A particle of mass 'm' is raised from the surface of earth to a height h = 2R. Find work done by some external agent in the process. Here, R is the radius of earth and `g` the acceleration due to gravity on earth's surface.

Explanation

To find the work done by an external agent in raising a particle of mass 'm' from the surface of the Earth to a height \( h = 2R \), where \( R \) is the radius of the Earth, we can follow these steps:

### Step 1: Understand the concept of gravitational potential energy
The gravitational potential energy \( U \) of an object at a height \( h \) above the Earth's surface is given by the formula:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth.

### Step 2: Determine the initial and final positions
- The initial position is at the surface of the Earth, where \( r = R \).
- The final position is at a height \( h = 2R \), so the distance from the center of the Earth at this height is \( r = R + h = R + 2R = 3R \).

### Step 3: Calculate the initial potential energy
At the surface of the Earth:
\[ U_i = -\frac{GMm}{R} \]

### Step 4: Calculate the final potential energy
At the height \( h = 2R \):
\[ U_f = -\frac{GMm}{3R} \]

### Step 5: Calculate the change in potential energy
The work done by the external agent is equal to the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values we calculated:
\[ W = \left(-\frac{GMm}{3R}\right) - \left(-\frac{GMm}{R}\right) \]
\[ W = -\frac{GMm}{3R} + \frac{GMm}{R} \]
To combine these fractions, we can write \( \frac{GMm}{R} \) as \( \frac{3GMm}{3R} \):
\[ W = -\frac{GMm}{3R} + \frac{3GMm}{3R} = \frac{2GMm}{3R} \]

### Step 6: Relate \( GM \) to \( g \)
We know that \( g = \frac{GM}{R^2} \). Therefore, \( GM = gR^2 \). Substituting this into our expression for work:
\[ W = \frac{2gR^2m}{3R} = \frac{2gRm}{3} \]

### Final Answer
The work done by the external agent in raising the particle to a height \( h = 2R \) is:
\[ W = \frac{2gRm}{3} \]