The work that must be done in lifting a body of weight P from the surface of the earth to a height h is

Options

A. `(PRh)/(R-h)`
B. `(R+h)/(PRh)`
C. `(PRh)/(R+h)`
D. `(R-h)/(PRh)`

Answer

To find the work done in lifting a body of weight \( P \) from the surface of the Earth to a height \( h \), we can follow these steps:

### Step 1: Understand the Concept of Work Done
The work done in lifting an object against gravity is equal to the change in its potential energy. When an object is lifted to a height \( h \), its potential energy increases.

### Step 2: Define the Initial and Final Positions
- **Initial Position**: The object is at the surface of the Earth, where the distance from the center of the Earth is \( r \) (the radius of the Earth).
- **Final Position**: The object is lifted to a height \( h \), making the distance from the center of the Earth \( r + h \).

### Step 3: Write the Formula for Potential Energy
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{G M m}{r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the object.

### Step 4: Calculate Initial and Final Potential Energy
- **Initial Potential Energy** at the surface:
\[ U_i = -\frac{G M m}{r} \]
- **Final Potential Energy** at height \( h \):
\[ U_f = -\frac{G M m}{r + h} \]

### Step 5: Calculate the Work Done
The work done \( W \) in lifting the object is the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values:
\[ W = \left(-\frac{G M m}{r + h}\right) - \left(-\frac{G M m}{r}\right) \]
\[ W = \frac{G M m}{r} - \frac{G M m}{r + h} \]

### Step 6: Simplify the Expression
Taking \( G M m \) common:
\[ W = G M m \left( \frac{1}{r} - \frac{1}{r + h} \right) \]
Finding a common denominator:
\[ W = G M m \left( \frac{(r + h) - r}{r(r + h)} \right) \]
\[ W = G M m \left( \frac{h}{r(r + h)} \right) \]

### Step 7: Relate Weight to Mass
The weight \( P \) of the object is given by:
\[ P = mg \]
Thus, \( mg = P \). We can express \( G M \) in terms of \( g \):
\[ g = \frac{G M}{r^2} \implies G M = g r^2 \]
Substituting this back into our work done equation:
\[ W = \frac{g r^2 m h}{r(r + h)} = \frac{P h r}{r + h} \]

### Final Answer
The work done in lifting a body of weight \( P \) from the surface of the Earth to a height \( h \) is:
\[ W = \frac{P h r}{r + h} \]