} \right)}^{2}}\]
Now we have been told that the moon’s radius is one-fourth the radius of the earth; this means that
$\implies$\[\dfrac{radiu{{s}_{earth}}}{radiu{{s}_{moon}}}=4\]
Also, the mass of the earth is eighty times that of the moon, so
$\implies$\[\dfrac{mas{{s}_{moon}}}{mas{{s}_{earth}}}=\dfrac{1}{80}\]
Substituting these values in our expression for \[g\], we get
\[\begin{align}
 & \dfrac{{{g}_{moon}}}{{{g}_{earth}}}=\dfrac{1}{80}\times {{(4)}^{2}} \\
 & \Rightarrow \dfrac{{{g}_{moon}}}{{{g}_{earth}}}=\dfrac{16}{80} \\
 & \Rightarrow {{g}_{moon}}=\dfrac{{{g}_{earth}}}{5} \\
\end{align}\]
Since the value \[{{g}_{earth}}\] is given to be \[g\], the values \[{{g}_{moon}}\] will be \[\dfrac{g}{5}\] .
Hence option (B) is the correct answer.

Additional Information: The expression for the acceleration due to gravity is derived from Newton’s law of gravitation formula. Newton’s law of gravitation gives us the force of gravity acting between two particles of a certain mass and kept at a certain distance from each other.

Note: We can simplify the calculation procedure of the above question by assuming the value of the earth’s mass and radius to have certain values. Then we can find the values of the mass of the moon and the radius of the moon with respect to those values. Upon substituting those values in the expression for the acceleration due to gravity, we’ll get very simple terms. The steps of the procedure will remain the same so don’t worry about that.