A body weight 1400 gram weight on the surface of earth. How will it weight on the surface of a planet whose mass is `(2)/(7)` and radius is `(1)/(3)` that of the earth ?

Options

A. 0.45 kg wt
B. 0.9 kg wt
C. 1.8 kg wt
D. 3.6 kg wt

Answer

To find the weight of a body on the surface of a new planet with given mass and radius in relation to Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the weight from grams to kilograms**:
- The weight of the body on Earth is given as 1400 grams. To convert this to kilograms, we divide by 1000.
\[ \text{Weight on Earth} = \frac{1400 \text{ grams}}{1000} = 1.4 \text{ kg} \]

**Hint**: Remember that 1 kg = 1000 grams.

2. **Identify the mass and radius of the new planet**:
- The mass of the new planet is given as \(\frac{2}{7}\) times the mass of Earth.
- The radius of the new planet is given as \(\frac{1}{3}\) times the radius of Earth.

3. **Use the formula for weight**:
- The weight of an object on the surface of a planet is given by:
\[ W = m \cdot g \]
where \(g\) is the acceleration due to gravity on that planet.

4. **Relate the gravitational acceleration on the new planet to that on Earth**:
- The acceleration due to gravity \(g\) on the surface of a planet is given by:
\[ g = \frac{G \cdot M}{R^2} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(R\) is the radius of the planet.

5. **Set up the ratio of gravitational accelerations**:
- The ratio of gravitational acceleration on the new planet to that on Earth can be expressed as:
\[ \frac{g_{\text{planet}}}{g_{\text{Earth}}} = \frac{\frac{G \cdot \left(\frac{2}{7} M_{\text{Earth}}\right)}{\left(\frac{1}{3} R_{\text{Earth}}\right)^2}}{\frac{G \cdot M_{\text{Earth}}}{R_{\text{Earth}}^2}} \]
- Simplifying this gives:
\[ \frac{g_{\text{planet}}}{g_{\text{Earth}}} = \frac{\frac{2}{7} M_{\text{Earth}} \cdot R_{\text{Earth}}^2}{M_{\text{Earth}} \cdot \left(\frac{1}{3} R_{\text{Earth}}\right)^2} = \frac{\frac{2}{7}}{\frac{1}{9}} = \frac{2}{7} \cdot 9 = \frac{18}{7} \]

6. **Calculate the weight on the new planet**:
- Now substituting back into the weight equation:
\[ W_{\text{planet}} = W_{\text{Earth}} \cdot \frac{g_{\text{planet}}}{g_{\text{Earth}}} \]
\[ W_{\text{planet}} = 1.4 \cdot \frac{18}{7} \]
- Performing the multiplication:
\[ W_{\text{planet}} = 1.4 \cdot \frac{18}{7} = \frac{1.4 \cdot 18}{7} = \frac{25.2}{7} \approx 3.6 \text{ kg} \]

### Final Answer:
The weight of the body on the surface of the new planet is approximately **3.6 kg**.

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