If R is radius of the earth, the height above the surface of the earth where the weight of a body is `36%` less than its weight on the surface of the earth is

Options

A. 4 R/5
B. R/5
C. R/6
D. R/4

Explanation

To solve the problem, we need to find the height \( h \) above the surface of the Earth where the weight of a body is 36% less than its weight on the surface of the Earth.

### Step-by-Step Solution:

1. **Understanding Weight Reduction**:
- Let \( W \) be the weight of the body on the surface of the Earth.
- If the weight is 36% less at height \( h \), then the weight at height \( h \) is:
\[ W_h = W - 0.36W = 0.64W \]

2. **Weight and Acceleration due to Gravity**:
- The weight of the body at height \( h \) can also be expressed in terms of the acceleration due to gravity at that height:
\[ W_h = m \cdot g_h \]
- On the surface of the Earth, the weight is:
\[ W = m \cdot g \]
- Therefore, we can write:
\[ \frac{W_h}{W} = \frac{g_h}{g} \]
- Substituting the values, we have:
\[ \frac{0.64W}{W} = \frac{g_h}{g} \]
- This simplifies to:
\[ \frac{g_h}{g} = 0.64 \]

3. **Expression for Gravitational Acceleration**:
- The acceleration due to gravity at height \( h \) is given by:
\[ g_h = \frac{GM}{(R + h)^2} \]
- The acceleration due to gravity on the surface is:
\[ g = \frac{GM}{R^2} \]
- Thus, we can set up the equation:
\[ \frac{g_h}{g} = \frac{GM/(R + h)^2}{GM/R^2} = \frac{R^2}{(R + h)^2} \]

4. **Setting Up the Equation**:
- From our previous result, we have:
\[ \frac{R^2}{(R + h)^2} = 0.64 \]
- Cross-multiplying gives:
\[ R^2 = 0.64(R + h)^2 \]

5. **Expanding and Rearranging**:
- Expanding the right side:
\[ R^2 = 0.64(R^2 + 2Rh + h^2) \]
- Rearranging gives:
\[ R^2 = 0.64R^2 + 1.28Rh + 0.64h^2 \]
- Moving all terms to one side:
\[ R^2 - 0.64R^2 - 1.28Rh - 0.64h^2 = 0 \]
- This simplifies to:
\[ 0.36R^2 - 1.28Rh - 0.64h^2 = 0 \]

6. **Using the Quadratic Formula**:
- This is a quadratic equation in \( h \):
\[ 0.64h^2 + 1.28Rh - 0.36R^2 = 0 \]
- Using the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 0.64 \), \( b = 1.28R \), and \( c = -0.36R^2 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (1.28R)^2 - 4(0.64)(-0.36R^2) \]
\[ = 1.6384R^2 + 0.9216R^2 = 2.56R^2 \]
- Thus:
\[ h = \frac{-1.28R \pm \sqrt{2.56R^2}}{2 \cdot 0.64} \]
\[ = \frac{-1.28R \pm 1.6R}{1.28} \]

7. **Calculating the Height**:
- Taking the positive root:
\[ h = \frac{0.32R}{1.28} = \frac{R}{4} \]

### Final Answer:
The height \( h \) above the surface of the Earth where the weight of a body is 36% less than its weight on the surface of the Earth is:
\[ h = \frac{R}{4} \]