Define binding energy and obtain an expression of binding energy of a satellite revolving in a circular orbit around the earth.

Binding Energy:
Binding energy is the energy required to disassemble a system of particles into individual components. In the context of a satellite, it refers to the energy needed to move a satellite from its orbit around the Earth to a position where it is no longer under the influence of Earth's gravitational field (i.e., at infinite distance).
Expression for Binding Energy of a Satellite Revolving in a Circular Orbit Around the Earth:
Consider a satellite of mass mm revolving in a circular orbit of radius rr around Earth, with mass MM. The gravitational force provides the centripetal force required for the satellite's circular motion.
Step 1: Gravitational Potential Energy
The gravitational potential energy (UU) of the satellite at a distance rr from the center of the Earth is given by the formula:
U=−GMmrU = - \frac{GMm}{r}
where:
• GG is the gravitational constant,
• MM is the mass of the Earth,
• mm is the mass of the satellite,
• rr is the distance between the center of the Earth and the satellite.
Step 2: Kinetic Energy of the Satellite
The kinetic energy (K.E.K.E.) of the satellite in its circular orbit is derived from the fact that the gravitational force provides the centripetal force required for its motion:
Fgravity=GMmr2F_{\text{gravity}} = \frac{GMm}{r^2} Fcentripetal=mv2rF_{\text{centripetal}} = \frac{mv^2}{r}
Equating these two forces gives:
GMmr2=mv2r\frac{GMm}{r^2} = \frac{mv^2}{r}
Solving for the speed vv of the satellite:
v2=GMrv^2 = \frac{GM}{r}
Thus, the kinetic energy K.E.K.E. is:
K.E.=12mv2=12m(GMr)=GMm2rK.E. = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{GM}{r}\right) = \frac{GMm}{2r}
Step 3: Total Mechanical Energy (E)
The total mechanical energy (EE) of the satellite is the sum of its kinetic and potential energies:
E=K.E.+U=GMm2r−GMmr=−GMm2rE = K.E. + U = \frac{GMm}{2r} - \frac{GMm}{r} = - \frac{GMm}{2r}
Step 4: Binding Energy
The binding energy (B.E.B.E.) is the energy required to move the satellite from its orbit (at a distance rr) to infinity, where the gravitational potential energy is zero. This is equal to the negative of the total mechanical energy:
B.E.=−E=GMm2rB.E. = -E = \frac{GMm}{2r}
Final Expression:
Thus, the binding energy of a satellite revolving in a circular orbit around the Earth is:
B.E.=GMm2rB.E. = \frac{GMm}{2r}
Where:
• GG is the gravitational constant,
• MM is the mass of the Earth,
• mm is the mass of the satellite,
• rr is the radius of the orbit.
This expression shows that the binding energy is directly proportional to the mass of the satellite and inversely proportional to the radius of its orbit.