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 Earth's surface, we can follow these steps:

### Step 1: Understand the concept of 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,
- \( M \) is the mass of the Earth,
- \( r \) is the distance from the center of the Earth to the mass \( m \) (which is equal to the radius of the Earth \( R \) when the mass is on the surface).

### Step 2: Calculate the potential energy at the Earth's surface
At the Earth's surface, the potential energy \( U \) can be expressed as:
\[ U = -\frac{G M m}{R} \]
Here, \( R \) is the radius of the Earth.

### Step 3: Determine the energy required to remove the mass to infinity
To remove the mass \( m \) from the Earth's surface to infinity, we need to provide enough energy to overcome the gravitational potential energy. At infinity, the potential energy is zero. Therefore, the energy \( E \) required is equal to the absolute value of the potential energy at the Earth's surface:
\[ E = -U = \frac{G M m}{R} \]

### Step 4: Substitute the expression for gravitational acceleration
We know that the gravitational acceleration \( g \) at the Earth's surface is given by:
\[ g = \frac{G M}{R^2} \]
From this, we can express \( G M \) as:
\[ G M = g R^2 \]
Substituting this into our expression for \( E \):
\[ E = \frac{g R^2 m}{R} = g R m \]

### Final Answer
Thus, the energy required to remove a body of mass \( m \) from the Earth's surface is:
\[ E = g R m \]

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