A body weight 7000gms on the surface of the earth. How much will it weight on the surface of a planet whose mass is 1/7 and radius half of the earth

Options

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

Answer

To find the weight of a body on the surface of a planet with a mass of \( \frac{1}{7} \) that of Earth and a radius of \( \frac{1}{2} \) that of Earth, we can follow these steps:

### Step 1: Understand the Weight on Earth
The weight of a body on the surface of the Earth is given as 7000 grams. We can convert this to kilograms for standard units:
\[ \text{Weight on Earth} = 7000 \text{ grams} = 7 \text{ kg} \]

### Step 2: Use the Formula for Acceleration due to Gravity
The formula for 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.

### Step 3: Calculate the Acceleration due to Gravity on the New Planet
Given:
- Mass of the new planet \( M' = \frac{1}{7} M \) (where \( M \) is the mass of Earth)
- Radius of the new planet \( R' = \frac{1}{2} R \) (where \( R \) is the radius of Earth)

Substituting these values into the formula for \( g' \) (acceleration due to gravity on the new planet):
\[ g' = \frac{G \cdot \left(\frac{1}{7} M\right)}{\left(\frac{1}{2} R\right)^2} \]
This simplifies to:
\[ g' = \frac{G \cdot \left(\frac{1}{7} M\right)}{\frac{1}{4} R^2} = \frac{4G \cdot M}{7R^2} = \frac{4}{7} g \]
where \( g \) is the acceleration due to gravity on Earth (approximately \( 9.8 \, \text{m/s}^2 \)).

### Step 4: Calculate the Value of \( g' \)
Now substituting the value of \( g \):
\[ g' = \frac{4}{7} \cdot 9.8 \approx 5.6 \, \text{m/s}^2 \]

### Step 5: Calculate the Weight on the New Planet
The weight of the body on the new planet can be calculated using the formula:
\[ \text{Weight} = m \cdot g' \]
Substituting the values:
\[ \text{Weight on new planet} = 7 \, \text{kg} \cdot 5.6 \, \text{m/s}^2 = 39.2 \, \text{N} \]
To convert this back to grams, we can use the fact that \( 1 \, \text{kg} = 1000 \, \text{grams} \):
\[ \text{Weight in grams} = 39.2 \, \text{N} \cdot \frac{1000 \, \text{grams}}{9.8 \, \text{N}} \approx 4000 \, \text{grams} \]

### Final Answer
The weight of the body on the surface of the new planet is approximately **4000 grams**.