Derive an expression for critical velocity of a satellite revolving around the earth in a circular orbit.

To derive the expression for the critical velocity of a satellite in a circular orbit around the Earth, we start by considering the forces acting on the satellite. The key force is the gravitational force, which provides the necessary centripetal force for circular motion.

Gravitational Force

The gravitational force \( F_g \) acting on the satellite can be expressed using Newton's law of gravitation:

F_g = \frac{G \cdot M \cdot m}{r^2}

• G is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).
• M is the mass of the Earth, about \( 5.972 \times 10^{24} \, \text{kg} \).
• m is the mass of the satellite.
• r is the distance from the center of the Earth to the satellite.

Centripetal Force

The centripetal force \( F_c \) required to keep the satellite in circular motion is given by:

F_c = \frac{m \cdot v^2}{r}

• v is the orbital velocity of the satellite.

Equating Forces

For the satellite to remain in orbit, the gravitational force must equal the centripetal force:

\frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r}

Simplifying the Equation

We can cancel the mass of the satellite \( m \) from both sides, as it does not affect the orbital velocity:

\frac{G \cdot M}{r^2} = \frac{v^2}{r}

Multiplying both sides by \( r \) gives:

G \cdot M = v^2 \cdot r

Finding Critical Velocity

Now, we can solve for the orbital velocity \( v \):

v^2 = \frac{G \cdot M}{r}

Taking the square root of both sides results in:

v = \sqrt{\frac{G \cdot M}{r}}

Final Expression

This equation represents the critical velocity of a satellite in a circular orbit around the Earth:

v = \sqrt{\frac{G \cdot M}{r}}

In this expression, \( r \) is the distance from the center of the Earth to the satellite, which includes the Earth's radius plus the altitude of the satellite above the Earth's surface.