A body of mass `m` is lifted up from the surface of earth to a height three times the radius of the earth `R`. The change in potential energy of the body is

Options

A. `3 mg R`
B. `(5)/(4) mgR`
C. `(3)/(4) mgR`
D. `2mgR`

Explanation

To find the change in potential energy of a body of mass \( m \) when it is lifted from the surface of the Earth to a height of \( 3R \) (where \( R \) is the radius of the Earth), we can follow these steps:

### Step 1: Understand the Initial and Final Positions
The initial position of the body is at the surface of the Earth, which is at a distance \( R \) from the center of the Earth. The final position is at a height of \( 3R \) above the surface of the Earth, which means the total distance from the center of the Earth at this height is \( R + 3R = 4R \).

### Step 2: Write 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 gravitational constant and \( M \) is the mass of the Earth.

### Step 3: Calculate Initial Potential Energy
At the surface of the Earth (initial position):
\[ U_i = -\frac{G M m}{R} \]

### Step 4: Calculate Final Potential Energy
At the height of \( 3R \) above the surface (final position):
\[ U_f = -\frac{G M m}{4R} \]

### Step 5: Calculate the Change in Potential Energy
The change in potential energy \( \Delta U \) is given by:
\[ \Delta U = U_f - U_i \]
Substituting the values we calculated:
\[ \Delta U = \left(-\frac{G M m}{4R}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{G M m}{4R} + \frac{G M m}{R} \]
\[ \Delta U = \frac{G M m}{R} - \frac{G M m}{4R} \]
Combining the fractions:
\[ \Delta U = \frac{G M m}{R} \left(1 - \frac{1}{4}\right) = \frac{G M m}{R} \cdot \frac{3}{4} \]
Thus, we have:
\[ \Delta U = \frac{3 G M m}{4R} \]

### Final Answer
The change in potential energy of the body when lifted to a height of \( 3R \) is:
\[ \Delta U = \frac{3 G M m}{4R} \]

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