1. Gravitation and Satellites
SACE Stage 2 Physics
Gravitation and Satellites

2. Gravitation and Satellites
Law of Universal Gravitation
For any two point masses in the universe there is a force of attraction acting on each mass along the line joining their centres. Each of these forces has the same magnitude and this magnitude is directly proportional to the product of their masses and inversely proportional to the square of their distance apart. F F m2 m1

3. Gravitation and Satellites
Law of Universal Gravitation
The forces obey Newton's Third Law. The forces do not "cancel" because they act on different objects.
G is the Universal Constant of Gravitation = 6.67 x N m2 kg-2.

4. Gravitation and Satellites
Law of Universal Gravitation
The gravitational attraction between two objects is weak, and so G is small. E.g., Calculate the gravitational force between two masses 1 kg and 10 kg a distance 1m apart.
Note: The force is determined by both masses and their distance apart.
The size (magnitude) of the force is the same for both objects (direction is opposite).

5. Gravitation and Satellites
The Value of g at the Earth’s Surface
A person standing on the Earth’s surface has weight,
Force on the person due to gravity,
Where m = the mass of a person on the Earths surface,
ME= the mass of the Earth
r = the radius of the Earth
and g = the acceleration felt by the person toward the Earth.

6. Gravitation and Satellites
The Value of g at the Earth’s Surface

7. Gravitation and Satellites
Example 1
Find the gravitational attraction between the Earth and the Sun.

8. Gravitation and Satellites
Example 1
Find the gravitational attraction between the Earth and the Sun.

9. Gravitation and Satellites
Example 2
Two bodies experience a gravitational force of 4.0 x 10-6N. Body A has a mass of 30 kg and the distance between their centres of mass is 5.0m.
Find the gravitational force between if,
The mass of body B is quadrupled.
The mass of body A is changed to 50 kg and the mass of body B is trebled.
The mass of body A is changed to 50 kg and the distance between them is reduced to 3.0m.

10. Gravitation and Satellites
Example 2
The mass of body B is quadrupled.
mB is multiplied by 4,  F is multiplied by 4
F = 4 x 4.0 x 10-6N = 1.6 x 10-5N.
2. The mass of body A is changed to 50 kg and the mass of body B is trebled.
mA is multiplied by 5/3 and mB is multiplied by 3
F is multiplied by 5/3 x 3
F = 5 x 4.0 x 10-6N = 2.0 x 10-5N

11. Gravitation and Satellites
Example 2
3. The mass of body A is changed to 50 kg and the distance between them is reduced to 3.0m.
mA is multiplied by 5/3 and r is multiplied by 3/5
F is multiplied 5/3 x (5/3)2 = 125/27
F = 125/27 x 4.0 x 10-6N = 1.85x 10-5N.

12. Gravitation and Satellites
Example 3
The mass of the Earth is x 1024kg. The mean radius of the Moons orbit around the Earth is x 108m. Calculate (in days) the expected period of rotation of the Moon about the Earth.

13. Gravitation and Satellites
Example 3

14. Gravitation and Satellites
Satellites in Circular Orbits
Planetary Orbits
Gravitational forces generally are weak.
If the masses are large enough, then the force is large.
The gravitational force between the sun and a planet provides the centripetal force to keep the planet in orbit.

15. Gravitation and Satellites
Satellites in Circular Orbits FG
Sun
Planet
F (=FG) r
Assume orbit is circular,
The gravitational attraction provides the centripetal acceleration.

16. Gravitation and Satellites
Satellites in Circular Orbits
Therefore,
where mp = mass of planet & ms = mass of sun

17. Gravitation and Satellites
Satellites in Circular Orbits
In order to measure the mass of the Sun, you simply
need values for the planets. The radius of orbit and
the period of revolution can both be measured with a telescope.

18. Gravitation and Satellites
Satellite speeds and radius of orbit
An expression for finding the speed of a satellite at radius r
There can only be one radius of orbit for a satellite for a given speed.

19. Gravitation and Satellites
Weather Satellites
Satellite Orbit
The centre of orbit of a satellite orbiting the Earth must coincide with the centre of the Earth.
Consider two pucks joined by a piece of string.

20. Gravitation and Satellites
Weather Satellites
If we replace one puck with another puck of twice the mass, the centre of rotation will shift towards the heavier mass. As we continue to increase the mass of the heavier puck so that the mass of the lighter puck (satellite) becomes negligible to the larger puck (Earth), the centre of rotation will approach the centre of the larger puck (Earth).

21. Gravitation and Satellites
Geostationary Satellite
A satellite that remains over a fixed point on the Earths surface is Geostationary.
Conditions,
Orbit must be equatorial
Orbit must be circular (constant speed)
Radius must corrospond to a period of 23h and 56min (one day)
Direction must be the same as the Earths rotation.

22. Gravitation and Satellites
Geostationary Satellite
Launching a Satellite into Orbit
The Earth rotates from West to East. It is very economical to launch satellites in this direction near the equator as this rotation can contribute to the final speed of the satellite.
Low Altitude Orbits (200 to 3000km)
To get greater resolution images, a low orbit satellite is used. If we reduce r, we must increase speed v which implies a smaller period T.

23. Gravitation and Satellites
Polar Orbit
A satellite that orbits pass the poles is in a polar orbit.
With a radius and period chosen so that as the Earth rotates under it, the satellite passes over the same location at the same time each day (actually passes over the same location twice a day).
By adjusting the satellites period, the rotational displacement can be chosen to be the actual field of view of the satellite and therefore the entire Earths surface can be seen.

24. Gravitation and Satellites
Example 4
A satellite is in polar orbit completing 14.2 orbits of the Earth in one day. Find,
Its height above the Earth in kilometres.
The orbital speed of the satellite in km/hr.
ME=5.977 x 1024kg, rE = x 106m.

25. Gravitation and Satellites
Example 4
1. Its height above the Earth in kilometres.

26. Gravitation and Satellites
Example 4
1. Its height above the Earth in kilometres.

27. Gravitation and Satellites
Example 4
The height above the Earths surface is given by,

28. Gravitation and Satellites
Example 4
The orbital speed of the satellite in km/hr.