If g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the surface of the earth to a height equal to the radius R of the earth, is

Options

A. `(1)/(2)mgR`
B. 2mg R (c )
C. mg R
D. `(1)/(4) mg R`

Explanation

To find the gain in potential energy of an object of mass \( m \) raised from the surface of the Earth to a height equal to the radius \( R \) of the Earth, we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy
The gravitational potential energy \( U \) of an object 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, and \( m \) is the mass of the object.

### Step 2: Calculate initial potential energy at the Earth's surface
At the Earth's surface, the distance \( r \) is equal to the radius of the Earth \( R \):
\[ U_{\text{initial}} = -\frac{G M m}{R} \]

### Step 3: Calculate potential energy at height \( R \)
When the object is raised to a height equal to the radius of the Earth, the distance from the center of the Earth becomes \( 2R \):
\[ U_{\text{final}} = -\frac{G M m}{2R} \]

### Step 4: Calculate the change in potential energy
The gain in potential energy \( \Delta U \) is given by the difference between the final and initial potential energies:
\[ \Delta U = U_{\text{final}} - U_{\text{initial}} \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{G M m}{2R} + \frac{G M m}{R} \]
\[ \Delta U = \frac{G M m}{R} - \frac{G M m}{2R} \]
\[ \Delta U = \frac{G M m}{2R} \]

### Step 5: Substitute \( G \) using \( g \)
We know that \( g = \frac{G M}{R^2} \). Thus, \( G M = g R^2 \). Substituting this into our equation for \( \Delta U \):
\[ \Delta U = \frac{g R^2 m}{2R} = \frac{g R m}{2} \]

### Final Answer
The gain in potential energy of the object when raised to a height equal to the radius of the Earth is:
\[ \Delta U = \frac{g R m}{2} \]