Question

At a height $H$ from the surface of the earth, the total energy of a satellite is equal to the potential energy of a body of equal mass at a height of $3R$ from the surface of the earth ($R$=Radius of the earth). Find the value of $H$.
A. $R$
B. $\dfrac{{4R}}{3}$
C. $3R$
D. $\dfrac{R}{3}$

Answer 1

Hint: A satellite is located at a height of $H$ from the surface of the earth, it will have some energy. That energy is equal to the potential energy of the body of equal mass located at a height of $3R$ from the surface. By using the energy formula, the energy of the satellite and the body can be obtained and by equating those two energies the value of $H$ can be calculated.

Useful formula:
Total energy of satellite located in space,
$T.E = - \dfrac{{GMm}}{{2{r_1}}}$
Where, $G$ is gravitational constant, $M$ is the mass of the earth, $m$ is the mass of the satellite and ${r_1}$ is the distance between the centre of earth to the satellite.

Potential energy of the object,
$P.E = - \dfrac{{GMm}}{{{r_2}}}$
Where, $G$ is gravitational constant, $M$ is the mass of the earth, $m$ is the mass of the body and ${r_2}$ is the distance between the centre of earth to the body.

Given data:
The radius of the earth is $R$
The distance between the surface of the earth and the object is $3R$
The height of satellite located from the surface of the earth is $H$

Step by step solution:
Total energy of satellite located in space,
$T.E = - \dfrac{{GMm}}{{2{r_1}}}$
Since, the distance between the centre of the earth to the satellite is ${r_1} = R + H$
Hence, $T.E = - \dfrac{{GMm}}{{2\left( {R + H} \right)}}$

Potential energy of the object,
$P.E = - \dfrac{{GMm}}{{{r_2}}}$
Since, the distance between the centre of the earth to the body is ${r_2} = R + 3R$
Hence,
$
  P.E = - \dfrac{{GMm}}{{\left( {R + 3R} \right)}} \\
  P.E = - \dfrac{{GMm}}{{4R}} \\
 $

From the question,
$T.E = P.E$
Substitute the values of energies in above relation, we get
$
   - \dfrac{{GMm}}{{2\left( {R + H} \right)}} = - \dfrac{{GMm}}{{4R}} \\
  2\left( {R + H} \right) = 4R \\
  R + H = 2R \\
  H = 2R - R \\
  H = R \\
 $

Hence, the option (A) is correct.

Note: It is clear from the question, there are two objects in space. One is a satellite and the other is a body. It is given that the total energy of the satellite is equal to the potential energy of the body. By equating both the energies, the value of the height of the satellite is obtained.