Rate of change of weight near the earth's surface varies with height h as

Options

A. `h^@`
B. `h^(-1)`
C. `h^(1//2)`
D. `h^(-2)`

Answer

To solve the problem of how the rate of change of weight near the Earth's surface varies with height \( h \), we can follow these steps:

### Step 1: Understand the Weight and Acceleration Due to Gravity
The weight \( W \) of an object is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity. At the surface of the Earth, \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

### Step 2: Determine the Expression for \( g \) at Height \( h \)
As we move to a height \( h \) above the Earth's surface, the acceleration due to gravity \( g_h \) can be expressed as:
\[ g_h = g \left(1 - \frac{2h}{R}\right) \]
where \( R \) is the radius of the Earth. This formula shows that \( g \) decreases as we move away from the surface.

### Step 3: Write the Weight at Height \( h \)
The weight of the object at height \( h \) is then:
\[ W_h = m \cdot g_h = m \cdot g \left(1 - \frac{2h}{R}\right) \]

### Step 4: Differentiate Weight with Respect to Height
To find the rate of change of weight with respect to height \( h \), we need to differentiate \( W_h \) with respect to \( h \):
\[ \frac{dW}{dh} = \frac{d}{dh} \left(m \cdot g \left(1 - \frac{2h}{R}\right)\right) \]
Since \( m \) and \( g \) are constants, we can factor them out:
\[ \frac{dW}{dh} = m \cdot g \cdot \frac{d}{dh} \left(1 - \frac{2h}{R}\right) \]

### Step 5: Calculate the Derivative
Now, we differentiate \( 1 - \frac{2h}{R} \):
\[ \frac{d}{dh} \left(1 - \frac{2h}{R}\right) = 0 - \frac{2}{R} = -\frac{2}{R} \]
Thus, we have:
\[ \frac{dW}{dh} = m \cdot g \cdot \left(-\frac{2}{R}\right) \]

### Step 6: Express the Result
This gives us:
\[ \frac{dW}{dh} = -\frac{2mg}{R} \]
This expression indicates that the rate of change of weight with respect to height \( h \) is a constant value, which does not depend on \( h \).

### Conclusion
The rate of change of weight near the Earth's surface varies with height \( h \) as \( h^0 \) (which is a constant). Therefore, the correct answer is:
\[ \text{Rate of change of weight} \propto h^0 \]