Space & Earth

Published on August 20th, 2011 | by Carl Mundy

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# The Basics of Gravitational Fields

Carl discusses the basics of gravitational fields, the equations used to describe them and how they can be represented on paper…

Gravity is all about masses attracting each other. Everything with a mass – you, me, the Earth etc. – has a gravitational field within which any other object will experience an attractive force. We do not notice our own gravitational fields because it is only objects with large masses that have a significant effect, such as the Earth, Sun and Moon – the gravitational interaction between these three bodies causes the ocean tides we observe here on Earth!

## Gravitational Fields

The force experienced by an object in a gravitational field is always attractive which is when the two objects are always pulled towards each other. We can describe the force an object experiences as a vector as it has both a value and direction. To work out the force we treat objects as point masses which is where we assume the objects behave as if all their mass is concentrated at their centres.

Gravitational field lines can be represented by lines emanating from the centre of an object with arrows showing the direction of the force that another mass would feel. An example of this is shown on the left. The closer the field lines are to each other, the larger the attractive force that would be experienced, as would be expected.

Luckily, Newton has done all the hard work for us and to calculate the force between two objects all we have to do is plug some numbers into Newton’s Law of Universal Gravitation. This law tells us the size of the force experienced between two objects with masses of $M$ and $m$ respectively. The form of this law that most people have been exposed to is the one shown below, where a negative sign (-) is sometimes used to show that the force is attractive.

In the above equation, $\mathrm{G}$ is the gravitational constant which is 6.67 × 10-11 Nm2kg-2 and $\mathrm{r}$ is the separation between the centres of the two objects, measured in meters. Newton’s Law of Universal Gravitation is an inverse-square law because the force decreases with the square of the separation. Inverse-square laws pop up in many areas of physics! The diagram above shows the force due to the big mass M acting on little mass m and the direction of that force shown by the arrow. The force due to the little mass acting on the big mass would be equal but in the opposite direction.

## Gravitational Potential Energy

When an object is moved towards or away from another object in its gravitational field, it’s gravitational potential energy (GPE) changes. The definition of GPE is that the potential energy is zero at an infinite distance ($\mathrm{r = \infty}$) from the big mass, $\mathrm{M}$. For example, if we took a small mass such as a satellite that was in Earth’s gravitational field and moved it away from the Earth, the satellite would gain potential energy. If we moved it closer to Earth, its GPE would decrease.

In other words, gravitational potential energy is the work done to move a mass m to a distance r away from the centre of the larger mass M. The GPE of any object can be calculated using the formula below which can be derived from Newton’s law of universal gravitation!

Where the gravitational constant is represented in the same way as before.

## Gravitational Potential

Simply put, gravitational potential is gravitational potential energy per unit mass. This means that we simply have to divide our expression for GPE by the small mass m. As with gravitational potential energy, gravitational potential is defined as being zero at a distance of infinity from the source of a gravitational field.

Where as on a map, contours show all the points that have the same height, in a gravitational field ,contours, or equipotentials as they are known, show all the points in a field which share the same potential. Shown in the diagram on the left, the equipotentials are shown in blue and are perpendicular to the field lines, shown again in black. For a spherical mass, like the Earth,  the equipotentials are spherical surfaces, or shells, that enclose the planet. If you were to travel along a line of equipotential you neither lose nor gain energy!

## Further Your Knowledge

Gravity is responsible for some of the largest structures known in the universe, as well as our own planet, its moon and the solar system we have spent the last fifty years or so exploring. Further your knowledge on gravity and gravitational fields by researching the following things!

• Gravity is a weak force, but why?
• Use Newton’s law of gravitation to calculate the force on the International Space Station in its orbit of 350km above the Earth’s surface and its mass of 417,000 kg.
• What does the graph of gravitational potential versus radius look like?

Header image courtesy Aimee Daniells.

#### About the Author

Astronomy PhD student from the UK with a passion for astronomy and science outreach projects. Involved with weekly science-based radio programme The Science Show on University Radio Nottingham (URN).