Question

Define the binding energy of a satellite. Obtain an expression for the binding energy of a satellite which is revolving around the earth at a certain height. State the factors affecting it.

Answer 1

Hint: The satellite is moving in the circumference of the circular orbit around the earth. At the equilibrium condition, the centripetal force will be equivalent to the gravitational force. The total energy will be the sum of kinetic energy and the potential energy.

Complete answer:
The minimum amount of energy needed for a satellite to escape from the gravitational influence of the earth is known as the binding energy of a satellite. $r$ be a radius of orbit. That is the sum of the radius of earth and the height at which satellite is orbiting the earth.
Let us assume that the mass of the satellite is $m$ and $r$ is the radius of the circular orbit. The satellite is moving in the circumference of the circular orbit around the earth.
Now, at the equilibrium condition, the centripetal force will be equivalent to the gravitational force. This can be written as,
$\text{centripetal force}=\text{gravitational force}$
That is,
$\dfrac{m{{v}^{2}}}{r}=\dfrac{GMm}{{{r}^{2}}}$
Cancelling the common terms in it,
$m{{v}^{2}}=\dfrac{GMm}{r}$
Multiplying with the half of it,
$\dfrac{1}{2}m{{v}^{2}}=\dfrac{GMm}{2r}$
The half of the product of the mass of the body and the square of the velocity of the particle will be equivalent to the kinetic energy of the particle. That is we can write that,
$KE=\dfrac{GMm}{2r}$
Now, the potential energy between satellite and the earth can be given as,
$PE=\dfrac{-GMm}{r}$
Here the negative sign represents the force acts between the satellite and earth is attractive. The total energy will be the sum of kinetic energy and the potential energy.
$TE=KE+PE$
Substituting the values in it,
$\begin{align}
  & TE=\dfrac{GMm}{2r}+\dfrac{-GMm}{r} \\
 & \Rightarrow TE=\dfrac{-GMm}{2r} \\
 & \therefore {{E}_{r}}=\dfrac{-GMm}{r} \\
\end{align}$
The factors at which the binding energy is dependent on will be the gravitational constant, mass of the earth, mass of the satellite and the radius. The question has been answered.

Note:
Here the negative sign represents that the satellite is bound to the earth by the attractive force and cannot leave it on its own. The potential energy of a body is the energy possessed by the alignment and orientation of a body. The kinetic energy of a body is explained as the energy possessed because of the motion of the body.