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 planets will be in ratio

Options

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

Answer

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

### Step 1: Understand the relationship between gravity, mass, and radius
The acceleration due to gravity \( g \) on the surface of a planet is given by the formula:
\[ g = \frac{GM}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 2: Express mass in terms of volume and density
The mass \( M \) of a planet can be expressed as:
\[ M = \text{Volume} \times \text{Density} = V \times d \]
For a spherical planet, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass can be written as:
\[ M = \frac{4}{3} \pi R^3 d \]

### Step 3: Set up the ratios
Let’s denote:
- Diameter of Planet 1: \( D_1 \) and Diameter of Planet 2: \( D_2 \)
- Mean density of Planet 1: \( d_1 \) and Mean density of Planet 2: \( d_2 \)

Given:
- The ratio of diameters \( D_1 : D_2 = 4 : 1 \)
- The ratio of densities \( d_1 : d_2 = 1 : 2 \)

From the diameter ratio, we can express the radii:
\[ R_1 = \frac{D_1}{2}, \quad R_2 = \frac{D_2}{2} \]
Thus,
\[ R_1 : R_2 = 4 : 2 = 2 : 1 \]

### Step 4: Calculate the mass ratios
Using the volume and density relationship:
\[ M_1 = \frac{4}{3} \pi R_1^3 d_1, \quad M_2 = \frac{4}{3} \pi R_2^3 d_2 \]
The ratio of masses \( M_1 : M_2 \) becomes:
\[ \frac{M_1}{M_2} = \frac{R_1^3 d_1}{R_2^3 d_2} \]

### Step 5: Substitute the ratios into the gravity formula
Now substituting the mass ratio into the gravity formula:
\[ \frac{g_1}{g_2} = \frac{M_1}{M_2} \cdot \frac{R_2^2}{R_1^2} \]
Substituting the expressions we derived:
\[ \frac{g_1}{g_2} = \frac{R_1^3 d_1}{R_2^3 d_2} \cdot \frac{R_2^2}{R_1^2} \]
This simplifies to:
\[ \frac{g_1}{g_2} = \frac{R_1 d_1}{R_2 d_2} \]

### Step 6: Substitute the known ratios
Now substituting the known ratios:
- \( R_1 : R_2 = 4 : 1 \) implies \( R_1 = 4k \) and \( R_2 = k \)
- \( d_1 : d_2 = 1 : 2 \) implies \( d_1 = m \) and \( d_2 = 2m \)

Substituting these into the ratio:
\[ \frac{g_1}{g_2} = \frac{4k \cdot m}{k \cdot 2m} = \frac{4}{2} = 2 \]

### Final Answer
Thus, the ratio of the acceleration due to gravity on the two planets is:
\[ g_1 : g_2 = 2 : 1 \]