GRAVITATIONAL FORCES AND FIELDS

Gravity is amazing! It is the first interaction that we experience, and it is an interaction that we take for granted. But without it, there would be no Earth, no Sun, no galaxies, no us!
You have probably seen pictures like Figure 17.1 before – astronauts can float in mid-air in the International Space Station (ISS). This is not because there is no gravity at this altitude, but because the ISS is effectively in freefall as it orbits around the Earth. The only astronauts who have experienced zero gravity are those who experienced it on their way to and from the Moon. Even then, there would be a small gravitational effect due to the pull of the Earth, the Sun and the Moon itself. This is sometimes referred to as microgravity.
All life on Earth evolved in the Earth’s gravitational field. The bodies of all animals evolved so that they could cope with the strains and stresses of the forces inherent in living in a gravitational field.
Astronauts spending long periods of time in microgravity, for example in the proposed trips to Mars, would find their bodies losing calcium and their muscles, which are used to supporting their weight, getting weaker.
One of the most amazing things about gravity is its role in the development of the Universe. Nebulae are great clouds of dust and (mostly) hydrogen gas in outer space. Irregularities in the density of the cloud lead to more material being attracted to the higher density areas, so this area becomes more and more concentrated as more material is attracted. The process gathers pace. The vast quantity of gravitational potential energy of the spread-out hydrogen becomes kinetic energy of hydrogen atoms that, once great enough, allows nuclear fusion to occur … and a star is born.
Why is this so amazing? Gravity is a very weak interaction; the electromagnetic interaction is far, far stronger. The electric repulsion between two protons in the nucleus of an atom is about 1036 times bigger (that is, 10 followed by 35 zeroes!) than the gravitational attraction.

Why does the gravitational interaction rule? The gravitational force is always attractive, whereas there are two types of charge (positive and negative) so the electromagnetic interaction can either be attractive or repulsive. In general, the negative and positive charges tend to cancel out, making any large scale object nearly electrically neutral.

We can represent the Earth’s gravitational field by drawing field lines, as shown in Figure 17.3.

The field lines show two things:
The arrows on the field lines show us the direction of the gravitational force on a mass placed in the field.
The spacing of the field lines indicates the strength of the gravitational field–the further apart they are, the weaker the field.
The drawing of the Earth’s gravitational field shows that all objects are attracted towards the centre of the Earth. This is true even if they are below the surface of the Earth. The gravitational force gets weaker as you get further away from the Earth’s surface – this is shown by the greater separation between the field lines. The Earth is almost a uniform spherical mass, although it does bulge a bit at the equator. The gravitational field of the Earth is as if its entire mass was concentrated at its centre; this is known as its centre of mass. As far as any object beyond the Earth’s surface is concerned, the Earth behaves as a point mass.
Figure 17.4 shows the Earth’s gravitational field closer to its surface. The gravitational field in and around a building on the Earth’s surface shows that the gravitational force is directed downwards everywhere and (because the field lines are very nearly parallel and evenly spaced) the strength of the gravitational field is virtually the same at all points in and around the building. This means that your weight is virtually the same everywhere in this gravitational field. Your weight does not become much less when you go upstairs.

We describe the Earth’s gravitational field as radial, since the field lines diverge (spread out) radially from the centre of the Earth. However, on the scale of a building, the gravitational field is uniform, since the field lines are equally spaced. Jupiter is a more massive planet than the Earth and so we would represent its gravitational field by showing more closely spaced field lines.

Newton’s law of gravitation

Newton used his ideas about mass and gravity to suggest a law of gravitation for two point masses (Figure 17.5).

Newton considered two point masses M and m separated by a distance r. Each point mass attracts the other with a force F. (According to Newton’s third law of motion, the point masses interact with each other and therefore exert equal but opposite forces on each other.)
Newton’s law of gravitation states that any two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation.
Note that the law refers to ‘point masses’ – you can alternatively use the term ‘particles’. Things are more complicated if we think about solid bodies that occupy a volume of space. Each particle of one body attracts every particle of the other body and we would have to add all these forces together to work out the force each body has on the other. Newton was able to show that two uniform spheres attract one another with a force that is the same as if their masses were concentrated at their centres (provided their centre-to-centre distance is greater than the sum of their radii).
According to Newton’s law of gravitation, we have:

force $ \propto \,$ product of the masses, or $F\, \propto \,Mm$
$force\, \propto \,\frac{1}{{distanc{e^2}}}$ or $F\, \propto \,\frac{1}{{{r^2}}}$

Therefore:

$F\, \propto \,\frac{{Mm}}{{{r^2}}}$

To make this into an equation, we introduce the gravitational constant G:

$F\, = \,\frac{{GMm}}{{{r^2}}}$

(The force is attractive, so F is in the opposite direction to r.)
The gravitational constant G is sometimes referred to as the universal gravitational constant because it is believed to have the same value, $6.67 \times {10^{ - 11}}\,N\,{m^2}\,k{g^{ - 2}}$, throughout the Universe. This is important for our understanding of the history and likely long-term future of the Universe.
The equation can also be applied to spherical objects (such as the Earth and the Moon) provided we remember to measure the separation r between the centres of the objects. You may also come across the equation in the form:

$F = \frac{{G{m_1}{m_2}}}{{{r^2}}}$

where ${m_1}$ and ${m_2}$ are the masses of the two bodies.

Let us examine this equation to see why it seems reasonable. First, each of the two masses is important.
Your weight (the gravitational force on you) depends on your mass and on the mass of the planet you happen to be standing on.
Second, the further away you are from the planet, the weaker its pull. Twice as far away gives onequarter of the force. This can be seen from the diagram of the field lines in Figure 17.6. If the distance is doubled, the lines are spread out over four times the surface area, so their concentration is reduced to one-quarter. This is called an inverse square law. Inverse square laws are common in physics, light or γ-rays spreading out uniformly from a point source also follow an inverse square law.

We measure distances from the centre of mass of one body to the centre of mass of the other (Figure 17.7). We treat each body as if its mass were concentrated at one point. The two bodies attract each other with equal and opposite forces, as required by Newton’s third law of motion. The Earth pulls on you with a force (your weight) directed towards the centre of the Earth; you attract the Earth with an equal force, directed away from its centre and towards you. Your pull on an object as massive as the Earth has little effect on it. The Sun’s pull on the Earth, however, has a very significant effect.