Have you ever wondered why a peach 🍑 falls from a tree to the earth and does not fly into space? This everyday observation, which we take for granted, is closely connected to one of the fundamental laws of physics - Newton's Law of Gravitation. This law, formulated by Isaac Newton in the 17th century, describes the attraction force between objects in the universe that have mass. So, all the objects you are familiar with.

This law, is super simple and at the same time so universal! It allows us to understand many observations quantitatively:

The movement of the celestial bodies and why the moon does not fall on the earth.

The movement of satellites to enable something like Google Maps.

The trajectories of asteroids to shoot them down in time before they wipe out humanity.

The structure of whole galaxies to understand how this universe came into being.

The trajectories of balls.

The launch of a SpaceX rocket by Elon Musk.

Let's consider an everyday example. Throw a ball into the air. What happens next? The ball goes up, away from the Earth, slows down as it reaches its highest point, momentarily hangs in the air, and eventually falls back to the ground. This seemingly simple event is actually the result of an invisible force acting on the ball - gravitational force exerted by the Earth on the ball.

The Newtonian Law of Gravitation describes the gravitational force \( F_{\text g} \) between an object with a mass \( \class{brown}{M} \) and another object with a mass \( \class{brown}{m} \), which interact with each other due to their masses. It states that every mass in the universe exerts an attractive force \( F_{\text g} \) on every other mass. This means that not only the earth attracts the ball, but also the ball attracts the earth - a mutual interaction.

The formula you can use to gain an understanding of gravitational attraction is as follows:

Formula anchor$$ \begin{align} F_{\text g} ~=~ G \, \class{brown}{M} \, \frac{\class{brown}{m}}{r^2} \end{align} $$

Here is \( G = 6.674 \cdot 10^{-11} \, \frac{\mathrm{m}^3}{ \mathrm{kg} \ \mathrm{s}^2 } \) the Gravitational constant. It is a natural constant and determines how strong two masses of one killogram attract each other at a distance of one meter. It therefore determines the strength of gravity in our universe:

Gravitational constant defines the strength of the gravitational force

The two masses, as you can see, attract each other with a very small gravitational force of \( 6.674 \cdot 10^{-11} \, \mathrm{N} \).

Gravitational force between more than two objects

Of course, our universe does not consist of only two objects. If we look at the Earth, for example, it is attracted not only by its nearest neighbor, the Moon, but also by the Sun and all the other planets in the solar system. Strictly speaking, it is attracted by all the masses in the universe. However, some masses are so far away (\(r\) is large) that their attraction is practically zero and can be neglected.

But in the case of the earth not only the attraction of the moon is not negligible, but also that of the sun. Let's assume that the earth is attracted by two celestial bodies, the sun and the moon. (We neglect the other planets for the time being).

What is the total gravitational force \( F_{\text g} \) acting on the Earth and where is the Earth being pulled towards? Towards the sun? Towards the moon? Somewhere in between? To answer these questions, you need the superposition principle. It states that the total attractive force \( F_{\text g} \) on the Earth, is the vectorial sum of the individual attractive forces:

Total gravitational force is the vectorial sum of the individual gravitational forces

Here the vector \( \boldsymbol{F}_{\text m} \) is the attraction of the moon to the earth and the vector \( \boldsymbol{F}_{\text s} \) is the attraction of the sun at the earth. Note that it is NOT the sum of the magnitudes, but the sum of the vectors! To determine the gravitational force acting on the earth, you must first be familiar with vector addition. Briefly summarized, how you do this graphically:

Draw the force vectors \( \boldsymbol{F}_{\text m} \) and \( \boldsymbol{F}_{\text s} \). Their length is equal to the magnitude and the direction is away from the Earth to the respective object, that is, toward the Moon and toward the Earth, as shown in illustration 2.

Move the force \( \boldsymbol{F}_{\text m} \) parallel so the beginning of vector \( \boldsymbol{F}_{\text m} \) is at the arrowhead of vector \( \boldsymbol{F}_{\text s} \).

Move the force \( \boldsymbol{F}_{\text s} \)parallel so the beginning of vector \( \boldsymbol{F}_{\text s} \) is at the arrowhead of vector \( \boldsymbol{F}_{\text m} \).

A parallelogram of forces is created. The total force \( \boldsymbol{F}_{\text g} \) points along the diagonal as shown in illustration 2.

How the exact vector addition works, you will learn in another lesson. This chapter should only clarify that the law of gravity is not limited to the attraction between TWO masses.

Gravitational field and the gravitational acceleration

The gravitational field \( g \) caused by a body of mass \( \class{brown}{M} \) is the gravitational force \( F_{\text g} \) per mass. In other words, the gravitational field indicates what gravitational force a small sample mass \( \class{brown}{m} \) (e.g., a ball) would experience if placed at a distance \( r \) from the large mass \( \class{brown}{M} \).

We get the gravitational field by dividing the gravitational force 1 by the test mass \( \class{brown}{m} \):

Gravitational field (acceleration field)

Formula anchor$$ \begin{align} g ~=~ G \, \class{brown}{M} \, \frac{1}{r^2} \end{align} $$

The gravitational field has the unit of force per mass, i.e. N/kg (Newton per kilogram). In SI units, this is equal to \( \mathrm{m}/\mathrm{s}^2 \) (meters per square second). The gravitational field is thus a acceleration field and indicates what gravitational acceleration a test mass at distance \(r\) from mass \( \class{brown}{M} \) would experience.

Table : Gravitational acceleration not far above the surface of the celestial body. For the distance of the celestial body to the test mass, the radius of the respective celestial body was used.

Celestial body

Gravitational acceleration \( \class{red}{g} \)

Mars

3.7

Venus

8.9

Earth

9.8

Jupiter

24.8

Sun

274

Gravitational potential and potential energy in the gravitational field

You get the potential energy (gravitational energy) \( W_{\text{pot}} \) of a test mass \( \class{brown}{m} \), which is at a distance \( r \) from the large mass \( \class{brown}{M} \), if you divide the gravitational force 1 by \( r \):

The potential energy is negative (has a minus sign) so that the test mass \( \class{brown}{m} \) has a smaller ("more negative") potential energy when it is closer to the mass \( \class{brown}{M} \). It's just a convention that physicists have established that way. If you are only interested in the magnitude of the potential energy, then you can omit the minus sign.

The potential energy of a test mass (e.g. ball) in the gravitational field of the massive body (e.g. planet) can be expressed with the help of the gravitational acceleration \( g \). Multiply both sides with the radius \( r \) in Eq. 6, then you get the following equation:

Gravitational acceleration times distance

Formula anchor$$ \begin{align} g \, r ~=~ G \, \class{brown}{M} \, \frac{1}{r} \end{align} $$

Now you can replace the righthand side of Eq. 8 with \( \class{brown}{m} \, g \, r \):

Potential energy using gravitational acceleration

Formula anchor$$ \begin{align} W_{\text{pot}} ~=~ -\class{brown}{m} \, g \, r \end{align} $$

If we use an approximation for the gravitational acceleration \( g \) for the earth, then \( r \) corresponds to the height (let's call it \( h \)) above the ground:

Formula anchor$$ \begin{align} W_{\text{pot}} ~=~ \class{brown}{m} \, g \, h \end{align} $$

We have omitted the minus sign because only the magnitude of the potential energy is given here.

The gravitational potential \( V \) is potential energy \( W_{\text{pot}} \) per mass \( \class{brown}{m} \). So divide equation 7 to get the gravitational potential:

Formula: Gravitational potential

Formula anchor$$ \begin{align} V ~=~ \frac{W_{\text{pot}}}{\class{brown}{m}} ~=~ -G \, \class{brown}{M} \, \frac{1}{r} \end{align} $$

The advantage of gravitational potential over potential energy is that it is independent of the test mass \( \class{brown}{m} \).

+ Perfect for high school and undergraduate physics students + Contains over 500 illustrated formulas on just 140 pages + Contains tables with examples and measured constants + Easy for everyone because without vectors and integrals