1.3 GRAVITATION AND SATELLITES

There is a mutual force of attraction between any two objects that is directly proportional to each of the masses,

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Newton’s Universal Law of Gravitation

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is inversely proportional to the square of the distance between their centres,

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Newton’s Universal Law of Gravitation

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and acts along the line joining their centres.

Newton’s Universal Law of Gravitation

𝐹=𝐺𝑚1𝑚2

𝑟2

= masses of objects (kg)

= distance between the centres of the objects (m)

N.m2.kg-2

Newton’s Universal Law of Gravitation

Calculate the gravitational force on a 450 kg satellite orbiting a planet of mass 7.4 x 1024 kg if the orbit radius is 6.5 x 106 m.

Example 1

Why does the Earth orbit the Sun and not the other way around?

Orbiting Planets

𝐹

𝑟

𝑀 𝑚

PROPORTIONALITIES

𝐹=𝐺𝑀𝑚𝑟2

Consider the satellite orbiting the planet from the previous example. What would happen to the magnitude of the force in the previous example if:

(a) the satellite’s mass was changed to 225 kg?

(b) the orbital radius tripled?

(c) the mass of the satellite was halved and its orbitalradius was reduced to 2/3 of its original size?

Example 2

What would happen to the force on the satellite if:

(a) The mass of the planet was trebled.

(b) The distance between the satellite and the planet was doubled.

(c) The distance between the satellite and the planet was halved.

(d) The mass of the planet was doubled and the mass of the satellite was quadrupled.

(e) The mass of the satellite was halved and its distance from the planet was reduced to ¾ of the original distance.

Practice Questions

The weight of an object equals the gravitational force between it and the planet.

= mass of planet (kg) = mass of object (kg) = radius of planet (m)

Note: does not depend on the object’s mass

𝑀

𝑚

Derivation: Acceleration due to Gravity

Find the acceleration due to gravity of Mars given that the mass of Mars is 6.4 x 1023 kg and its radius is 3.4 x 106 m.

Example 3

Application: Satellites

𝑀

𝑚Derivation: Speed of a Satellite

The gravitational force causes the centripetal acceleration.

= mass of planet (kg) = mass of satellite (kg) = radius of orbit (m) = speed of the satellite (m.s-1)

𝑀

𝑚Derivation: Speed of a Satellite

𝑣=√𝐺𝑀𝑟Note:The speed only depends on the mass of the central planet and the orbital radius. For a fixed radius, there is only one possible speed.

𝑀

𝑚Derivation: Period of a Satellite

Consider the two equations for the velocity of a satellite.

𝑀

𝑚Derivation: Period of a Satellite

𝑇 2=4 𝜋2𝑟3

𝐺𝑀

Note:The period only depends on the mass of the central planet and the orbital radius.

Possible Orbits

For a satellite to stay in a circular orbit:

𝑎𝐹• There must be a centripetal

acceleration towards the centre of the orbit.

• The gravitational force (which is directed towards the centre of the Earth), supplies this acceleration.

• Hence, the centre of the Earth must coincide with the centre of the satellite’s orbit.

Possible Orbits

Launching Low Altitude Satellites

When launched from the equator, satellites areusually launchedfrom west to east.

The speed of the launch site adds to the final speed of the satellite.

Thus, less fuel is needed.

Geostationary Satellites

These satellites stay over the same point on the Earth’s surface.

https://www.youtube.com/watch?v=FsfcIEmR_b0

Geostationary Satellites

In order to remain above the same point, there are certain orbital requirements:

• Rotation West to East

The satellite must orbit in the same direction that the Earth rotates to stay above the same point.

• Equatorial Orbit

The satellite must orbit in a plane perpendicular to the Earth’s rotational axis and the centre of orbit

must coincide with the centre of the Earth.

Polar orbit satellite video

rotational axis

Geostationary Satellites

Geostationary Satellites

• Large Radius

The period of the satellite must match the period of the Earth (24 hours). Hence, the orbital radius of the satelliteis fixed. Note: mass of the Earth = kg.

Polar orbit satellite video

Geostationary Satellites

Given that the radius of the Earth is m,

height above Earth’s surface radius of orbit - radius of the Earth

https://www.youtube.com/watch?v=FsfcIEmR_b0

Polar Orbits

These satellites orbit over the poles.

polar orbits

geostationaryorbit

Comparing Orbits

Satellite Uses

Geostationary

Mainly used for communications as reception points on the Earth’s surface can remain fixed since the satellite’s relative position does not change. Sometimes used for weather monitoring.

Satellite Uses

Polar

Mainly used for surveillance and meteorology because:

• Low altitude provides a closer view of the Earth, providing photos with a higher resolution.

• The smaller orbital radius means these satellites have a shorter period and higher speed they can photograph more of the Earth in a

shorter time.

• They travel around the Earth many times a day (e.g. 12). Since the Earth rotates perpendicular to the satellite’s orbit, patchwork photos can be constructed.