A satellite can be in a geostationary orbit around earth in an orbit of radius `r`. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit aroun earth radius

Options

A. `r/2`
B. `r/(2sqrt2)`
C. `r/(4)^(1//3)`
D. `r/(2)^(1//3)`

Explanation

To solve the problem of determining the new radius of a geostationary satellite when the angular velocity of the Earth doubles, we can follow these steps:

### Step 1: Understand the initial conditions
A geostationary satellite orbits the Earth with an orbital radius \( r \) and has an angular velocity \( \omega \) that matches the Earth's rotation. The period of the satellite's orbit \( T \) is equal to the rotational period of the Earth (24 hours).

### Step 2: Relate angular velocity and period
The relationship between the angular velocity \( \omega \) and the period \( T \) is given by:
\[ T = \frac{2\pi}{\omega} \]
If the angular velocity of the Earth doubles, the new angular velocity \( \omega' \) becomes:
\[ \omega' = 2\omega \]
The new period \( T' \) corresponding to this angular velocity is:
\[ T' = \frac{2\pi}{\omega'} = \frac{2\pi}{2\omega} = \frac{T}{2} \]

### Step 3: Apply Kepler's Third Law
According to Kepler's Third Law, the square of the period of orbit \( T \) is proportional to the cube of the radius \( r \):
\[ T^2 \propto r^3 \]
This can be expressed as:
\[ \frac{T'^2}{T^2} = \frac{r'^3}{r^3} \]
Substituting \( T' = \frac{T}{2} \):
\[ \left(\frac{T}{2}\right)^2 = \frac{r'^3}{r^3} \]
This simplifies to:
\[ \frac{T^2}{4} = \frac{r'^3}{r^3} \]

### Step 4: Rearranging the equation
From the above equation, we can rearrange it to find \( r' \):
\[ r'^3 = \frac{T^2}{4} \cdot r^3 \]
Taking the cube root of both sides:
\[ r' = r \left(\frac{1}{2}\right)^{2/3} \]

### Step 5: Final expression
Thus, we can express \( r' \) as:
\[ r' = r \cdot \frac{1}{2^{2/3}} = r \cdot \frac{1}{\sqrt[3]{4}} \]

### Conclusion
The new radius \( r' \) of the geostationary orbit when the angular velocity of the Earth doubles is:
\[ r' = r \cdot \frac{1}{\sqrt[3]{4}} \]