Gravity and Gravitational Fields (VCE SSCE Physics): Revision Notes

Gravity and Gravitational Fields

Introduction: The nature of gravity

Gravity is one of the fundamental forces in physics. It is the gravitational force that holds you to Earth, keeps the Moon in orbit around Earth, and holds Earth in orbit around the Sun. This same force extends throughout our Milky Way Galaxy, holding together billions of stars. In our Local Group, three large galaxies (the Triangulum Galaxy, the Milky Way, and the Andromeda Galaxy) are all held together by gravity.

At the centre of our galaxy sits a supermassive black hole containing the mass of 4.3 million suns, where gravity is so strong that even light cannot escape. Throughout the known universe, approximately $10^{11}$ galaxies are all governed by gravity.

Historical understanding of gravity

The study of gravity has a rich history spanning cultures and millennia:

• Ancient Greece (400 BC): The philosopher Aristotle wrote early discussions about gravitation
• 7th century: The Indian astronomer Brahmagupta described gravity as an attractive force
• Early 17th century: Galileo Galilei discovered that all objects accelerate equally in free fall under gravity's influence
• Mid-17th century: Isaac Newton formulated his law of universal gravitation

Aboriginal and Torres Strait Islander knowledge

The Yolngu people of the Northern Territory developed detailed knowledge of the Moon's influence on tides. Their stories explain the Moon's phases and accurately link the Moon with changing tides. This knowledge allowed them to predict ocean phenomena such as the time and height of the next tide. This represents a deep practical understanding that contrasts with even the great astronomer Galileo Galilei, who did not believe the Moon was connected to tidal phenomena.

What are gravitational fields?

Masses attract each other even when they are not touching - this is called action at a distance. In 1849, the English scientist Michael Faraday proposed the concept of a field while studying electromagnetism. Faraday's field referred to "a region in the vicinity of a magnet affected by some force."

Field: A region where an object feels a force, such as gravitational, electric, or magnetic force. More precisely, a field is a physical quantity that has a value at each point in space.

Types of fields: Scalar and vector fields

Fields can be classified into two main types:

Scalar fields

A scalar field assigns a single numerical value to each point in space. An example is a heat map used in sports analytics to show a player's movement during a game. These maps indicate the frequency and range of movement but do not show direction.

Vector fields

Gravitational, electric, and magnetic fields are all vector fields. They provide both:

• The magnitude (strength) of the field at each point
• The direction of the field at that point

Field line conventions

Monopoles and dipoles

Monopole

A monopole is an object that is either:

• A source of a field (field lines point away from it), OR
• A sink of a field (field lines point towards it)

But not both.

Dipole

A dipole describes a source and a sink paired together, like the north and south poles of a bar magnet.

Gravitational monopole

Mass is a gravitational monopole. Since gravity is always an attractive force, mass always acts as a sink - field lines always point towards it (right diagram above).

Gravitational field properties

A gravitational field is the region around an object where other objects will experience a gravitational force. This field:

• Exists in the space around every mass or group of masses
• Extends out in all directions
• Decreases in magnitude as distance from the object increases

Because the gravitational field gets weaker the further out it is from the mass, it is described as a non-uniform gravitational field.

Gravitational field strength

Gravitational field strength ($g$) is measured in newtons per kilogram (Nkg⁻¹). It tells us the force acting on each kilogram of mass at that point in the field.

The diagram above shows circles of equal gravitational field strength encircling Earth, with arrows indicating the direction of the gravitational force.

Uniform gravitational fields

Close to Earth's surface, the gravitational field can be approximated as uniform. A uniform gravitational field has:

• Constant strength at all points
• The same direction at all points

When representing a uniform field, all field lines are:

• Evenly spaced
• Parallel to each other

Gravitational field strength values

The average magnitude of the uniform gravitational field strength measured across Earth's surface is 9.81 Nkg⁻¹. On the Moon's surface, this value is only 1.61 Nkg⁻¹ (approximately one-sixth of Earth's value).

The table below shows surface gravitational field strength for planets, our Moon, and the Sun in our solar system, in ascending order:

Body	$g$ (Nkg⁻¹)
Moon	1.6
Mercury	3.6
Mars	3.7
Uranus	8.7
Venus	8.9
Saturn	9.0
Earth	9.8
Neptune	11.0
Jupiter	23.1
Sun	274

Variations in Earth's gravitational field

The gravitational field $g$ varies by about 0.5% depending on location on Earth's surface:

• At the top of Mount Everest: 9.77 Nkg⁻¹
• At the poles: 9.83 Nkg⁻¹

Application: Using gravity mapping to study climate change

NASA has mapped Earth's gravity using the GRACE (Gravity Recovery and Climate Experiment) twin spacecraft. These satellites flew in tandem for 15 years in a polar orbit with a period of 99 minutes, completing almost 80,000 orbits.

GRACE measured changes in the local pull of gravity as water shifts around Earth due to changing seasons, weather, and climate processes. The mission:

• Monitored ice mass loss from Earth's ice sheets
• Improved understanding of sea level rise and ocean circulation
• Provided insights into shrinking or growing groundwater resources
• Monitored dry soils contributing to drought
• Tracked changes in the solid Earth

Mountain ranges like the Himalayas have high concentrations of mass and therefore stronger gravitational fields (shown in warm colours on the map).

The inverse square law

The diagram shows light radiating from a source passing through frames at single, double, and triple distances. Each frame is the same size, representing a unit of area. Close to the source, at radius $r = 1$, the light energy intensity is $E$. Further away, at $r = 3$, the light energy intensity is now one-ninth as strong: $\frac{E}{3^2}$.

Gravitational field strength formula

The strength of a gravitational field, $g$, decreases with the square of the distance, $r$, from the mass, $M$, creating the field. Isaac Newton demonstrated this in 1666 by analysing Johannes Kepler's data concerning planetary orbits around the Sun.

Formula 3A-1: Gravitational field strength

$g \propto \frac{M}{r^2} \text{ or } g = G\frac{M}{r^2}$

Where:

• $g$ = Gravitational field strength (Nkg⁻¹ or ms⁻²)
• $G$ = Universal gravitational constant = 6.67 × 10⁻¹¹ Nm²kg⁻²
• $M$ = Mass (kg)
• $r$ = Distance from the centre of the mass (m)

Gravitational constant ($G$): The universal gravitational constant has a value of $6.67 \times 10^{-11}$ Nm²kg⁻².

Calculating Earth's surface gravity

Substituting Earth's data into the formula:

• Radius of Earth, $R_E = 6.37 \times 10^6$ m
• Mass of Earth, $M_E = 5.97 \times 10^{24}$ kg

$g = (6.67 \times 10^{-11}) \frac{(5.97 \times 10^{24})}{(6.37 \times 10^6)^2} = 9.81 \text{ Nkg}^{-1}$

This matches the average gravitational field strength measured at Earth's surface.

Using the point mass approximation

When we treat Earth's mass as a point mass (all mass concentrated at the centre), we can calculate field strength at different distances:

• At Earth's surface ($R_E$): $g = 9.81$ Nkg⁻¹
• At two Earth radii ($2R_E$): $g = \frac{9.81}{4} = 2.45$ Nkg⁻¹
• At three Earth radii ($3R_E$): $g = \frac{9.81}{9} = 1.09$ Nkg⁻¹

Worked example: International Space Station

The International Space Station (ISS) orbits at a height of 408 km above Earth's surface, in a weaker gravitational field than at Earth's surface.

Graph of gravitational field strength vs distance

This graph shows how gravitational field strength decreases with distance above Earth's surface (measured in Earth radii, $R_E$). Note the characteristic inverse square relationship - the field strength drops rapidly at first, then more gradually at greater distances.

Force due to gravity

The gravitational force acting on an object can be determined by multiplying the mass of the object by the gravitational field strength at that point:

$F_g = mg$

Where:

• $F_g$ = Gravitational force (N)
• $m$ = Mass of object (kg)
• $g$ = Gravitational field strength (Nkg⁻¹)

Acceleration due to gravity

If the gravitational force is the only force acting on an object ($F_g = F_{net}$), then:

$mg = ma$

Acceleration due to gravity: The rate at which a falling object will accelerate in a gravitational field. Equivalent to the gravitational field strength; measured in ms⁻².

All objects fall at the same rate

If no other forces such as air resistance are acting, all objects close to Earth's surface will fall at the same acceleration rate, regardless of their mass. Galileo originally discovered this in the mid-17th century.

This is easily demonstrated by dropping a coin and a feather:

• In air: The coin beats the feather because of air resistance acting on the feather
• In a vacuum: When virtually all the air is removed, both land at the same time

Astronaut David Scott famously demonstrated this on the airless Moon by dropping a feather and a hammer simultaneously. Both fell at the same rate and landed on the Moon's surface at the same time.

The normal force in gravitational fields

We do not directly feel the force of gravity, $F_g$. Instead, we feel normal forces that give us a sense of gravity.

The normal force, $F_N$, is what keeps us from sinking into the ground. When standing on firm ground, the surface provides an equal but opposite force on our feet:

$F_N = -F_g$

Newton's law of universal gravitation

Combining the gravitational field strength formula ($g = G\frac{M}{r^2}$) with the gravitational force formula ($F_g = mg$) gives us Newton's law of universal gravitation.

Formula 3A-2: Newton's law of universal gravitation

$F_g = G\frac{m_1m_2}{r^2}$

Where:

• $F_g$ = Force due to gravity (N)
• $G$ = Universal gravitational constant = 6.67 × 10⁻¹¹ Nm²kg⁻²
• $m_1$ = Mass of the first object (kg)
• $m_2$ = Mass of the second object (kg)
• $r$ = Distance between the centres of mass of $m_1$ and $m_2$ (m)

Newton's law of universal gravitation: The attractive gravitational force between two masses, whose centres of mass are a distance $r$ apart, is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Newton's third law and gravity

Newton's discovery was not just that Earth's gravity creates a force on an apple pulling it down to Earth, but that the apple's gravity creates an equal force on Earth pulling it up!

• The apple accelerates down to Earth at 9.81 ms⁻²
• Earth accelerates up towards the apple at $1.67 \times 10^{-24}$ ms⁻² (not practically detectable)

However, the interaction between the Moon and Earth is more noticeable:

• Earth's force on the Moon causes it to orbit Earth
• The Moon's equal force on Earth is observable in the tides

The Moon's orbit and circular motion

Newton believed that the force of attraction between Earth and the apple, and between Earth and the Moon, was the same kind of force. He reasoned that the Moon continuously "falls" towards Earth instead of travelling off into space.

Earth's gravity provides a centripetal force on the Moon that maintains its circular path around Earth. This centripetal force causes a centripetal acceleration towards Earth.

Centripetal force: A net force directed towards the centre of the circular path of an object.

Centripetal acceleration: Acceleration directed towards the centre of the circular path of an object.

Newton's calculation for the Moon

Newton calculated the centripetal acceleration of the Moon from observation:

• Ratio of Moon's orbital radius to Earth's radius: 60
• Moon's orbital radius: $r = 3.82 \times 10^8$ m
• Moon's orbital period: $T = 2.36 \times 10^6$ s (27 days, 7 hours, 43 minutes)

Using the formula for centripetal acceleration:

$a = \frac{4\pi^2r}{T^2} = \frac{4\pi^2(3.82 \times 10^8)}{(2.36 \times 10^6)^2} = 2.72 \times 10^{-3} \text{ ms}^{-2}$

This calculation helped Newton demonstrate that gravity follows an inverse square law:

$\frac{g_{\text{moon}}}{g_{\text{apple}}} = \frac{2.72 \times 10^{-3}}{9.8} = \frac{1}{3600} = \frac{1}{60^2}$

The Moon experiences $\frac{1}{60^2}$ of the gravitational field strength because it is 60 times further from Earth's centre than an apple on the surface.

Applications of Newton's law

Planetary motion

The motion of planets around the Sun, and the interaction of planets on each other, can be understood using Newton's law of universal gravitation. The gravitational attractions of the Sun on the planets and the planets on each other explain almost all features of their motions.

Discovery of Neptune

Solar system data

The table below provides modern values for various bodies in our solar system:

Body	Mass (kg)	Radius (m)	Mean orbital radius (m)	Orbital period (s)
Sun	$1.99 \times 10^{30}$	$6.96 \times 10^8$	–	–
Mercury	$3.30 \times 10^{23}$	$2.44 \times 10^6$	$5.79 \times 10^{10}$	$7.60 \times 10^6$
Venus	$4.87 \times 10^{24}$	$6.05 \times 10^6$	$1.08 \times 10^{11}$	$1.94 \times 10^7$
Earth	$5.97 \times 10^{24}$	$6.37 \times 10^6$	$1.50 \times 10^{11}$	$3.16 \times 10^7$
Mars	$6.42 \times 10^{23}$	$3.40 \times 10^6$	$2.28 \times 10^{11}$	$5.94 \times 10^7$
Jupiter	$1.90 \times 10^{27}$	$7.15 \times 10^7$	$7.78 \times 10^{11}$	$3.74 \times 10^8$
Saturn	$5.68 \times 10^{26}$	$6.03 \times 10^7$	$1.43 \times 10^{12}$	$9.29 \times 10^8$
Uranus	$8.68 \times 10^{25}$	$2.59 \times 10^7$	$2.87 \times 10^{12}$	$2.64 \times 10^9$
Neptune	$1.02 \times 10^{26}$	$2.48 \times 10^7$	$4.50 \times 10^{12}$	$5.17 \times 10^9$
Pluto	$1.30 \times 10^{22}$	$1.88 \times 10^6$	$5.91 \times 10^{12}$	$7.82 \times 10^9$

Newton's gravitational discoveries gave theoretical support to Kepler's detailed planetary observations.

Limitations and refinements

Precise measurements of Mercury's motion showed it "wobbled" faster than could be accounted for by gravitational interactions with other known planets. Astronomers searched in vain for other bodies to explain this discrepancy.

Skills: Using the inverse square law for gravity

The inverse square law is useful for determining gravitational field strength at different locations when you know the field strength at one location, even if you don't know the mass creating the field.

Remember!