Weight of a body on the surfaces of two planets is the same. If their densities are `d_1 and d_2` then the ratio of their radius

Options

A. `d_1/d_2`
B. `d_2/d_1`
C. `d_1^2/d_2^2`
D. `d_2^2/d_1^2`

Answer

To solve the problem, we need to find the ratio of the radii of two planets given that the weight of a body on their surfaces is the same and their densities are \(d_1\) and \(d_2\).

### Step-by-Step Solution:

1. **Understanding the Weight on the Surface of the Planets**:
The weight of a body on the surface of a planet is given by the formula:
\[ W = mg \]
where \(m\) is the mass of the body and \(g\) is the acceleration due to gravity on the surface of the planet.

2. **Acceleration Due to Gravity**:
The acceleration due to gravity \(g\) on the surface of a planet can be expressed as:
\[ g = \frac{GM}{R^2} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet.

3. **Relating Mass and Density**:
The mass \(M\) of a planet can also be expressed in terms of its density \(d\) and volume \(V\):
\[ M = d \cdot V \]
The volume of a spherical planet is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, we can write:
\[ M = d \cdot \frac{4}{3} \pi R^3 \]

4. **Substituting Mass in the Gravity Formula**:
Substituting the expression for mass into the formula for \(g\):
\[ g = \frac{G \left(d \cdot \frac{4}{3} \pi R^3\right)}{R^2} \]
Simplifying this gives:
\[ g = \frac{4}{3} \pi G d R \]

5. **Setting Up the Equation for Two Planets**:
For two planets with densities \(d_1\) and \(d_2\) and radii \(R_1\) and \(R_2\), we have:
\[ g_1 = \frac{4}{3} \pi G d_1 R_1 \quad \text{and} \quad g_2 = \frac{4}{3} \pi G d_2 R_2 \]
Since the weight of the body is the same on both planets, we have \(g_1 = g_2\).

6. **Equating the Two Expressions**:
Setting the two expressions for \(g\) equal to each other:
\[ \frac{4}{3} \pi G d_1 R_1 = \frac{4}{3} \pi G d_2 R_2 \]
The constants \(\frac{4}{3} \pi G\) cancel out, leading to:
\[ d_1 R_1 = d_2 R_2 \]

7. **Finding the Ratio of the Radii**:
Rearranging the equation gives:
\[ \frac{R_1}{R_2} = \frac{d_2}{d_1} \]

### Final Answer:
The ratio of the radii of the two planets is:
\[ \frac{R_1}{R_2} = \frac{d_2}{d_1} \]