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 Initial and Final States
- The initial position of the particle is at the surface of the Earth, where the potential energy (PE) can be calculated using the formula:
\[ PE_{\text{initial}} = -\frac{G M m}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

- The final position of the particle is at a height \( h = 2R \) above the surface. The distance from the center of the Earth to the particle at this height is \( R + h = R + 2R = 3R \).

### Step 2: Calculate the Final Potential Energy
- The potential energy at the final position is given by:
\[ PE_{\text{final}} = -\frac{G M m}{3R} \]

### Step 3: Calculate the Change in Potential Energy
- The work done by the external agent is equal to the change in potential energy:
\[ W = PE_{\text{final}} - PE_{\text{initial}} \]
Substituting the values we have:
\[ W = \left(-\frac{G M m}{3R}\right) - \left(-\frac{G M m}{R}\right) \]
\[ W = -\frac{G M m}{3R} + \frac{G M m}{R} \]

### Step 4: Simplify the Expression
- To combine the terms, we find a common denominator:
\[ W = \left(-\frac{G M m}{3R} + \frac{3G M m}{3R}\right) = \frac{2G M m}{3R} \]

### Step 5: Relate to Gravitational Acceleration
- We know that \( g = \frac{G M}{R^2} \), thus \( G M = g R^2 \). Substituting this into the work done:
\[ W = \frac{2(g R^2) m}{3R} = \frac{2g m R}{3} \]

### Step 6: Substitute \( h = 2R \)
- Since \( h = 2R \), we can express the work done in terms of \( h \):
\[ W = \frac{2g m}{3} \cdot \frac{h}{2} = \frac{gh m}{3} \]

### Final Result
- Therefore, the work done by the external agent in raising the particle to a height \( h = 2R \) is:
\[ W = \frac{2g m}{3} \]