A satellite is orbiting just above the surface of the earth with period T. If d is the average density of the earth and G is the universal constant of gravitation, the quantity `(3pi)/(Gd)` represents its

Options

A. periodic time
B. square of the period
C. cube of the periodic time
D. square root of the period

Explanation

To solve the problem, we need to derive the relationship between the period \( T \) of a satellite orbiting just above the surface of the Earth and the average density \( d \) of the Earth, using the universal gravitational constant \( G \).

### Step-by-Step Solution:

1. **Understanding the Period of a Satellite**:
The period \( T \) of a satellite in orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{R^3}{GM}} \]
where \( R \) is the radius of the Earth and \( M \) is the mass of the Earth.

2. **Expressing Mass in Terms of Density**:
The mass \( M \) of the Earth can be expressed in terms of its density \( d \) and volume. The volume \( V \) of the Earth (assuming it is a sphere) is:
\[ V = \frac{4}{3} \pi R^3 \]
Therefore, the mass \( M \) can be written as:
\[ M = d \cdot V = d \cdot \frac{4}{3} \pi R^3 \]

3. **Substituting Mass into the Period Formula**:
Now, substitute \( M \) into the period formula:
\[ T = 2\pi \sqrt{\frac{R^3}{G \left( d \cdot \frac{4}{3} \pi R^3 \right)}} \]

4. **Simplifying the Expression**:
Simplifying the expression inside the square root:
\[ T = 2\pi \sqrt{\frac{R^3}{\frac{4}{3} \pi G d R^3}} = 2\pi \sqrt{\frac{3}{4\pi G d}} \]
The \( R^3 \) terms cancel out.

5. **Final Expression for Period**:
Thus, we have:
\[ T = 2\pi \sqrt{\frac{3}{4\pi G d}} \]
Squaring both sides gives:
\[ T^2 = 4\pi^2 \cdot \frac{3}{4\pi G d} = \frac{3\pi}{G d} \]

6. **Identifying the Quantity**:
From the equation \( T^2 = \frac{3\pi}{G d} \), we can see that:
\[ \frac{3\pi}{G d} = T^2 \]
Therefore, the quantity \( \frac{3\pi}{G d} \) represents the square of the period \( T^2 \).

### Conclusion:
The quantity \( \frac{3\pi}{Gd} \) represents the **square of the period** of the satellite orbiting just above the surface of the Earth.