Question

Mars has a diameter of approximately $0.5$ times that of earth and mass is $0.1$ times that of earth. The surface gravitational field strength on mars as compared to that of earth is a factor of
A. $0.1$
B. $0.2$
C. $2.0$
D. $0.4$

Answer 1

Hint: In order to solve this question, we should know about gravitational field strength of a planet, Gravitational field strength is a measure of acceleration on a body falling freely under the force of gravity on a planet, we will use the general formula of gravitational field strength and will compare of these two planets mars and earth.

Formula Used:
If $M$ is the mass of a planet and $R$ is the radius of a planet and $G$ be the gravitational constant and $g$ represents the gravitational field strength then \[g = \dfrac{{GM}}{{{R^2}}}\].

Complete step by step answer:
According to the question we have given, assume ${M_{mars}}(and){M_{earth}}$ denote the mass of mars and earth. and ${R_{mars}}(and){R_{earth}}$ denote the radii of these two planets.
${M_{mars}} = 0.1{M_{earth}}$
and diameters of two planets can be written as $R = \dfrac{D}{2}$ and it is given that
${D_{mars}} = 0.5{D_{earth}}$
$\Rightarrow \dfrac{{{D_{mars}}}}{2} = \dfrac{{0.5({D_{earth}})}}{2}$
$\Rightarrow {R_{mars}} = 0.5{R_{earth}}$
Now for planet earth using the formula \[g = \dfrac{{GM}}{{{R^2}}}\] we have,
\[{g_{earth}} = \dfrac{{G{M_{earth}}}}{{{R_{earth}}^2}}\]
for planet mars we have,
\[{g_{mars}} = \dfrac{{G{M_{mars}}}}{{{R_{mars}}^2}}\]
putting the values of parameters we get,
\[{g_{mars}} = \dfrac{{(0.1)G{M_{earth}}}}{{{{(0.5)}^2}{R_{earth}}^2}}\]
\[\Rightarrow {g_{mars}} = \dfrac{{0.1}}{{0.25}}{g_{earth}}\]
\[\Rightarrow {g_{mars}} = \dfrac{{10}}{{25}}{g_{earth}}\]
\[\therefore {g_{mars}} = 0.4{g_{earth}}\]

Hence, the correct option is D.

Note: It should be remembered that, the value of acceleration due to gravity on each planet is different and larger the acceleration due to gravity of a planet the more the energy needed to escape the gravitational field of that planet. Free fall means when a body is released to fall under the force of gravity with zero initial velocity.