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))`

Answer

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

### Step 1: Understand the condition for a geostationary satellite
A geostationary satellite has an angular velocity equal to that of the Earth. The angular velocity of the satellite (\( \omega_s \)) is given by:

\[ \omega_s = \sqrt{\frac{GM}{r^3}} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the radius of the orbit of the satellite.

### Step 2: Set up the equation for the new angular velocity
If the angular velocity of the Earth doubles, we denote the new angular velocity as \( \omega' = 2\omega \). For the satellite to remain geostationary with this new angular velocity, we can set up the following equation:

\[ \omega' = \sqrt{\frac{GM}{R^3}} \]

where \( R \) is the new radius of the satellite's orbit.

### Step 3: Relate the two angular velocities
Since the new angular velocity is double the original, we can write:

\[ 2\sqrt{\frac{GM}{r^3}} = \sqrt{\frac{GM}{R^3}} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:

\[ 4 \cdot \frac{GM}{r^3} = \frac{GM}{R^3} \]

### Step 5: Simplify the equation
We can cancel \( GM \) from both sides (assuming \( G \) and \( M \) are constant):

\[ 4 \cdot \frac{1}{r^3} = \frac{1}{R^3} \]

### Step 6: Rearrange to find the relationship between \( R \) and \( r \)
Rearranging the equation gives:

\[ R^3 = \frac{r^3}{4} \]

### Step 7: Take the cube root to find \( R \)
Taking the cube root of both sides results in:

\[ R = \frac{r}{\sqrt[3]{4}} \]

### Conclusion
Thus, the new radius \( R \) for the geostationary orbit when the Earth's angular velocity doubles is:

\[ R = \frac{r}{4^{1/3}} \]