An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kelper's Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show usnig dimensional analysis, that `T = (k)/(R ) sqrt((r^3)/(g)),` Where k is a dimensionless constant and g is acceleration due to gravity.

Explanation

To show that \( T = \frac{k}{R} \sqrt{\frac{r^3}{g}} \) using dimensional analysis, we will follow these steps:

### Step 1: Identify the quantities involved
We have:
- \( T \): Time period of the satellite (dimension: [T])
- \( r \): Radius of the orbit (dimension: [L])
- \( g \): Acceleration due to gravity (dimension: [L T^{-2}])
- \( R \): Radius of the planet (dimension: [L])
...