Gravity Change Calculation: Doubled Earth Radius

The acceleration due to gravity, denoted by '$g$', depends on the mass ($M$) and radius ($R$) of the celestial body. The formula is:

$g = \frac{GM}{R^2}$

where $G$ is the universal gravitational constant.

Initial Conditions

Let the original radius of the Earth be $R_{original}$ and the original acceleration due to gravity be $g_{original}$.

$g_{original} = \frac{GM}{R_{original}^2}$

New Conditions

The problem states that the radius doubles, so the new radius $R_{new} = 2 \times R_{original}$. The mass $M$ remains the same.

Calculating New Gravity ($g_{new}$)

Substitute the new radius into the formula:

$g_{new} = \frac{GM}{R_{new}^2}$

$g_{new} = \frac{GM}{(2 \times R_{original})^2}$

$g_{new} = \frac{GM}{4 \times R_{original}^2}$

Comparing New and Original Gravity

Now, relate $g_{new}$ to $g_{original}$:

$g_{new} = \frac{1}{4} \left( \frac{GM}{R_{original}^2} \right)$

Since $g_{original} = \frac{GM}{R_{original}^2}$, we can substitute:

$g_{new} = \frac{1}{4} g_{original}$

Conclusion

The new value of $g$ is one-fourth the original value.