The value of g varies with distance (r ) above the earth 's surface as

Options

A. `gprop(1)/(r^(2))`
B. `gpropr`
C. `gpropr^(2)`
D. `gpropr^(@)`

Answer

To determine how the value of gravitational acceleration \( g \) varies with distance \( r \) above the Earth's surface, we can follow these steps:

### Step 1: Understand the relationship between \( g \) and \( r \)
The gravitational acceleration \( g \) at a distance \( r \) from the center of the Earth can be expressed using the formula:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

### Step 2: Define the variables
- Let \( R \) be the radius of the Earth.
- When we are at the surface of the Earth, \( r = R \).
- When we are above the surface, \( r > R \).

### Step 3: Analyze the formula for \( g \)
When we are above the Earth's surface, the distance \( r \) from the center of the Earth is greater than \( R \). Therefore, we can write:
\[ g = \frac{GM}{r^2} \]
As \( r \) increases (i.e., as we move further away from the Earth's surface), the value of \( g \) decreases because \( g \) is inversely proportional to the square of the distance \( r \).

### Step 4: Identify the relationship
From the formula, we can see that:
\[ g \propto \frac{1}{r^2} \]
This indicates that as the distance \( r \) increases, the gravitational acceleration \( g \) decreases.

### Step 5: Conclusion
Thus, the value of \( g \) varies with distance \( r \) above the Earth's surface as:
\[ g \propto \frac{1}{r^2} \]
This means that gravitational acceleration decreases with the square of the distance from the center of the Earth.