Question

When the radius of earth is reduced by 1% without changing the mass, then the acceleration due to gravity will
A. increase by 2%
B. decrease by 1.5%
C. increase by 1%
D. decrease by 1%

Answer 1

Hint: As a first step recall the expression for acceleration due to gravity. Then understand what a 1% decrease in radius of earth means. Now make necessary changes in the radius of earth in the expression for acceleration due to gravity due to this 1% decrease and then compare with the original expression. Now you could find the percentage change in g.
Formula used:
Expression for acceleration due to gravity,
$g=\dfrac{GM}{{{R}^{2}}}$

Complete step by step answer:
In the question we are given that the radius of earth is reduced by 1% without changing the mass and we are asked to find the percentage change in the acceleration due gravity due to this reduction in radius of earth.
Let us mathematically express reduction by 1% in radius. If the radius becomes $R'$ after reduction then,
$R'=R-\dfrac{R}{100}$
$\Rightarrow R'=\dfrac{99R}{100}$
$\therefore R'=0.99R$ ……………………………… (1)
Now let us recall the standard expression for acceleration due to gravity g. g is given by,
$g=\dfrac{GM}{{{R}^{2}}}$
Where, G is the universal gravitational constant given by,
$G=6.67\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}}$
Also, M and R are the mass and radius of earth.
So, for the mass of earth and universal gravitational constant remaining constant with the radius reduced by 1%, the acceleration due to gravity becomes,
$g'=\dfrac{GM}{R{{'}^{2}}}$
Substituting (1) we have,
$g'=\dfrac{GM}{{{\left( 0.99R \right)}^{2}}}$
$\Rightarrow g'=1.02\dfrac{GM}{{{R}^{2}}}$
$\therefore g'=1.02g$ ………………………………………….. (2)
Now we are asked to find the percentage change in acceleration due to gravity, which is given by,
Percentage change in$g=\dfrac{g'-g}{g}\times 100$
$\dfrac{g'-g}{g}\times 100=\dfrac{1.02g-g}{g}\times 100=0.02\times 100$
Therefore, we see that the percentage change in the acceleration due to gravity (g) when the radius of earth is reduced by 1% without changing the mass will be 2% increase.

Hence, option A is the right answer.

Note:
Basically acceleration due to gravity g is the acceleration of any object that is under the sole influence of gravity. The numerical value is accurately known to be $9.8m{{s}^{-2}}$, that is, any free falling object changes its velocity by 9.8m/s every consecutive second. However, we do observe variations in this value with the variations due to latitude, altitude and also the local geographical structure of the region.