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)`

Explanation

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

### Step 1: Understand the Weight and Its Variation
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. As we move to a height \( h \) above the Earth's surface, the value of \( g \) changes.

### Step 2: Express \( g \) at Height \( h \)
The acceleration due to gravity at a height \( h \) can be approximated as:
\[ g_h = g \left(1 - \frac{2h}{r}\right) \]
where \( g \) is the acceleration due to gravity at the surface of the Earth, and \( r \) is the radius of the Earth.

### Step 3: Write the Weight at Height \( h \)
Substituting \( g_h \) into the weight formula, we have:
\[ W = m \cdot g_h = m \cdot g \left(1 - \frac{2h}{r}\right) \]

### Step 4: Differentiate Weight with Respect to Height
To find the rate of change of weight with respect to height \( h \), we differentiate \( W \) 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 take them outside the differentiation:
\[ \frac{dW}{dh} = m \cdot g \cdot \frac{d}{dh} \left(1 - \frac{2h}{r}\right) \]

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

### Step 6: Final Expression
This gives us the rate of change of weight as:
\[ \frac{dW}{dh} = -\frac{2mg}{r} \]

### Step 7: Conclusion About Dependence on Height
From the expression \( \frac{dW}{dh} = -\frac{2mg}{r} \), we can see that the rate of change of weight does not depend on \( h \) (height) because \( h \) does not appear in the final expression. Therefore, we can conclude that:
\[ \frac{dW}{dh} \propto h^0 \]
which means it is independent of height.