The masses of two planets are in the ratio 1 : 2. Their radii are in the ratio 1 : 2. The acceleration due to gravity on the planets are in the ratio. (a) 1 : 2 (b) 2 : 1 (c) 3 : 5 (d) 5 : 3

To figure out the acceleration due to gravity on the two planets, we can use the formula for gravitational acceleration, which is given by:

Understanding Gravitational Acceleration

The formula for the acceleration due to gravity (g) on the surface of a planet is expressed as:

g = G * (M / R^2)

where:

• G is the gravitational constant,
• M is the mass of the planet, and
• R is the radius of the planet.

Applying the Given Ratios

In this case, we know the following ratios:

• The masses of the planets are in the ratio 1:2, so we can let the mass of the first planet be M and the mass of the second be 2M.
• The radii are in the ratio 1:2, meaning we can let the radius of the first planet be R and the radius of the second be 2R.

Calculating the Acceleration Due to Gravity

Now, we can find the acceleration due to gravity on each planet:

• For the first planet:

g₁ = G * (M / R²)

• For the second planet:

g₂ = G * (2M / (2R)²)

This simplifies to:

g₂ = G * (2M / 4R²) = (G * (M / 2R²))

Finding the Ratio of Acceleration

Now, we can find the ratio of the gravitational accelerations:

g₁ : g₂ = (G * (M / R²)) : (G * (M / 2R²))

The G and M terms cancel out, leaving us with:

g₁ : g₂ = (1/R²) : (1/(2R²))

This simplifies to:

g₁ : g₂ = 2 : 1

Final Answer

The ratio of the accelerations due to gravity on the two planets is therefore 2:1. The correct answer is (b) 2:1.