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

Explanation

To find the work done against the gravitational force to take a particle of mass 10g far away from a uniform sphere of mass 100kg and radius 10cm, we can follow these steps:

### Step 1: Understand the Problem
We need to calculate the work done to move a particle from a distance \( r \) (the surface of the sphere) to infinity, where the gravitational potential energy becomes zero.

### Step 2: Identify the Variables
- Mass of the particle, \( m = 10 \text{ g} = 0.01 \text{ kg} \) (since 1 g = 0.001 kg)
- Mass of the sphere, \( M = 100 \text{ kg} \)
- Radius of the sphere, \( r = 10 \text{ cm} = 0.1 \text{ m} \)
- Gravitational constant, \( G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2 \)

### Step 3: Calculate the Initial Gravitational Potential Energy
The gravitational potential energy \( U \) between two masses is given by the formula:
\[ U = -\frac{G M m}{r} \]
Substituting the values:
\[ U = -\frac{(6.67 \times 10^{-11}) \times (100) \times (0.01)}{0.1} \]

### Step 4: Simplify the Calculation
Calculating the above expression:
\[ U = -\frac{(6.67 \times 10^{-11}) \times (100) \times (0.01)}{0.1} = -\frac{6.67 \times 10^{-11} \times 1}{0.1} = -6.67 \times 10^{-10} \text{ J} \]

### Step 5: Calculate the Work Done
The work done \( W \) to move the particle to infinity is equal to the change in potential energy:
\[ W = U_{\text{final}} - U_{\text{initial}} \]
Since \( U_{\text{final}} = 0 \) (at infinity):
\[ W = 0 - (-6.67 \times 10^{-10}) = 6.67 \times 10^{-10} \text{ J} \]

### Step 6: Final Answer
Thus, the work done against the gravitational force to take the particle far away from the sphere is:
\[ W = 6.67 \times 10^{-10} \text{ J} \]