The diameters of two planets are in the ratio `4:1` and their mean densities in the ratio `1:2` The acceleration due to gravity on the particles wil be in ratio.

Options

A. `1:2`
B. `2:3`
C. `2:1`
D. `4:1`

Explanation

To find the ratio of the acceleration due to gravity on two planets based on their diameters and mean densities, we can follow these steps:

### Step 1: Understand the relationship between gravity, density, and radius.
The formula for acceleration due to gravity \( g \) on the surface of a planet is given by:
\[ g = \frac{4}{3} \pi R \rho \]
where \( R \) is the radius of the planet and \( \rho \) is the mean density of the planet.

### Step 2: Express the ratios of the diameters and densities.
Given:
- The diameters of the two planets are in the ratio \( 4:1 \).
- Therefore, the radii \( R_1 \) and \( R_2 \) (since radius is half of the diameter) will be in the ratio:
\[ R_1 : R_2 = 4 : 1 \implies R_1 = 4R \text{ and } R_2 = R \]

- The mean densities are in the ratio \( 1:2 \):
\[ \rho_1 : \rho_2 = 1 : 2 \implies \rho_1 = \rho \text{ and } \rho_2 = 2\rho \]

### Step 3: Set up the ratio of gravitational accelerations.
Using the formula for gravity, we can express the gravitational accelerations for both planets:
\[ g_1 = \frac{4}{3} \pi R_1 \rho_1 \quad \text{and} \quad g_2 = \frac{4}{3} \pi R_2 \rho_2 \]

### Step 4: Substitute the values into the ratio.
Now substituting the values of \( R_1, R_2, \rho_1, \) and \( \rho_2 \):
\[ g_1 = \frac{4}{3} \pi (4R)(\rho) = \frac{16}{3} \pi R \rho \]
\[ g_2 = \frac{4}{3} \pi (R)(2\rho) = \frac{8}{3} \pi R \rho \]

### Step 5: Calculate the ratio \( \frac{g_1}{g_2} \).
Now, we can find the ratio of the gravitational accelerations:
\[ \frac{g_1}{g_2} = \frac{\frac{16}{3} \pi R \rho}{\frac{8}{3} \pi R \rho} \]
The \( \frac{4}{3} \pi R \rho \) terms cancel out:
\[ \frac{g_1}{g_2} = \frac{16}{8} = 2 \]

### Step 6: State the final ratio.
Thus, the ratio of the acceleration due to gravity on the two planets is:
\[ g_1 : g_2 = 2 : 1 \]

### Final Answer:
The acceleration due to gravity on the two planets will be in the ratio \( 2:1 \).

---