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# Gravitational Fields

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You must have heard the phrase what goes up must come down. In physics terms, as Sir Isaac Newton discovered, what goes away from the earth will be pushed back towards it. So, when you throw a ball in the air, whether it goes straight up, to the left, or to the right, it eventually falls back to the ground. And this is true regardless of the height from which the ball is released.

In fact, it takes a significant amount of force, such as the engines of an aeroplane and the lift force produced by the wings, to keep pushing an object away from the earth. Without these forces, it would fall out of the sky.

## What is gravity?

Gravity is a force that attracts all objects that have a mass to each other. As the earth has a mass, it attracts other objects towards it. The same is true for other objects, which similarly attract each other towards themselves, including the earth. Even we attract the earth towards ourselves with the force of gravity.

But why isnt this obvious? Why dont we see other objects attract each other, given that they all have mass? We will consider this in what follows.

### Gravity is not just a force but a force field

A force field is a region in which an object experiences a non-contact force.

Force fields cause an interaction between objects and particles without the objects touching each other. In the case of gravity, that interaction happens between masses. Any object will experience an attractive force if you put it in the gravitational field of another object.

### The vector representation of force fields

Force fields can be represented as a system of vectors, as in this diagram, in which the arrows represent the gravitational field on the earth.

Figure 1. Gravitational field lines. Source: Sjlegg, Wikimedia Commons (Public domain).

The earths gravitational field is radial, which means that the lines of force intersect at the centre of the earth.

As the diagram shows, the field lines are closer together at the surface of the earth. This indicates that the gravitational force is stronger here. Where the lines move further apart from each other, the force decreases.

## How do we calculate the force of gravity?

Have a look at the equation below, which represents Newtons law of gravitation:

• F = magnitude of the gravitational force.
• G = gravitational constant.
• r = distance between the centres of two masses.
• = mass of one of the objects.
• = mass of the other object.

Newtons gravitational field: when two bodies are placed in a gravitational field, they experience a force that is the product of the two masses and the inverse square of the distance from the centre of both masses.

The constant G is a gravitational constant, which has a very small value:

Calculate the gravitational force between two 3kg spheres that are 2m apart.

The mass of both objects is 3kg. So m1 and m2 are 3kg, while r is 2m, with G being 6.67 * 10 ^ -11 Nm ^ 2 / kg ^ 2. Putting in all the values gives us:

The gravitational constant G, which, as we said, has a very small value, is the reason why objects dont fly and collide with each other. It is also the reason why the earth is not attracted to us but we to it. After all, our mass is negligible compared to that of the earth.

Figure 2. Force F acting on m2 due to m1. Source: Usama Adeel, StudySmarter.

The distance between the two objects has more impact than their masses because Newtons gravitational equation follows an inverse square law. This means that if the distance doubles, the force is one-quarter of the strength of the original force.

## What is the gravitational force of a single mass?

The force of a single mass is its gravitational field strength, which is defined as force per unit mass when it is placed in a gravitational field.

• g is measured in units of newtons per kilogram ().
• F is the force experienced by mass m when it is placed in a gravitational field.

As the gravitational field on the earths surface is almost uniform, we can assume g to be constant. Hence, g is just the acceleration of mass m in a gravitational field.

### Point masses

Point masses are objects that behave as if all mass is concentrated at their centre. Uniform shapes have a point mass.

The significance of point masses is that they have a radial gravitational field. In this, the field lines radiate from its centre. For point masses, our earlier equation becomes:

• m = mass of the object (kg).
• G = gravitational constant ().
• r = distance from the centre (m).

### The gravitational field strength of other planets

The gravitational force depends on the mass of the planet. Mars, for instance, has a gravitational field strength of 3.71 N/kg because it is only about half the diameter of the earth. But, and here comes the interesting part, your weight also depends on the gravitational force g.

### Mass and weight

Your mass is the same wherever you go in the universe. What differs is your weight, which depends not only on your mass but also on gravity. So, for instance, if you weigh 99.8kg on earth, you would only weigh 37.74kg on Mars.

The moon has a gravitational force of 1.62 N/kg. This is why on the moon, it is easier to fly than to walk. On Mars, walking becomes a bit easier but is still a challenge because of the low gravitational pull.

### Mass and distance

The tides that form on the surface of the earth show how both mass and distance affect the gravitational force.

We get tides on the earths surface because of the gravitational pull of the moon and the sun. And although the sun has far more mass than the earth, the distance between the two plays a significant role due to the inverse square proportionality. As the moon is much closer to the earth, the earths oceans respond to the moon revolving around the earth, which causes the tides. The sun does have an impact, too, but the tides produced by the sun are much smaller.

## Gravitational Fields - Key takeaways

• Gravity is all about masses attracting one another.
• A gravitational field is a force field in which an object experiences a force.
• Only large masses, such as the sun, the moon, and other planets, have a significant gravitational force.
• Newtons law of gravitation is:

• The law of gravitation is an inverse square law, which says that the gravitational force decreases when the distance between the objects increases.

• Gravitational field strength is force per unit mass acting on an object placed in a gravitational field.

• In a radial field, the gravitational field can be represented as:

## Frequently Asked Questions about Gravitational Fields

The gravitational field strength on earth is 10 N/kg.

Gravitational field strength is calculated as follows:

g=F/m

It is measured in Newtons per kilogram (N/kg).

A gravitational field is a region where an object experiences a gravitational force due to the presence of another object.

This can be calculated using the equation below:

weight = mass * gravitational field strength

w=m*g

## Final Gravitational Fields Quiz

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Newton’s second law of motion is the law that relates the net force to the rate of change of momentum.

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Newton’s first law of motion implies that moving objects in space move with constant velocity.

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Newton’s law of gravity states that the force of attraction between two bodies is the same for each of them.

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Rockets move because they expel particles with momenta, and, due to Newton’s third law of motion, momentum is generated on the rocket.

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Is the formulation ‘the total force equals the mass times the acceleration’ always true?

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No, ‘the total force equals the mass times the acceleration’ is only true if the mass is constant.

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According to Newton’s laws, can we decelerate light by exerting a force?

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According to Newton’s laws, we cannot decelerate light by exerting a force because light does not have mass.

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Why do objects lying on a surface remain still (not move) according to Newton’s laws?

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Objects lying on a surface do not move because there is no net force acting on them to change their resting state.

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What is the inertial mass?

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The inertial mass is the name given to the mass appearing in Newton’s laws.

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Why do objects moving at a certain speed usually stop at some point? Choose the correct answer.

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Because there are no ideal conditions for Newton’s first law to be applied. There is always some force, usually friction.

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Explain in terms of Newton’s third law of motion why we can swim? Choose the correct answer.

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When we move our arms and legs in the water, we displace particles of water that exert a force back at us, causing us to move.

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If an object is spinning in circles in a no-friction circular motion and the circular motion suddenly stops, what happens to the object? Choose the correct answer.

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If an object suddenly stops spinning, it will remain moving in a straight line tangent to the original circle at constant velocity.

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What is the principle of equivalence?

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It is the principle that states that the mass measuring the intensity of the gravitation interaction and the inertial mass are the same.

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Do the masses in Newton’s law of gravitation play an equivalent role?

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Yes, exchanging them amounts to a global sign in the formula, which accounts for the opposite direction consistent with Newton’s third law.

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The gravitational field strength is sourced by mass.

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The gravitational field strength is sourced by mass and affects masses.

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Newton’s description of the gravitational field strength is compatible with more modern descriptions.

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Radial dependence is a sign of isotropy, i.e. equivalence of all spatial directions.

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The gravitational field strength on the Sun is greater than on Jupiter.

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Who stated the first rigorous formulation of the gravitational field strength?

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Newton stated the first rigorous formulation of the gravitational field strength.

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What are two characteristics of a theory to describe the gravitational field strength?

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The two characteristics of a theory to describe the gravitational field strength are

• sourced by masses
• attractive (decaying with distance from the source)

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Do masses play an equivalent role when computing the forces caused by the gravitational field strength?

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Yes, masses play an equivalent role when computing the forces caused by the gravitational field strength because one can describe the attractive force from the point of view of one mass or the other.

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Can you name another theory about gravitational field strength that is more fundamental than Newton’s?

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Einstein’s theory of general relativity.

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Is the gravitational field strength constant on the surface of planets and stars?

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Yes, the gravitational field strength is constant on the surface of planets and stars because they have an approximately spherical shape and the surface is at a constant radial distance.

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Is Earth’s gravitational field strength constant when an object is inside it and travelling towards its core?

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No, Earth’s gravitational field strength is not constant when an object is inside it and travelling towards its core because the radial distance is decreasing (as well as the amount of mass attracting towards the core).

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Is Earth’s gravitational field strength constant while getting away from the surface?

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No, Earth’s gravitational field strength is not constant while getting away from the surface because the radial distance is increasing.

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Which astronomical object in the Solar System has the biggest surface gravitational field strength?

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The Sun has the biggest surface gravitational field strength (it is the biggest astronomical object in the Solar System, but this is not the only variable because its mass is important too).

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Which planet in the Solar System has the biggest surface gravitational field strength?

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Jupiter has the biggest surface gravitational field strength (it is the biggest planet in the Solar System, (but this is not the only variable because its mass is important too).

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Does the gravitational field strength have an infinite range?

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Yes, the gravitational field strength has infinite range. It can reach arbitrarily long distances with a non-zero value of intensity.

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The gravitational field has an associated gravitational potential field.

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The gravitational potential energy is sourced by masses.

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The gravitational potential energy decreases with radial distance.

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Gravitational potential energy explains why we get tired when going up a hill.

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The gravitational potential energy can be approximated through a simpler equation for points close to the earth’s surface.

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What is the feature of the gravitational field that ensures the existence of a gravitational potential?

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The gravitational field is conservative.

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Why does the gravitational potential have spherical symmetry?

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Because there are no special spatial directions.

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Is the gravitational potential a vector field?

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No, it is a scalar field measuring the energy.

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What is the name of the principle that says that bodies tend to states with the lowest possible energy?

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The stability principle.

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Does the gravitational potential fully capture the dynamics of a body under the influence of a gravitational field?

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Yes, it does.

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On which principle is the gravitational potential energy based to capture the dynamics of a body under the influence of a gravitational field?

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The stability principle.

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Is the formula for the gravitational potential energy over the surface of the earth valid for all points outside the earth?

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No, it is just an approximation.

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Is the gravitational potential energy zero at large distances?

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Almost, it approaches asymptotically zero.

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How would you compute the gravitational field from a gravitational potential?

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By differentiating with respect to the radial distance.

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Is the gravitational potential energy finite at zero radial distance?

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No, it approaches minus infinity.

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The gravitational force is a spherically symmetric interaction.

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In a circular motion, speed is constant.

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In a circular motion caused by the gravitational interaction, the speed allows us to know the radial distance of the orbit.

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For an elliptical orbit, the object will usually have points where the potential energy is larger and smaller.

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Are orbits caused by the gravitational interaction between bodies?

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Yes, the attraction force causes objects to move around the gravitational source and to escape from its influence or remain trapped by it.

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Does the gravitational potential energy formula apply to complex orbits (not only circular)?

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Yes, it is independent of the type of orbit.

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Do planetary satellites have approximate circular orbits?

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Yes, they are closed orbits whose initial conditions allow us to treat them as circular.

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