Question

If both the mass and radius of the earth decrease by 1%, the value of
This question has multiple correct options
A. acceleration due to gravity would decrease by nearly 1%.
B. acceleration due to gravity would increase by 1%.
C. escape velocity from the earth’s surface would decrease by 1%.
D. the gravitational potential energy of a body on earth’s surface will remain unchanged

Answer 1

Hint: To solve this question, we will use the relationship of the mass and radius of the earth with the acceleration due to gravity and the escape velocity and the gravitational potential energy. That are: $g = \dfrac{{GM}}{{{R^2}}}$ and $V = - \dfrac{{GM}}{R}$ and ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $.

Complete step-by-step solution -
Formula used: $g = \dfrac{{GM}}{{{R^2}}}$ and $V = - \dfrac{{GM}}{R}$ and ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $.
Provided that, if the earth's mass and radius decreases by 1%,
First we'll check how this impacts acceleration due to gravity, g
As we do know, $g = \dfrac{{GM}}{{{R^2}}}$
Here the acceleration due to gravity is g, the gravitational constant is G, the earth's mass is M and the earth's radius is R.
If the earth's mass and radius decrease by 1%,
Earth's mass is to become, $M'$ = $M - \dfrac{1}{{100}}M = 0.99M$
Likewise earth's radius will become, $R'$ = 0.99R.
Now, the acceleration due to gravity will become,
$g' = \dfrac{{GM'}}{{{{(R')}^2}}}$
\[ \Rightarrow g' = \dfrac{{G(0.99M)}}{{{{(0.99R)}^2}}} = 1.01\dfrac{{GM}}{{{R^2}}}\]
\[ \Rightarrow g' = 1.01g\].
Thus we can say that acceleration due to gravity will increase by about 1 percent.
Now we are going to check for escape velocity ${v_e}$,
We know it, ${v_e} = \sqrt {\dfrac{{2GM}}{R}} $.
Likewise, if the earth's mass and radius falls by 1%
Then escape velocity is
${v_e}^\prime = \sqrt {\dfrac{{2GM'}}{{R'}}} $.
$
   \Rightarrow {v_e}^\prime = \sqrt {\dfrac{{2G(0.99M)}}{{(0.99R)}}} = \sqrt {\dfrac{{2GM}}{R}} \\
   \Rightarrow {v_e}^\prime = {v_e} \\
$
We may therefore assume that the escape velocity remains constant if we reduce the earth's mass and radius by 1%.
Now we're going to test for potential energy from gravity, V
We are mindful of that, $V = - \dfrac{{GM}}{R}$
It is,
$
   \Rightarrow V' = - \dfrac{{GM'}}{{R'}} \\
   \Rightarrow V' = - \dfrac{{G(0.99M)}}{{(0.99R)}} = - \dfrac{{GM}}{R} \\
   \Rightarrow V' = V \\
$
Gravitational energy potential thus remains unchanged.
Thus, we can assume that if the earth's mass and radius decreases by 1 percent, then the acceleration value due to gravity will increase by 1 percent and a body's gravitational potential energy will remain unchanged on the earth's surface.
So, choice (B) and choice (D) are the correct answers.

Note: If we ask questions of this kind, we must first determine the correct relationship that will be used. We will then obtain the new values for the terms listed in the query. Then bring the new values into those ties, we can get the idea that either the value will increase or decrease, or that it will remain the same. So if they increase or decrease, then by what amount. We will get the answer through that.