A particle of mass 10g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them, to take the particle 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 solve the problem of finding the work done against the gravitational force to take a particle of mass 10 g away from a uniform sphere of mass 100 kg and radius 10 cm, we will follow these steps:

### Step 1: Convert the masses and radius to SI 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 \( r = 10 \, \text{cm} = 10 \times 10^{-2} \, \text{m} = 0.1 \, \text{m} \)

### Step 2: Understand the work done against gravitational force
The work done \( W \) to move the particle from the surface of the sphere to infinity can be calculated using the change in gravitational potential energy. The formula for gravitational potential energy \( U \) between two masses is given by:
\[ U = -\frac{G M m}{r} \]
where \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).

### Step 3: Calculate the initial potential energy at the surface of the sphere
Substituting the values into the potential energy formula:
\[ U_i = -\frac{G M m}{r} \]
\[ U_i = -\frac{(6.67 \times 10^{-11}) \times (100) \times (0.01)}{0.1} \]

### Step 4: Simplify the expression
Calculating the numerator:
\[ = (6.67 \times 10^{-11}) \times (100) \times (0.01) = 6.67 \times 10^{-11} \times 1 = 6.67 \times 10^{-11} \]
Now substituting back into the potential energy equation:
\[ U_i = -\frac{6.67 \times 10^{-11}}{0.1} = -6.67 \times 10^{-10} \, \text{J} \]

### Step 5: Calculate the final potential energy at infinity
At infinity, the potential energy \( U_f = 0 \).

### Step 6: Calculate the work done
The work done \( W \) is the change in potential energy:
\[ W = U_f - U_i = 0 - (-6.67 \times 10^{-10}) = 6.67 \times 10^{-10} \, \text{J} \]

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

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