How long will a satellite, placed in a circular orbit of radius that is `(1/4)^(th)` the radius of a geostationary satellite, take to complete one revolution around the earth?

Options

A. 12 hours
B. 6 hours
C. 3 hours
D. 4 days

Explanation

To solve the problem of how long a satellite in a circular orbit with a radius that is one-fourth the radius of a geostationary satellite takes to complete one revolution around the Earth, we can use Kepler's Third Law of planetary motion. Here's a step-by-step solution:

### Step 1: Understand the relationship between time period and radius
According to Kepler's Third Law, the square of the time period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit. Mathematically, this can be expressed as:
\[ T^2 \propto r^3 \]
This means:
\[ T^2 = k \cdot r^3 \]
where \( k \) is a constant.

### Step 2: Define the known values
For a geostationary satellite:
- The time period \( T_1 \) is 24 hours (or 86400 seconds).
- The radius \( r_1 \) is the radius of the geostationary orbit.

For the satellite in question:
- The radius \( r_2 \) is \( \frac{1}{4} r_1 \).

### Step 3: Set up the equations using Kepler's Law
From Kepler's law, we can write:
1. For the geostationary satellite:
\[ T_1^2 = k \cdot r_1^3 \]
2. For the satellite in question:
\[ T_2^2 = k \cdot r_2^3 \]

### Step 4: Substitute \( r_2 \) into the equation
Since \( r_2 = \frac{1}{4} r_1 \), we can substitute this into the second equation:
\[ T_2^2 = k \cdot \left(\frac{1}{4} r_1\right)^3 \]
\[ T_2^2 = k \cdot \frac{1}{64} r_1^3 \]

### Step 5: Relate \( T_2 \) to \( T_1 \)
Now, we can relate \( T_2^2 \) to \( T_1^2 \):
From the first equation, we know:
\[ T_1^2 = k \cdot r_1^3 \]
Now, dividing the second equation by the first:
\[ \frac{T_2^2}{T_1^2} = \frac{k \cdot \frac{1}{64} r_1^3}{k \cdot r_1^3} \]
This simplifies to:
\[ \frac{T_2^2}{T_1^2} = \frac{1}{64} \]

### Step 6: Solve for \( T_2 \)
Taking the square root of both sides:
\[ \frac{T_2}{T_1} = \frac{1}{8} \]
Thus:
\[ T_2 = \frac{T_1}{8} \]
Substituting \( T_1 = 24 \) hours:
\[ T_2 = \frac{24 \text{ hours}}{8} = 3 \text{ hours} \]

### Conclusion
The time taken by the satellite to complete one revolution around the Earth is **3 hours**.

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