Question

Derive expression for orbital velocity?

Answer 1

Derive the orbital velocity, as we concern ourselves with the following two concepts:
Gravitational Force: it is very important to keep the universe functioning. Is a force that exists among all material objects in the universe. For any two objects or particles having nonzero mass, the force of gravity tends to attract them toward each other.
Centripetal Force: Is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as “a force by which bodies are drawn or impelled, or in any way tend, toward a point as to a center.

Complete step by step solution:
For the derivation, let us consider a satellite of mass $m$ revolving around the Earth in a circular orbit of the radius $r$ at a height $h$ from the surface of the Earth. Suppose $M$ and $R$ are the mass and radius of the Earth respectively, then $r = R + h$ .
To revolve the satellite, centripetal force of $\dfrac{{m{v_0}^2}}{r}$ is needed which is provided by the gravitational force $G\dfrac{{Mm}}{{{r^2}}}$ between the satellite and the Earth.
So, equating both the equations, we get
$\dfrac{{m{v_0}^2}}{r} = G\dfrac{{Mm}}{{{r^2}}}$
${v_0}^2 = \dfrac{{GM}}{r} = \dfrac{{GM}}{{R + h}}$
Simplifying the above equation further, we get
${v_0} = \sqrt {\dfrac{{GM}}{{R + h}}}$ ……. $(1)$
But $GM = g{R^2}$ , Where $g$ is the acceleration due to gravity.
Therefore,
${V_0} = \sqrt {\dfrac{{g{R^2}}}{{R + h}}}$
From the above equation, we get
${v_0} = R\sqrt {\dfrac{g}{{R + h}}}$
Let $g$ ’ be the acceleration due to gravity (at a height h from the surface)
$g' = \dfrac{{GM}}{{{{\left( {R + h} \right)}^2}}}$
Solve it. We get,
$\dfrac{{GM}}{{\left( {R + h} \right)}} = g'\left( {R + h} \right) = g'r$ …. $(2)$
Substituting $(2)$ in $(1)$ , we get
${v_0} = \sqrt {g'r} = \sqrt {g'\left( {R + h} \right)}$

Note:
Satellites, the object is not to escape Earth’s gravity. But balance it. Orbital velocity is velocity needed to achieve balance between the gravity’s pull on satellite and inertia of the satellite’s motion- the satellite’s tendency to keep going.