The relationship between density, radius, and acceleration due to gravity on the surface of planets:

The density, radius, and acceleration due to gravity are interconnected properties of a planet. In order to understand the relationship between them, let's break down the problem into smaller parts:

1. Density:
Density is defined as the mass of an object divided by its volume. In the case of planets, we can consider the average density, which is the total mass of the planet divided by its total volume. Mathematically, it can be expressed as:

Density = Mass / Volume

2. Volume:
The volume of a planet can be calculated using the formula for the volume of a sphere, which is given by:

Volume = (4/3) * π * (radius)^3

3. Mass:
The mass of a planet can be calculated using the formula:

Mass = Density * Volume

4. Acceleration due to gravity:
The acceleration due to gravity on the surface of a planet is given by Newton's law of universal gravitation:

Acceleration due to gravity = (G * Mass) / (radius)^2

Where G is the gravitational constant.

Proving the ratio of acceleration due to gravity:

Let's consider two planets with the same density but radii in the ratio of 1:2. We can denote the radii as r1 and r2, where r2 = 2r1.

1. Density:
Since the density is the same for both planets, we can denote it as ρ1 = ρ2 = ρ.

2. Volume:
Using the volume formula, we can calculate the volumes of the two planets as:

Volume1 = (4/3) * π * (r1)^3
Volume2 = (4/3) * π * (r2)^3 = (4/3) * π * (2r1)^3 = (4/3) * π * 8 * (r1)^3 = 8 * (4/3) * π * (r1)^3 = 8 * Volume1

3. Mass:
Since the density is the same for both planets, the masses can be calculated as:

Mass1 = ρ * Volume1
Mass2 = ρ * Volume2 = ρ * 8 * Volume1 = 8 * Mass1

4. Acceleration due to gravity:
Using Newton's law of universal gravitation, we can calculate the acceleration due to gravity for the two planets as:

Acceleration due to gravity1 = (G * Mass1) / (r1)^2
Acceleration due to gravity2 = (G * Mass2) / (r2)^2 = (G * 8 * Mass1) / (2r1)^2 = (G * 8 * Mass1) / 4 * (r1)^2 = 2 * (G * Mass1) / (r1)^2 = 2 * Acceleration due to gravity1

Conclusion:

From the calculations above, we can see that the ratio of the acceleration due to gravity on the surface of the two planets is 1:2. This means that the planet with the larger radius has twice the acceleration due to gravity