If the height of a satellite from the earth is negligible in comparison to the radius of the earth R , the orbital velocity of the satellite is

Options

A. `gR `
B. `g//R2`
C. `sqrt(g//R)`
D. `sqrt(gR)`

Answer

To find the orbital velocity of a satellite when its height from the Earth's surface is negligible compared to the radius of the Earth (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Forces on the Satellite**:
- A satellite in orbit experiences two main forces: the gravitational force pulling it towards the Earth and the centrifugal force due to its circular motion.

2. **Gravitational Force**:
- The gravitational force acting on the satellite can be expressed as:
\[ F_{\text{gravity}} = \frac{G M m}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite (which is approximately equal to the radius of the Earth \( R \) when height \( h \) is negligible).

3. **Centrifugal Force**:
- The centrifugal force acting on the satellite due to its circular motion is given by:
\[ F_{\text{centrifugal}} = \frac{m v^2}{r} \]
where \( v \) is the orbital velocity of the satellite.

4. **Setting Forces Equal**:
- For the satellite to remain in a stable orbit, these two forces must be equal:
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G M}{r^2} = \frac{v^2}{r} \]

5. **Rearranging the Equation**:
- Multiplying both sides by \( r \):
\[ \frac{G M}{r} = v^2 \]

6. **Finding the Orbital Velocity**:
- Taking the square root of both sides gives us the formula for the orbital velocity:
\[ v = \sqrt{\frac{G M}{r}} \]

7. **Using the Relation for Gravitational Acceleration**:
- We know that the gravitational acceleration \( g \) at the Earth's surface is given by:
\[ g = \frac{G M}{R^2} \]
- Therefore, we can express \( G M \) as:
\[ G M = g R^2 \]

8. **Substituting into the Velocity Formula**:
- Substituting \( G M \) into the orbital velocity equation:
\[ v = \sqrt{\frac{g R^2}{r}} \]
- Since \( r \) is approximately equal to \( R \) when the height \( h \) is negligible:
\[ v = \sqrt{g R} \]

### Final Result:
Thus, the orbital velocity of the satellite when its height is negligible compared to the radius of the Earth is:
\[ v = \sqrt{g R} \]