Assuming that tha gravitational potential energy of an object at infinity is zero, the change in potential energy (final - initial) of an object of mass m, when taken to a height h from the surface of earth (of radius R), is given by

Options

A. `(GMm)/(R+H)`
B. `-(GMm)/(R+H)`
C. `(GMmh)/(R(R+H))`
D. mgh

Explanation

To find the change in gravitational potential energy (ΔU) of an object of mass \( m \) when taken to a height \( h \) from the surface of the Earth (with radius \( R \)), we can follow these steps:

### Step 1: Understand the Gravitational Potential Energy Formula
The gravitational potential energy \( U \) of an object in a gravitational field is given by the formula:
\[ U = -\frac{G M m}{r} \]
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 to the object.

### Step 2: Calculate the Initial Potential Energy (at the surface)
At the surface of the Earth (point A), the distance \( r \) is equal to the radius of the Earth \( R \). Therefore, the initial potential energy \( U_A \) is:
\[ U_A = -\frac{G M m}{R} \]

### Step 3: Calculate the Final Potential Energy (at height \( h \))
When the object is raised to a height \( h \) above the surface, the distance from the center of the Earth becomes \( R + h \). Thus, the final potential energy \( U_B \) is:
\[ U_B = -\frac{G M m}{R + h} \]

### Step 4: Find the Change in Potential Energy
The change in potential energy \( \Delta U \) is given by:
\[ \Delta U = U_B - U_A \]
Substituting the expressions for \( U_B \) and \( U_A \):
\[ \Delta U = \left(-\frac{G M m}{R + h}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{G M m}{R + h} + \frac{G M m}{R} \]

### Step 5: Combine the Terms
To combine the terms, we can find a common denominator:
\[ \Delta U = G M m \left( \frac{1}{R} - \frac{1}{R + h} \right) \]
\[ = G M m \left( \frac{(R + h) - R}{R(R + h)} \right) \]
\[ = G M m \left( \frac{h}{R(R + h)} \right) \]

### Final Expression
Thus, the change in potential energy when the object is taken to a height \( h \) from the surface of the Earth is:
\[ \Delta U = \frac{G M m h}{R(R + h)} \]