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Understanding how the mass and radius of a planet affect its gravitational field strength is crucial.

The mass of a planet (M) and its radius (R) directly influence the gravitational field strength.
Gravitational field strength (\textbf{g}) is essentially the force per unit mass experienced by an object in the planet's gravitational field.

According to the equation \( g = \frac{GM}{R^2} \), the field strength is proportional to the planet's mass.
However, it is inversely proportional to the square of the planet's radius.

This means that:

• If the planet's mass increases, the gravitational field strength increases.
• If the planet's radius increases, the gravitational field strength decreases.

In the exercise, planet A has twice the mass and twice the radius of planet B.
By substituting these values into the formula, one can understand how variations in mass and radius influence gravitational field strength.

The gravitational constant (G) is a fundamental physical constant used in the calculation of gravitational forces.

It's value is approximately 6.674×10^{-11} N m²/kg².
Despite the planets' differing masses and radii, this value remains constant.

Including G in the formula \( g = \frac{GM}{R^2} \), ensures that gravitational calculations are accurate and consistent across different celestial bodies.
It's crucial to remember that while mass and radius can vary, G provides the steady foundation necessary for these computations.

For instance, if we didn't have a constant such as G, calculating the gravitational field strength for various planets would lack consistency.
Therefore, understanding the role of the gravitational constant helps in appreciating its importance in maintaining uniformity in our gravitational measurements and calculations.

Calculating ratios helps to compare different quantities against each other. In this exercise, we determine the ratio of gravitational field strengths.

Given planets A and B, the gravitational field strength on planet B is given by \( g_B = \frac{G M_B}{R_B^2} \).
For planet A, with twice the mass and radius of planet B, its field strength is \( g_A = \frac{G (2M_B)}{(2R_B)^2} = \frac{2G M_B}{4R_B^2} = \frac{G M_B}{2R_B^2} \).

To find the ratio of the gravitational field strength on planet A to that on planet B, we use the formula:
\[ \frac{g_A}{g_B} = \frac{\frac{G M_B}{2R_B^2}}{\frac{G M_B}{R_B^2}} = \frac{1}{2} \]. This simplifies the comparison by canceling out common factors.

Thus, the gravitational field strength on planet A is half that of planet B.
Ratio calculations can greatly simplify complex problems by providing relative measures.
They offer insight into how varying a few parameters can drastically alter outcomes.