Question

If the radius of the earth shrinks by 1%, its mass remaining the same, what is the change in acceleration due to gravity on the surface of the earth?
(A) Decrease by 2%
(B) Decrease by 0.5%
(C) Increase by 2%
(D) Increase by 0.5%

Answer 1

Hint: Percent error formula is the absolute value of the difference of the measured value and the actual value divided by the actual value and multiplied by 100 i.e.
 $ %Error\text{ }in\text{ g; }\Delta \text{g}=\dfrac{g'-g}{g}\times 100 $
Where g’ is the new value and g is the actual value of acceleration due to gravity.

Complete step by step solution
Here the value of radius R shrinks by 1% so,
 $ \Delta R=-1 $
 $ \begin{align}
  & \dfrac{R'-R}{R}\times 100=-1 \\
 & R'-R=\dfrac{-R}{100} \\
 & R'=R-\dfrac{R}{100} \\
 & R'=0.99R \\
\end{align} $
Now the new value of g i.e. g’ becomes:
 $ \begin{align}
  & g'=\dfrac{GM}{R{{'}^{2}}} \\
 & g'=\dfrac{GM}{{{\left( 0.99R \right)}^{2}}} \\
 & g'=\dfrac{g}{{{\left( 0.99 \right)}^{2}}} \\
\end{align} $
Percentage change in g is
 $ \begin{align}
  & \Delta g=\dfrac{g'-g}{g}\times 100 \\
 & =\dfrac{\dfrac{g}{{{\left( 0.99 \right)}^{2}}}-g}{g}\times 100 \\
 & =\dfrac{g\left( \dfrac{1}{{{(0.99)}^{2}}}-1 \right)}{g}\times 100 \\
 & =\dfrac{1-0.9801}{0.9801}\times 100 \\
 & =\dfrac{0.0199}{0.9801}\times 100 \\
 & =0.0203\times 100 \\
\end{align} $
Here the positive sign indicates that the value of g is decreasing.
Therefore the value of acceleration due to gravity is increased by 2%.

Note
Alternate method:
Percentage error in any term Z having formula $ Z=\dfrac{{{A}^{p}}{{B}^{q}}}{{{C}^{r}}} $ is given by:
 $ \dfrac{\Delta Z}{Z}\times 100=\left( p\dfrac{\Delta A}{A}+q\dfrac{\Delta B}{B}-r\dfrac{\Delta C}{C} \right)\times 100 $
Here
 $ \begin{align}
  & g=\dfrac{GM}{{{R}^{2}}} \\
 & g\text{ }\alpha \text{ }\dfrac{1}{{{R}^{2}}} \\
 & \dfrac{\Delta g}{g}\times 100=-2\dfrac{\Delta R}{R}\times 100 \\
\end{align} $
As $ \begin{align}
  & \dfrac{\Delta R}{R}\times 100=-1 \\
 & \\
\end{align} $
So,
 $ \begin{align}
  & \dfrac{\Delta g}{g}\times 100=-2\times \left( -1 \right) \\
\end{align} $
Therefore the value of g increases by 2%.