What will be gain in potential energy of a body of mass m at a height equal to three times the radius 'R' of the earth ?

Options

A. mgR
B. 2mgR
C. `(mgR)/(3)`
D. `(3mgR)/(4)`

Answer

To find the gain in potential energy of a body of mass \( m \) at a height equal to three times the radius \( R \) of the Earth, we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy
The gravitational potential energy \( U \) at a height \( h \) above the Earth's surface is given by the formula:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the body, and \( r \) is the distance from the center of the Earth to the body.

### Step 2: Determine the height above the Earth's surface
Given that the height \( h \) is equal to three times the radius of the Earth, we have:
\[ h = 3R \]

### Step 3: Calculate the distance from the center of the Earth
The distance \( r \) from the center of the Earth to the body at height \( h \) is:
\[ r = R + h = R + 3R = 4R \]

### Step 4: Calculate the initial potential energy at the surface of the Earth
At the surface of the Earth (where \( h = 0 \)):
\[ U_{\text{initial}} = -\frac{GMm}{R} \]

### Step 5: Calculate the potential energy at height \( h \)
At height \( h = 3R \) (where \( r = 4R \)):
\[ U_{\text{final}} = -\frac{GMm}{4R} \]

### Step 6: Calculate the gain in potential energy
The gain in potential energy \( \Delta U \) is given by:
\[ \Delta U = U_{\text{final}} - U_{\text{initial}} \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{GMm}{4R}\right) - \left(-\frac{GMm}{R}\right) \]
\[ \Delta U = -\frac{GMm}{4R} + \frac{4GMm}{4R} = \frac{3GMm}{4R} \]

### Step 7: Relate \( GM \) to \( g \)
We know that \( g = \frac{GM}{R^2} \), hence \( GM = gR^2 \). Substituting this into our expression for gain in potential energy:
\[ \Delta U = \frac{3gR^2 m}{4R} = \frac{3gRm}{4} \]

### Final Answer
Thus, the gain in potential energy of a body of mass \( m \) at a height equal to three times the radius of the Earth is:
\[ \Delta U = \frac{3gRm}{4} \]