A particle of mass 'm' is raised to a height `h = R` from the surface of earth. Find increase in potential energy. `R =` radius of earth. `g =` acceleration due to gravity on the surface of earth.

Explanation

To find the increase in potential energy of a particle of mass 'm' raised to a height 'h' equal to the radius of the Earth 'R', we can follow these steps:

### Step 1: Understand the Initial Potential Energy
The gravitational potential energy (U) at a distance 'r' from the center of the Earth is given by the formula:
\[ U = -\frac{GMm}{r} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the particle,
- \( r \) is the distance from the center of the Earth.

At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth \( R \). Thus, the initial potential energy \( U_1 \) when the particle is at the surface is:
\[ U_1 = -\frac{GMm}{R} \]

### Step 2: Calculate the Final Potential Energy
When the particle is raised to a height \( h = R \), the new distance from the center of the Earth becomes \( r = R + R = 2R \). The potential energy \( U_2 \) at this height is:
\[ U_2 = -\frac{GMm}{2R} \]

### Step 3: Find the Change in Potential Energy
The increase in potential energy \( \Delta U \) is given by the difference between the final and initial potential energies:
\[ \Delta U = U_2 - U_1 \]
Substituting the values we found:
\[ \Delta U = \left(-\frac{GMm}{2R}\right) - \left(-\frac{GMm}{R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{GMm}{2R} + \frac{GMm}{R} \]
\[ \Delta U = \frac{GMm}{R} - \frac{GMm}{2R} \]
\[ \Delta U = \frac{GMm}{2R} \]

### Step 4: Express in Terms of 'g'
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{GM}{R^2} \]
Substituting \( GM \) from the equation of \( g \):
\[ \Delta U = \frac{gR^2m}{2R} = \frac{gRm}{2} \]

### Final Answer
Thus, the increase in potential energy when the particle is raised to a height equal to the radius of the Earth is:
\[ \Delta U = \frac{gRm}{2} \]