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 \( h = R \) (where \( R \) is the radius of the Earth), we can follow these steps:

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

### Step 2: Calculate Initial Potential Energy
When the particle is at the surface of the Earth, the distance from the center of the Earth is \( r = R \). Therefore, the initial potential energy \( U_i \) is:
\[ U_i = -\frac{G M m}{R} \]

### Step 3: Calculate Final Potential Energy
When the particle is raised to a height \( h = R \), the new distance from the center of the Earth becomes \( r = R + R = 2R \). Thus, the final potential energy \( U_f \) is:
\[ U_f = -\frac{G M m}{2R} \]

### Step 4: Calculate the Increase in Potential Energy
The increase in potential energy \( \Delta U \) is given by the difference between the final and initial potential energies:
\[ \Delta U = U_f - U_i \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \]
\[ \Delta U = -\frac{G M m}{2R} + \frac{G M m}{R} \]
\[ \Delta U = \frac{G M m}{R} - \frac{G M m}{2R} \]
\[ \Delta U = \frac{G M m}{2R} \]

### Step 5: Relate to Acceleration due to Gravity
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
Thus, we can express \( G M \) in terms of \( g \):
\[ G M = g R^2 \]
Substituting this back into our expression for \( \Delta U \):
\[ \Delta U = \frac{g R^2 m}{2R} = \frac{g R m}{2} \]

### Final Result
The increase in potential energy when the particle is raised to a height \( h = R \) from the surface of the Earth is:
\[ \Delta U = \frac{g R m}{2} \]