Question

A Skylab of mass $m\,{\text{kg}}$ is first launched from the surface of the Earth in a circular orbit of radius $2R$ (from the centre of the Earth) and then it is shifted from this circular orbit to another circular orbit of radius $3R$. The minimum energy required to place the lab in the first orbit and to shift the lab from first orbit to second orbit is
A. $\dfrac{3}{4}mgR,\,\dfrac{{mgR}}{6}$
B. $\dfrac{3}{4}mgR,\,\dfrac{{mgR}}{{12}}$
C. $mgR,\,mgR$
D. $2mgR,\,mgR$

Answer 1

Use the formula for the total energy of an object on the surface of the Earth. Using this formula, calculate the total energy of the Skylab on the surface of the Earth, in the first circular orbit and in the second circular orbit. Take the difference of total energy in first orbit and total energy on the surface of the Earth to calculate the first energy value and take the difference between the energy of Skylab in second and first orbit.

Formulae used:
The total energy $E$ of an object on the surface of the Earth is given by
$E = - \dfrac{{GMm}}{{2r}}$ …… (1)
Here, $G$ is a universal gravitational constant, $M$ is mass of the Earth, $m$ is mass of the object and $r$ is distance from the centre of the Earth.
The relation between the universal gravitational constant $G$ and acceleration due to gravity $g$ is
$GM = g{R^2}$ …… (2)
Here, $M$ is the mass of the planet and $R$ is the radius of the planet.

Complete step by step answer:
We have given that the mass of the Skylab is $m$ and it is launched from the surface of the Earth to the first circular orbit or radius $2R$ from the centre of the Earth. This Skylab is then shifted to the second circular orbit of radius $3R$ from the centre of the Earth.We have asked to calculate the energy required to place the Skylab in the first circular orbit and then energy required to shift this Skylab from the first to second circular orbit.

Let us first calculate the energy required to place the Skylab in the first circular orbit.The total energy ${E_1}$ of the Skylab at the surface of the Earth is the sum of potential energy and kinetic energy. But the kinetic energy is zero. Hence, the total energy is equal to potential energy
${E_1} = - \dfrac{{GMm}}{R}$
Substitute $2R$ for $r$ in equation (1).
${E_2} = - \dfrac{{GMm}}{{2\left( {2R} \right)}}$
$\Rightarrow {E_2} = - \dfrac{{GMm}}{{4R}}$
Here, ${E_2}$ is the total energy of the Skylab in the first orbit.

The energy required to place the Skylab in the first orbit is
$\Rightarrow {E_2} - {E_1} = \left( { - \dfrac{{GMm}}{{4R}}} \right) - \left( { - \dfrac{{GMm}}{R}} \right)$
$\Rightarrow {E_2} - {E_1} = \dfrac{{GMm}}{R} - \dfrac{{GMm}}{{4R}}$
$\Rightarrow {E_2} - {E_1} = \dfrac{{3GMm}}{{4R}}$
Substitute $g{R^2}$ for $GM$ in the above equation.
$\Rightarrow {E_2} - {E_1} = \dfrac{{3\left( {g{R^2}} \right)m}}{{4R}}$
$\Rightarrow {E_2} - {E_1} = \dfrac{3}{4}mgR$
Hence, the energy required to place the Skylab in the first orbit is $\dfrac{3}{4}mgR$.

Substitute $3R$ for $r$ in equation (1).
${E_3} = - \dfrac{{GMm}}{{2\left( {3R} \right)}}$
$\Rightarrow {E_3} = - \dfrac{{GMm}}{{6R}}$
Here, ${E_3}$ is the total energy of the Skylab in the second orbit.
The energy required to place the Skylab in the first orbit is
$\Rightarrow {E_3} - {E_2} = \left( { - \dfrac{{GMm}}{{6R}}} \right) - \left( { - \dfrac{{GMm}}{{4R}}} \right)$
$\Rightarrow {E_3} - {E_2} = \dfrac{{GMm}}{{4R}} - \dfrac{{GMm}}{{6R}}$
$\Rightarrow {E_3} - {E_2} = \dfrac{{GMm}}{{12R}}$
Substitute $g{R^2}$ for $GM$ in the above equation.
$\Rightarrow {E_3} - {E_2} = \dfrac{{\left( {g{R^2}} \right)m}}{{12R}}$
$\therefore {E_3} - {E_2} = \dfrac{{mgR}}{{12}}$
Hence, the energy required to shift the Skylab from first to second circular orbit is $\dfrac{{mgR}}{{12}}$.

Hence, the correct option is B.

Note: The students should keep in mind that we are using the formula for total energy of the Skylab from the centre of the Earth. One can also calculate the individual energies of the Skylab instead of using the formula for the total energy and then calculate the energy difference to calculate the energy required to place the Skylab in the first and second circular orbit.