Gravitation

Earth Satellites

What comes to your mind when you think of the satellites? Yes, you may definitely think of the MOON. Do you know what are earth satellites and how do these satellites orbit the earth? Let us study about earth satellites in detail.

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Earth Satellites

What is an Earth Satellite? An object revolving around the earth is earth satellite. Do you why the reason why do the satellite’s orbit? The satellites orbit due to the 1st law of motion which states that an object is at rest or in a state of motion unless acted upon by an external force.

So when we talk about a planet and a satellite, when the satellite is orbiting around the planet is because of two reasons. The first reason is that there is a gravitational force between the satellite and the planet. The second reason is that it just wants to speed past the planet. It just wants to go out of the orbit. Satellites are classified into two types.

  • Natural Satellites
  • Artifical Satellites

Natural Satellites

The satellites there have existed in nature on their own are natural satellites. No efforts have been put to discover these satellites. For example the Moon is a natural satellite of the earth.

Earth Satellites

In fact, the Moon is the only natural satellite for earth that has existed on its own and keeps on revolving around the earth.

Artificial Satellites

Artificial satellites are the objects that are intentionally placed by humans which orbits the earth for practical uses. These artificial satellites are built for various purposes. They are used for:

  • Communication satellites are used for wide communications. eg. Mobile phones.
  • Television broadcast
  • Navigation
  • Military support
  • Weather observations
  • Scientific Research.

Time period of Earth Satellites

Let us derive an expression to determine the time taken by the satellite to complete one rotation around the earth. Suppose a satellite keeps on revolving around them in a circular orbit. So as it moves in circular motion, there is a centripetal force acting on it.

F=  \( \frac{mv²}{r²} \)

F=  \( \frac{mv²}{R_e + h} \),  ‘ h ‘ is the distance above earth’s surface.

This centripetal force will act towards the centre. Now there is another gravitational force between the earth and the satellite that is,

  • m = mass of the satellite
  • Me = mass of the Sun

FG = \( \frac{GmM_e} {(R_e + h)^2}\)

Now F= FG, implies

\( \frac{mv²} {R_e + h}\) =  \( \frac{GmM_e} {(R_e + h)^2}\)

⇒  v² =  \( \frac{-GM_e} {R_e + h}\)

⇒  v = \( \sqrt[]{\frac{GM_e}{R_e+h}} \)  = Velocity

We want to calculate the time period of the satellite. We know that, satellite covers a distance of 2π ( Re+ h ) in one revolution

T = \( \frac{distance}{velocity} =  \frac{2π (R_e + h )} {v}\)

T = \( \frac{2π (R_e + h )^{3/2}} {√{GM_e}}\)

Thus this is the time period taken by the satellite to revolve around the earth.

The Energy of Orbiting Satellites

We know that m is the mass of the satellite and the velocity with which it moves is v. So what is the kinetic energy of the satellite? It is given by \( \frac{1}{2} \) mv²

As we know v = \( \sqrt{\frac{GM_em} {R_e + h}}\)

So, the kinetic energy is, \( \frac{1}{2} \) \( \frac{GM_em}{R_e + h} \)

Now the potential energy is, \( \frac{- GM_em} {R_e + h}\)

Toatl energy = kinetic energy + potential energy

\( \frac{1}{2} \) \( \frac{GM_em}{R_e + h} \) + \( \frac{- GM_em}{R_e + h} \)

Total energy = \( \frac{- GM_em}{2(R_e + h)} \)

Solved Question For You

Q1. Out of the following statements, the one which correctly describes a satellite orbiting about the earth is

  1. There is no force acting on the satellite.
  2. The acceleration and velocity of the satellite are roughly in the same direction.
  3. Satellite is always accelerating around the earth.
  4. The satellite must fall back to earth when its fuel is exhausted.

Ans: C. When the satellite is revolving around the earth, it is because of the gravitational force towards the earth that acts as a centripetal force. Since the initial speed is less than the escape speed, earth’s gravity pulls the satellite towards the centre of the earth. So the satellite is always accelerating around the earth.

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