Two metallic spheres A and B are made up of same material and also have identical surfaces. Both spheres are heated to same temperature and are then placed in a chamber which is at a lower temperature, thermally insulated from each other. If ratio of masses of A and B is 3:1, then the ratio of their initial rate of cooling is

Options

A. `(1)/(sqrt3)`
B. `sqrt(3)//1 `
C. `1/3`
D. `(1/3)^(1/3)`

Explanation

To solve the problem, we need to find the ratio of the initial rates of cooling of two metallic spheres A and B, given that their masses are in the ratio of 3:1. We will use the principles of heat transfer and the Stefan-Boltzmann law.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Let the mass of sphere A be \( m_A \) and the mass of sphere B be \( m_B \).
- Given that \( \frac{m_A}{m_B} = \frac{3}{1} \), we can write \( m_A = 3m \) and \( m_B = m \) for some mass \( m \).

2. **Understand the Cooling Process**:
- Both spheres are made of the same material and have identical surfaces, meaning they have the same emissivity \( \epsilon \), and surface area \( A \).
- The initial rate of cooling can be described using Newton's Law of Cooling or the Stefan-Boltzmann Law, which states that the rate of heat loss \( \frac{dQ}{dt} \) is proportional to the surface area and the temperature difference to the fourth power.

3. **Apply the Stefan-Boltzmann Law**:
- The rate of heat loss for each sphere can be expressed as:
\[ \frac{dQ_A}{dt} = \epsilon \sigma A (T^4 - T_{env}^4) \]
\[ \frac{dQ_B}{dt} = \epsilon \sigma A (T^4 - T_{env}^4) \]
- However, since we are looking for the initial rate of cooling, we can simplify this to:
\[ \text{Rate of cooling} \propto \frac{1}{m^{1/3}} \]
- This is derived from the fact that the rate of cooling depends inversely on the cube root of the mass when all other factors are constant.

4. **Calculate the Ratio of Initial Rates of Cooling**:
- The initial rate of cooling for sphere A is:
\[ R_A \propto \frac{1}{(m_A)^{1/3}} = \frac{1}{(3m)^{1/3}} = \frac{1}{3^{1/3} m^{1/3}} \]
- The initial rate of cooling for sphere B is:
\[ R_B \propto \frac{1}{(m_B)^{1/3}} = \frac{1}{(m)^{1/3}} \]
- Now, we can find the ratio of the rates of cooling:
\[ \frac{R_A}{R_B} = \frac{\frac{1}{3^{1/3} m^{1/3}}}{\frac{1}{m^{1/3}}} = \frac{1}{3^{1/3}} \]

5. **Final Result**:
- Thus, the ratio of the initial rates of cooling of spheres A and B is:
\[ \frac{R_A}{R_B} = \frac{1}{3^{1/3}} \]

### Conclusion:
The ratio of the initial rate of cooling of spheres A and B is \( \frac{1}{3^{1/3}} \).