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9.2 Space: 3. The Solar System is held together by gravity

Syllabus reference (October 2002 version)
3. The Solar System is held together by gravity	Students learn to: describe a gravitational field in the region surrounding a massive object in terms of its effects on other masses in it define Newton's Law of Universal Gravitation: discuss the importance of Newton's Law of Universal Gravitation in understanding and calculating the motion of satellites identify that a slingshot effect can be provided by planets for space probes	Students: present information and use available evidence to discuss the factors affecting the strength of the gravitational force solve problems and analyse information using

Extract from Physics Stage 6 Syllabus (Amended October 2002). © Board of Studies, NSW.

Prior Learning: Preliminary module 8.5 (subsections 5 and 6)

present information and use available evidence to discuss the factors affecting the strength of the gravitational force

Identify sources of information and use them to gather and present information about factors affecting the strength of the gravitational field surrounding objects. Make sure you collect information to reference your sources.
Your discussion should initially focus on showing how the mass of an object and the distance from the centre of the mass are the two determinants of the strength at any particular point.
Be sure that in the construction of your discussion, you draw on available evidence to coherently and logically illustrate the factors that affect a gravitational field. This will provide you with opportunities to include in your discussion other factors that affect the gravitational field. For example, the distribution of mass is important in determining the uniformity of the field. The variations in the strength of the Earth's gravitational field due to uneven mass distributions within the Earth's crust are used as indicators of possible oil and natural gas reservoirs. Check that you correctly use scientific principles and ideas. It is helpful to your audience if you use appropriate methods to acknowledge sources of information.

describe a gravitational field in the region surrounding a massive object in terms of its effects on other masses in it

The strength of a gravitational field around a massive object is proportional to the value of it's mass and decreases in inverse proportion to the square of the distance from the centre of the object. So, if other smaller masses are in the field, the force of attraction to the massive object that they experience will depend on their mass and the distance to the centre of the massive object.

define Newton's Law of Universal Gravitation

In simplistic terms, what Newton said was that an object attracts every other object in the universe. The two factors that determine the force of the attraction are:
- the mass of each of the two objects
- the distances between their centres of mass.
In mathematical terms, this is:

where,
- F = force of attraction between objects
- G = universal gravitational constant (which is equal to 6.67 x 10^-11 N m² kg^-2)
- m₁ = mass of object 1
- m₂ = mass of object 2
- d = distance between their centres of mass

solve problems and analyse information using

Solve problems by choosing a strategy that will allow you to calculate the value of an unknown variable from the known data. This will probably involve rearranging the equation, substituting values for known variables and calculating the unknown. Make sure that all variables are expressed in SI units, using scientific notation where appropriate.
You can analyse information about gravitational force by looking at the relationship between any two of the variables, while holding the other variables constant.

Sample problem

A geostationary satellite of mass 600 kg orbits at a distance of 35 800 km. Calculate the force of gravity that keeps the satellite in orbit.

Data:	G = 6.67 x 10^-11 N m² kg^-2 m₁ = mass of Earth = 6.0 x 10²⁴ kg m₂ = mass of satellite = 6.0 x 10² kg d = 3.58 x 10⁴ km = 3.58 x 10⁷ m

F = G m₁ m₂ / d²

= (6.67 x 10^-11)( 6.0 x 10²⁴)( 6.0 x 10²) / (3.58 x 10⁷)²

= 187 N

Sample analysis

Describe the way in which gravitational force varies with distance for any two objects.

For any two objects chosen, for example, Earth and a Space Shuttle, the two masses m₁ and m₂ remain constant. G is also constant, therefore the equation can be reduced to:

F = k / d²

where k is a constant.

Therefore gravitational force is inversely proportional to the distance between the objects. Force diminishes very rapidly with distance at first, but diminishes more and more slowly as distance increases.

discuss the importance of Newton's Law of Universal Gravitation in understanding and calculating the motion of satellites

Background

Gravitation provides the centripetal force that produces the circular motion that is the satellite’s orbit around a planet. Therefore, it can be said that:

Gravitational force = Centripetal force

This equation shows that the required orbital velocity of a satellite depends upon the mass and the orbital radius. Note: this equation is NOT stipulated by the syllabus.

The following expressions can be derived, from the relationships described in the two lots of background information above:
The second equation describes the relationship between the radius of an orbit and the period for any (and all) satellites, natural and artificial, orbiting the earth. This second equation is stipulated in the syllabus. It is also a more useful version of Kepler’s third law.

identify that a slingshot effect can be provided by planets for space probes

The slingshot effect is also known as a planetary swing by or a gravity-assist manoeuvre. It is performed to achieve an increase in velocity relative to the Sun and/or a change of direction.
A spacecraft is aimed close to a planet. As it approaches, the spacecraft is caught by the gravitational field of the planet, and swings around it. The speed acquired is then sufficient to throw the spacecraft back out again, away from the planet. By controlling the approach, the outcome of the manoeuvre can be manipulated.

Extra information outside the syllabus:

The manoeuvre can be analysed as an elastic collision, even though no actual contact occurs. A slingshot manoeuvre can therefore be used to change the spaceship's trajectory and velocity relative to the Sun, though the spacecraft's speed relative to the planet on effectively entering and leaving it's gravitational field, will remain the same-as it must according to the law of conservation of energy.
To a first approximation, from a large distance, the spacecraft appears to have bounced off the planet. The planet will have slowed very marginally, losing an equivalent amount of kinetic energy. (Recall that E_k = ½mv² and the mass of a planet is very large so that the change in velocity of the planet will be very small.)