Question

The radius of the earth is about 6400 km, and that of Mars is about 3200 km. The mass of the earth is about ten times the mass of Mars. An object weighs 200N on the surface of the earth. Its weight on the surface of the mars would be.
A.6N
B.29N
C.40N
D.80N

Answer 1

Hint: In this question, we need to determine the weight of the object on Mars such that its weight on the Earth is measured as 200 N. For this, we need to follow relation between the gravitational forces between the different masses and are separated by a distance.

Complete step by step answer:
The radius of the earth \[{R_e} = 6400km\]
The radius of the mars \[{R_m} = 3200km\]
So we can conclude that the radius of the earth is doubled the radius of mars
\[{R_e} = 2{R_m} - - (i)\]
Now it is given that mass of the earth is about ten times the mass of Mars, which can be written as \[{M_e} = 10{M_m} - - (ii)\], where the mass of the earth is \[{M_e}\], and that of mars is \[{M_m}\]
We know that the gravitational force acting on own mass is given by the formula
\[g = \dfrac{{G \bullet M}}{{{r^2}}}\]
So the gravitational force acting on earth's surface will be
\[{g_e} = \dfrac{{G \bullet {M_e}}}{{r_e^2}} - - (iii)\]
And the gravitational force acting on mars surface will be
\[{g_m} = \dfrac{{G \bullet {M_m}}}{{r_m^2}} - - (iv)\]
Now since the radius of the earth is doubled of mars and the mass of the earth is ten times that of mars, so we can write equation (iii) as
\[
  {g_e} = \dfrac{{G \bullet 10{M_m}}}{{{{\left( {2{r_m}} \right)}^2}}} \\
  \Rightarrow{g_e} = \dfrac{{G \bullet 10{M_m}}}{{4r_m^2}} \\
   = 2.5\dfrac{{G \bullet {M_m}}}{{r_m^2}} \\
 \]
So by using equation (iv), we can write
\[{g_e} = 2.5{g_m} - - (v)\]
Hence we can say gravity on earth is 2.5 times that of gravity on mars, and the weight of an object is dependent on the force of gravity.
So the mass of the object on the surface of the mass
\[
  {g_m} = \dfrac{{{g_e}}}{{2.5}} \\
  \Rightarrow {g_m} = \dfrac{{200}}{{2.5}} \\
     \therefore {g_m} = 80N \\
 \]
Hence weight on the surface of the mars will be \[ = 80N\]

So, the correct answer is Option D.

Note:
Students must note here that the force of gravity on a body is directly proportional to the mass of the body and inversely proportional to the distance of the object. The gravitational force acting on own mass is given by the formula \[g = \dfrac{{GMm}}{{{r^2}}}\], and if the force is acting on the body itself then it is given as \[g = \dfrac{{GM}}{{{r^2}}}\].