A body weighs 700gm-wt on the surface of the earth. How much will it weigh on the surface of a planet whose mass is 1/7 and radius half of the earth

Options

A. 200gm wt
B. 400 gm wt
C. 500 gm wt
D. 300 gm wt

Answer

To solve the problem, we need to determine the weight of a body on the surface of a planet with specific mass and radius characteristics compared to Earth. Here’s a step-by-step solution:

### Step 1: Understand the given data
- Weight of the body on Earth, \( W_E = 700 \, \text{g-wt} \)
- Mass of the planet, \( M_P = \frac{1}{7} M_E \) (where \( M_E \) is the mass of Earth)
- Radius of the planet, \( R_P = \frac{1}{2} R_E \) (where \( R_E \) is the radius of Earth)

### Step 2: Calculate the acceleration due to gravity on the planet
The formula for acceleration due to gravity \( g \) 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.

For Earth:
\[ g_E = \frac{G \cdot M_E}{R_E^2} \]

For the planet:
\[ g_P = \frac{G \cdot M_P}{R_P^2} \]

Substituting the values of \( M_P \) and \( R_P \):
\[ g_P = \frac{G \cdot \left(\frac{1}{7} M_E\right)}{\left(\frac{1}{2} R_E\right)^2} \]
\[ g_P = \frac{G \cdot \left(\frac{1}{7} M_E\right)}{\frac{1}{4} R_E^2} \]
\[ g_P = \frac{4G \cdot M_E}{7R_E^2} \]

### Step 3: Relate \( g_P \) to \( g_E \)
Since \( g_E = \frac{G \cdot M_E}{R_E^2} \), we can express \( g_P \) in terms of \( g_E \):
\[ g_P = \frac{4}{7} g_E \]

### Step 4: Substitute the value of \( g_E \)
Given that \( g_E \approx 9.8 \, \text{m/s}^2 \):
\[ g_P = \frac{4}{7} \cdot 9.8 \approx 5.6 \, \text{m/s}^2 \]

### Step 5: Calculate the weight on the planet
The weight \( W_P \) on the planet can be calculated using the formula:
\[ W_P = m \cdot g_P \]
Where \( m \) is the mass of the body. Since the weight on Earth is given as \( 700 \, \text{g-wt} \), we can convert this to mass:
\[ m = \frac{700 \, \text{g-wt}}{g_E} = \frac{700}{9.8} \approx 71.43 \, \text{g} \]

Now substituting \( m \) and \( g_P \):
\[ W_P = 71.43 \cdot 5.6 \approx 400 \, \text{g-wt} \]

### Final Answer
The weight of the body on the surface of the planet is approximately **400 g-wt**.

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