A satellite of mass m is orbiting the earth in a circular orbit at a height 2R abvoe earth surface where R is the radius of the earth. If it starts losing energy at a constant rate `beta`, then it will fall on the earth surface after the time `t=(GMm)/(nbetaR)`. Assuming that the satellite is approximately in a circular orbit at all times, find the value of n.

Answer

To solve the problem, we need to analyze the energy changes of a satellite in a circular orbit around the Earth as it loses energy at a constant rate and eventually falls to the Earth's surface.

### Step-by-Step Solution:

1. **Understanding the Initial and Final Conditions:**
- The satellite is at a height of \(2R\) above the Earth's surface. Therefore, the distance from the center of the Earth is \(3R\) (since the radius of the Earth is \(R\)).
- The satellite will eventually fall to the Earth's surface, which is at a distance of \(R\) from the center of the Earth.

2. **Total Energy of the Satellite:**
- The total mechanical energy \(E\) of a satellite in orbit is given by:
\[ E = -\frac{GMm}{2r} \]
- For the initial position (at \(3R\)):
\[ E_i = -\frac{GMm}{2 \times 3R} = -\frac{GMm}{6R} \]
- For the final position (at \(R\)):
\[ E_f = -\frac{GMm}{2R} \]

3. **Change in Energy:**
- The change in energy as the satellite falls from its initial position to the final position can be expressed as:
\[ E_f - E_i = -\frac{GMm}{2R} - \left(-\frac{GMm}{6R}\right) \]
- Simplifying this:
\[ E_f - E_i = -\frac{GMm}{2R} + \frac{GMm}{6R} = -\frac{3GMm}{6R} + \frac{GMm}{6R} = -\frac{2GMm}{6R} = -\frac{GMm}{3R} \]

4. **Rate of Energy Loss:**
- The rate of energy loss is given as \(-\beta\). Therefore, we can write:
\[ \frac{dE}{dt} = -\beta \]
- This implies:
\[ dE = -\beta dt \]

5. **Integrating the Energy Change:**
- Integrating from the initial energy \(E_i\) to the final energy \(E_f\):
\[ E_f - E_i = -\beta t \]
- Substituting the change in energy:
\[ -\frac{GMm}{3R} = -\beta t \]
- Rearranging gives:
\[ t = \frac{GMm}{3\beta R} \]

6. **Comparing with Given Time Expression:**
- The problem states that the time \(t\) can also be expressed as:
\[ t = \frac{GMm}{n\beta R} \]
- By comparing both expressions for \(t\):
\[ \frac{GMm}{3\beta R} = \frac{GMm}{n\beta R} \]
- This leads to:
\[ n = 3 \]

### Final Answer:
Thus, the value of \(n\) is \(3\).