Time period of revolution of a nearest satellite around a planet of radius R is T . Period of revolution around another planet, whose radius is 3R but having same density is

Options

A. `T`
B. `3T`
C. `9T`
D. `3sqrt(3)T`

Answer

To find the period of revolution of a satellite around a planet with a radius of \(3R\) and the same density as the first planet, we can follow these steps:

### Step 1: Understand the relationship between the period, radius, and density
The time period \(T\) of a satellite orbiting a planet can be expressed using the formula:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where \(r\) is the orbital radius, \(G\) is the gravitational constant, and \(M\) is the mass of the planet.

### Step 2: Express the mass in terms of density
The mass \(M\) of a planet can be expressed in terms of its density \(\rho\) and volume. The volume \(V\) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass \(M\) can be written as:
\[ M = \rho V = \rho \left(\frac{4}{3} \pi r^3\right) \]

### Step 3: Substitute mass into the period formula
Substituting the expression for mass into the period formula, we get:
\[ T = 2\pi \sqrt{\frac{r^3}{G \cdot \rho \left(\frac{4}{3} \pi r^3\right)}} \]
This simplifies to:
\[ T = 2\pi \sqrt{\frac{3}{4\pi G \rho}} \]
Notice that this shows the time period \(T\) is independent of the radius \(r\) when density \(\rho\) is constant.

### Step 4: Calculate the period for the second planet
For the second planet with radius \(3R\) and the same density \(\rho\), the time period \(T_2\) can be expressed similarly:
\[ T_2 = 2\pi \sqrt{\frac{3}{4\pi G \rho}} \]
Since this expression is the same as that for the first planet, we have:
\[ T_2 = T \]

### Conclusion
Thus, the period of revolution around the second planet is the same as that of the first planet:
\[ T_2 = T \]