Find work done in shifting a body of mass m from a height h above the earth's surface to a height 2h above the earth's surface..

Explanation

To find the work done in shifting a body of mass \( m \) from a height \( h \) above the Earth's surface to a height \( 2h \) above the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Potential Energy Formula**:
The gravitational potential energy \( U \) of a mass \( m \) at a height \( h \) above the Earth's surface is given by:
\[ U(h) = -\frac{GMm}{R + h} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth,
- \( h \) is the height above the Earth's surface.

2. **Calculate Initial Potential Energy \( U_i \)**:
The initial potential energy at height \( h \) is:
\[ U_i = U(h) = -\frac{GMm}{R + h} \]

3. **Calculate Final Potential Energy \( U_f \)**:
The final potential energy at height \( 2h \) is:
\[ U_f = U(2h) = -\frac{GMm}{R + 2h} \]

4. **Determine the Work Done \( W \)**:
The work done in moving the mass from height \( h \) to height \( 2h \) is equal to the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values of \( U_f \) and \( U_i \):
\[ W = \left(-\frac{GMm}{R + 2h}\right) - \left(-\frac{GMm}{R + h}\right) \]
Simplifying this gives:
\[ W = -\frac{GMm}{R + 2h} + \frac{GMm}{R + h} \]
\[ W = GMm \left(\frac{1}{R + h} - \frac{1}{R + 2h}\right) \]

5. **Combine the Fractions**:
To combine the fractions, we find a common denominator:
\[ W = GMm \left(\frac{(R + 2h) - (R + h)}{(R + h)(R + 2h)}\right) \]
Simplifying the numerator:
\[ W = GMm \left(\frac{h}{(R + h)(R + 2h)}\right) \]

6. **Final Expression for Work Done**:
Thus, the work done in shifting the body from height \( h \) to height \( 2h \) is:
\[ W = \frac{GMmh}{(R + h)(R + 2h)} \]