A particle of mass 10g is kept on the surface of a uniform sphere of masss 100kg and radius 10cm. Find the work to be done against the gravitational force between them to take the particel far away from the sphere (you may take `G = 6.67xx10^(-11) Nm^2 /kg^2)`

Options

A. `13.34xx10^(-10)J`
B. `3.33xx10^(-10)J`
C. `6.67xx10^(-9)J`
D. `6.67xx10^(-10)J`

Explanation

To find the work done against the gravitational force to take a particle of mass 10 g far away from a uniform sphere of mass 100 kg and radius 10 cm, we can use the concept of gravitational potential energy.

### Step-by-Step Solution:

1. **Convert Units**:
- The mass of the particle, \( m = 10 \, \text{g} = 10 \times 10^{-3} \, \text{kg} = 0.01 \, \text{kg} \).
- The mass of the sphere, \( M = 100 \, \text{kg} \).
- The radius of the sphere, \( r = 10 \, \text{cm} = 10 \times 10^{-2} \, \text{m} = 0.1 \, \text{m} \).

2. **Gravitational Potential Energy Formula**:
The gravitational potential energy \( U \) between two masses is given by:
\[ U = -\frac{G M m}{r} \]
where \( G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \).

3. **Calculate Initial Gravitational Potential Energy**:
Substitute the values into the formula:
\[ U = -\frac{(6.67 \times 10^{-11}) (100) (0.01)}{0.1} \]
Simplifying this:
\[ U = -\frac{(6.67 \times 10^{-11}) (1)}{0.1} = -6.67 \times 10^{-10} \, \text{J} \]

4. **Final Gravitational Potential Energy**:
When the particle is taken far away from the sphere, the gravitational potential energy approaches zero:
\[ U_{\text{final}} = 0 \]

5. **Calculate Work Done**:
The work done \( W \) against the gravitational force is the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} = 0 - (-6.67 \times 10^{-10}) = 6.67 \times 10^{-10} \, \text{J} \]

### Final Answer:
The work done against the gravitational force to take the particle far away from the sphere is:
\[ W = 6.67 \times 10^{-10} \, \text{J} \]