Question

If the distance between the sun and the earth is increased by three times, then the gravitational force between the two will

• A. remain constant
• B. decrease by $63$%
• C. increase by $63$%
• D. decrease by $89$%

Answer 1

The correct option is D decrease by $89$%

$\text{Step 1: Gravitational force of attraction :}$

Initial gravitational force between sun and earth, $F=\frac{G{M}_s{M}_e}{r^2}$ $....\left(1\right)$

where, $r$ is the distance between sun and earth.

When distance is increased to $r^{\prime }=3r$, then force become $F^{\prime }$

$F^{\prime }=\frac{G{M}_s{M}_e}{r^{\prime 2}}=\frac{G{M}_s{M}_e}{9r^2}$ $....\left(2\right)$

$\text{Step 2: Percentage decrease in force:}$

From equation $\left(1\right)$ and $\left(2\right)$

$\frac{F^{\prime }}{F}=\frac{1}{9}$

Percentage decrease in force= $\frac{F-F^{\prime }}{F}\times 100\%$ $=(1-\frac{1}{9})\times 100\%=\frac{8}{9}\times 100\%=88.88\%\approx 89\%$

Hence, Option(D) is correct.