A particle is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done per unit mass against the gravitational force between them, to take the particle far away from the sphere (you may take `h=6.67 xx 10^(-11) "Nm"^(2) "kg"^(-2)`)

Options

A. `13.34 xx 10^(-10) J`
B. `3.33 xx 10^(-10) J`
C. `6.67 xx 10^(-9)J`
D. `6.67 xx 10^(-8) J`

Explanation

To find the work done per unit mass against the gravitational force to take a particle far away from a uniform sphere, we can follow these steps:

### Step 1: Understand the Problem
We have a uniform sphere with a mass \( M = 100 \, \text{kg} \) and a radius \( R = 10 \, \text{cm} = 0.1 \, \text{m} \). We need to calculate the work done to move a particle of unit mass (1 kg) from the surface of the sphere to infinity.

### Step 2: Gravitational Potential Energy Formula
The gravitational potential energy \( U \) at a distance \( r \) from a mass \( M \) is given by the formula:
\[ U = -\frac{GMm}{r} \]
where:
- \( G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \) (gravitational constant),
- \( M \) is the mass of the sphere,
- \( m \) is the mass of the particle,
- \( r \) is the distance from the center of the sphere to the particle.

### Step 3: Calculate the Potential Energy at the Surface
At the surface of the sphere, the distance \( r \) is equal to the radius \( R \):
\[ r = R = 0.1 \, \text{m} \]
Substituting the values into the potential energy formula:
\[ U = -\frac{(6.67 \times 10^{-11}) \times (100) \times (1)}{0.1} \]
Calculating this gives:
\[ U = -\frac{6.67 \times 10^{-9}}{0.1} = -6.67 \times 10^{-8} \, \text{J} \]

### Step 4: Work Done to Move the Particle to Infinity
To move the particle from the surface of the sphere to infinity, we need to do work equal to the negative of the potential energy at the surface:
\[ W = -U = 6.67 \times 10^{-8} \, \text{J} \]

### Step 5: Work Done Per Unit Mass
Since we are interested in the work done per unit mass, we divide the total work done by the mass of the particle:
\[ \text{Work done per unit mass} = \frac{W}{m} = \frac{6.67 \times 10^{-8}}{1} = 6.67 \times 10^{-8} \, \text{J/kg} \]

### Final Answer
The work done per unit mass against the gravitational force to take the particle far away from the sphere is:
\[ \boxed{6.67 \times 10^{-8} \, \text{J/kg}} \]