Satellites

 

Satellites

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To launch an artificial satellite in space, multi-stage rockets are used. It raises the satellite to a predetermined height, projects it in the right direction with a velocity which enables it to revolve around the Earth. This velocity is called the escape velocity. Escape velocity is the velocity with which a body should be projected to enable it to escape from the gravitational influence of the Earth.

Consider a body of mass m at a distance r from the centre of the Earth.

If this body is raised by a distance dr against the force then the work done

dw = F dr

Total work done in moving an object from the surface of the Earth to infinity is

If v_e is the escape velocity then

Substituting for Gm in the equation, we get

If g = 9.8ms^-2 and R = 6.4 x 106m

Then v_e = 11.2 x 103ms^-1

Motion of a satellite

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Orbital velocity

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The velocity with which, an object circles the Earth in a circular path is known as orbital velocity.

Let a body of mass m be projected from the Earth's surface so that it revolves around it. Let its height be h from the surface of the Earth.

The centripetal force required make the satellite go around in a circular orbit, is equal to  and it acts towards the centre of the Earth and this force is made available by the gravitational force of attraction   which also acts towards the centre of the Earth.

The orbital velocity can be calculated taking g = 9.8 ms^-1 and

R = 6.4 x 10⁶m

v = 7.9 x 10³ms^-1

Period of revolution or time period of a satellite

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It is the time taken by a satellite to complete one revolution around the Earth. Circumference of a circle of radius R + h is 2p (R+h).

Let T be the time period, v the velocity and h the height of the satellite above the Earth's surface.

Substituting in the equation, we get

Substituting for GM in the equation, we get

Height of a satellite (h)

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We know

\ T²   µ r³ and this is consistent with Kepler's law which states that the squares of the orbital periods of the planets are proportional to the cubes of there mean distances from the sun.

This is the required expression for height of a satellite.

Uses of artificial satellites

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They can be used for

• weather forecasting

• studying the atmosphere

• telecommunication

• mapping the Earth

Geostationary satellite

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Geostationary satellite

A satellite whose period of revolution is 24 hours, is a geostationary satellite.

It always appears to be at a fixed point in space, because the period of rotation of the Earth about its own axis is also equal to 24 hours.

Knowing T = 24 hours, g = 9.8 ms^-1, the height of a geostationary satellite is calculated to be 36000km.

• Its orbital velocity is 3.1 km/s.

• Its plane of orbit is the equatorial plane.

• It revolves from west to east which is similar to the Earth's movement.

• It is very useful in telecommunication.

Polar satellites

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Polar satellites

Satellites in polar orbits are used for environmental and earth resources' survey. They move over the polar regions and cover the whole of the Earth's surface in a few weeks.