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Space: 1. The Earth’s gravitational field

Syllabus reference (October 2002 version)
1. The Earth has a gravitational field that exerts a force on objects both on it and around it	Students learn to: define weight as the force on an object due to a gravitational field explain that a change in gravitational potential energy is related to work done define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field	Students: perform an investigation and gather information to determine a value for acceleration due to gravity using pendulum motion or computer assisted technology and identify reason for possible variations from the value 9.8ms^-2 gather secondary information to predict the value of acceleration due to gravity on other planets analyse information using the expression to determine the weight force for a body on Earth and for the same body on other planets

Extract from Physics Stage 6 Syllabus (Amended October 2002). © Board of Studies, NSW.
[Edit: 21 Aug 08]

Prior learning: Preliminary modules 8.4 (subsection 2), 8.5 (subsection 4)

Background: A gravitational field surrounds all masses. The strength of the field is defined by the vector g (units: N kg^-1), since g = F/m. Because of the way field strength is defined, for the earth’s gravitational field, g has the same numerical value as the acceleration experienced by a free falling object, a_g (units: m s^-2).

define weight as the force on an object due to a gravitational field

Weight is the force on an object in a gravitational field. It is a vector quantity and the measurement unit is the Newton (N).

explain that a change in gravitational potential energy is related to work done

Consider the work done in moving an object from the Earth's surface to a height, h metres.
- W = F.d
- Therefore, W = Fg.d (where Fg is the weight of the object)
- Fg = mg (as weight of the object = mass x acceleration due to gravity)
- Therefore, W = mg.d
- Therefore, W = mgh (h is the distance the object has been moved)
- Gravitational Potential Energy Zona Land, Edward A. Zobel

perform an investigation and gather information to determine a value for acceleration due to gravity using pendulum motion or computer assisted technology and identify reason for possible variations from the value 9.8 ms^-2

You may be performing an investigation that has been planned by your teacher. There are several suitable investigations that will achieve this purpose. One suitable investigation is described here.

A procedure for determining a value for acceleration due to gravity

A value for acceleration due to gravity can easily and accurately be measured by observing the motion of a pendulum.

Construct a pendulum at least one metre long, attached at its top to a support (such as a clamp connected to a retort stand) and with a small mass tied to its lower end to act as the pendulum bob.
Measure the length (l) of your pendulum, from its point of attachment to the centre of mass of its bob.
Pull the pendulum aside and release it so that it starts swinging. Using a stopwatch (or other device for measuring time), begin timing at an extreme of the pendulum’s motion and time ten full swings (one swing = back and forth) of the pendulum. Divide this time by ten to get a value for the average period (T) of the motion. Using this averaging technique tends to minimise random errors.

The period of a pendulum depends upon the length (l) and the value of acceleration due to gravity (g), as described in the following equation:

Rearranging this equation gives an expression that can be used to calculate g.
Substitute your values for l and T into this equation to determine a value for g.

As you gather information during your investigation, you may need to carry out repeat trials to confirm the reliability of your results. Also, you may want to use other, more accurate, timing devices or procedures to minimise the effect of random errors.
You need to be able to identify reasons for possible variations from the value 9.8 ms^-2. Variations can be as a result of two general factors; either from experimental errors or from actual variations in the value of acceleration due to gravity as an effect of local variations in the nature of the Earth’s crust.

Experimental errors might occur as a result of things like the accuracy of equipment and human reaction time in the use of equipment.

The value of acceleration due to gravity at the surface of the Earth varies from the usually accepted value of 9.8 m s^-2, due to a number of factors:
- The Earth’s lithosphere varies in structure, thickness and density. Thickness variations are a product of the source and history of the material. Oceanic crust is thinner than continental crust. Continental crust is thickest under mountain ranges. Density variations occur due to the presence of concentrated and large mineral deposits or petroleum gas and related liquids trapped in sedimentary rocks and structures. All of these variations can influence local values of g.
- The Earth’s globe is flattened at the poles. This means that the distance of the surface from the centre of the Earth is less at the poles, which increases the local value of g.
- The spinning Earth also affects the value of g. At the equator, the spin effect is greatest resulting in a lowering of the value of g. As you travel from the equator to the poles, the spin effect on g shrinks to zero.
- As a result of the above, the value of g at the surface of the Earth varies between 9.782 m s^-2 at the equator and 9.832 m s^-2 at the poles.
- The value of g reduces with altitude above the surface of a planet, becoming zero only at an infinite distance. At low Earth orbit altitude, the value of g is approximately 8.9 ms^-2.

gather secondary information to predict the value of acceleration due to gravity on other planets

Try to gather information from astronomy reference books or authoritative web sites. You will be looking for information that will assist you to predict the value of acceleration due to gravity at the surface of other planets. Note that the gas giants Jupiter, Saturn, Uranus and Neptune, do not possess a surface upon which you could stand, so that the value of g for those planets will be theoretical only. There are, however, many moons within our solar system that are large enough to possess a significant gravitational field.

You may be able to find information about the radius (r) and mass (m) of planets, and use that information to calculate a predicted value of g by using the following equation:

(G is the universal gravitational constant)
Once you have gathered the information, summarise and collate the information in a table, such as the following. Some sample values have been included.

Planet	Value of g
Mercury
Venus	8.9
Earth	9.8
Mars
Jupiter	24.8
Saturn
Uranus
Neptune
Pluto

analyse information using the expression F = mg to determine the weight force for a body on Earth and for the same body on other planets

The magnitude of the weight force on an object can be calculated using a slightly altered form of Newton’s second law: F = mg. The direction of the weight force is the same as the direction of the gravitational field at the location of the object.
The value of acceleration due to gravity at the surface of the Earth is 9.8 m s^-2. The weight of a 65 kg person can therefore be calculated as follows:
The table below shows the acceleration due to gravity at the surface of other rocky planets in the solar system, plus the Moon. Analyse the data by calculating the weight force of the same 65 kg person at each location. Click on the link below to check your answers.

Planet surface	Value of g	Weight force
Mercury	4.0
Venus	8.9
Mars	3.7
Pluto	0.6
the Moon	1.6

Answers

define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field

The gravitational potential energy of an object at some point within a gravitational field is equivalent to the work done in moving the object from an infinite distance to that point.
It can be shown mathematically that the gravitational energy, E_p, of an object with mass, m₁, a distance, r, from the centre of a planet of mass, m₂, is given by:
A graph of the E_p surrounding a planet looks like this:

When lifting an object against a gravitational field, e.g. launching a rocket, work is done on the object, that is, energy is transferred to the object. The object’s gravitational potential energy, E_p, that is, the energy it has due to its position within the gravitational field, increases as a result.

When an object moves toward the source of the gravitational field, such as when dropping a stone, energy due to position in a field is transformed into kinetic energy (the stone speeds up).

Hence the position of lowest E_p in the gravitational field surrounding a planet is at the surface of the planet.

An object only has zero E_p when it is no longer within the gravitational field, that is, a very large distance away. (Mathematically, distance must be infinite.)