The period of a satellite in a circular orbit of radius R is T. What is the period of another satellite in a circular orbit of radius 4 R ?

Options

A. 4 T
B. T/8
C. T/4
D. 8 T

Explanation

To find the period of a satellite in a circular orbit of radius \(4R\), we can use the formula for the period of a satellite in a circular orbit, which is given by:

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

where:
- \(T\) is the period of the satellite,
- \(r\) is the radius of the orbit,
- \(G\) is the gravitational constant,
- \(M\) is the mass of the Earth.

### Step 1: Write the formula for the period of the first satellite
For the first satellite with radius \(R\), the period \(T\) is:

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

### Step 2: Write the formula for the period of the second satellite
For the second satellite with radius \(4R\), we denote its period as \(T'\):

\[ T' = 2\pi \sqrt{\frac{(4R)^3}{GM}} \]

### Step 3: Simplify the expression for \(T'\)
Calculating \((4R)^3\):

\[ (4R)^3 = 64R^3 \]

Now, substituting this back into the formula for \(T'\):

\[ T' = 2\pi \sqrt{\frac{64R^3}{GM}} \]

### Step 4: Factor out the square root
We can simplify this further:

\[ T' = 2\pi \sqrt{64} \sqrt{\frac{R^3}{GM}} \]

Since \(\sqrt{64} = 8\), we have:

\[ T' = 8 \cdot 2\pi \sqrt{\frac{R^3}{GM}} \]

### Step 5: Substitute \(T\) back into the equation
We know that \(T = 2\pi \sqrt{\frac{R^3}{GM}}\), so we can substitute \(T\) into the equation:

\[ T' = 8T \]

### Conclusion
Thus, the period of the satellite in a circular orbit of radius \(4R\) is:

\[ T' = 8T \]