Gravitational potential at a height R from the surface of the earth will be [Take M = mass of the earth, R = radius of the earth]

Options

A. `(-GM)/(2R)`
B. `(-GM)/(R )`
C. `(-GM)/(4R)`
D. `-GM`

Answer

To find the gravitational potential at a height \( R \) from the surface of the Earth, we can follow these steps:

### Step 1: Understand the formula for gravitational potential
The gravitational potential \( U \) at a distance \( r \) from the center of a mass \( M \) is given by the formula:
\[ U = -\frac{GM}{r} \]
where \( G \) is the gravitational constant.

### Step 2: Define the parameters
In this case:
- \( M \) is the mass of the Earth.
- \( R \) is the radius of the Earth.
- The height \( h \) from the surface of the Earth is also \( R \).

### Step 3: Calculate the distance from the center of the Earth
The total distance from the center of the Earth to the point at height \( R \) is:
\[ r = R + h = R + R = 2R \]

### Step 4: Substitute the values into the gravitational potential formula
Now, substituting \( r = 2R \) into the gravitational potential formula:
\[ U = -\frac{GM}{2R} \]

### Step 5: Conclusion
Thus, the gravitational potential at a height \( R \) from the surface of the Earth is:
\[ U = -\frac{GM}{2R} \]