The energy required to remove a body of mass m from earth's surfac is/are equal to

Options

A. `-(GMm)/(R )`
B. `mgR`
C. `-mgR`
D. none of these

Answer

To find the energy required to remove a body of mass \( m \) from the surface of the Earth to an infinite distance away, we need to calculate the work done against the gravitational force. Here’s a step-by-step solution:

### Step 1: Understand the Concept of 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:
\[ U = -\frac{GMm}{r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the object,
- \( r \) is the distance from the center of the Earth.

### Step 2: Calculate the Potential Energy at Earth's Surface
At the surface of the Earth, the distance \( r \) is equal to the radius of the Earth (\( R \)):
\[ U_{\text{surface}} = -\frac{GMm}{R} \]

### Step 3: Determine the Potential Energy at Infinite Distance
At an infinite distance from the Earth, the potential energy is:
\[ U_{\text{infinity}} = 0 \]

### Step 4: Calculate the Change in Potential Energy
The work done (or energy required) to remove the mass \( m \) from the surface of the Earth to an infinite distance is equal to the change in potential energy:
\[ W = U_{\text{infinity}} - U_{\text{surface}} = 0 - \left(-\frac{GMm}{R}\right) = \frac{GMm}{R} \]

### Step 5: Relate to Gravitational Acceleration
We can express \( \frac{GM}{R} \) in terms of gravitational acceleration \( g \) at the surface of the Earth:
\[ g = \frac{GM}{R} \]
Thus, the energy required to remove the mass \( m \) becomes:
\[ W = mg \]

### Final Answer
The energy required to remove a body of mass \( m \) from the Earth's surface to an infinite distance is:
\[ W = mgR \]