Newton's Law of Gravitation
Earth and Space

Gravity and space - Physics narrative

Physics Narrative for 11-14

A Physics Narrative presents a storyline, showing a coherent path through a topic. The storyline developed here provides a series of coherent and rigorous explanations, while also providing insights into the teaching and learning challenges. It is aimed at teachers but at a level that could be used with students.

It is constructed from various kinds of nuggets: an introduction to the topic; sequenced expositions (comprehensive descriptions and explanations of an idea within this topic); and, sometimes optional extensions (those providing more information, and those taking you more deeply into the subject). 

Core ideas of the Earth in Space topic: 

  • Structure of the solar system
  • Contents of the universe
  • Scales of distance, duration and mass 
  • Orbiting and rotating
  • Gravity forces and 'down'
  • Day and night
  • Phases of the moon
  • Seasons

The ideas outlined within this subtopic include:

  • The gravitational force and field
  • Action at a distance
  • Down-ness
  • Weight and mass.

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The gravitational force

Newtons Law of Gravitation
Earth and Space

The gravitational force

Physics Narrative for 11-14

Introducing gravity

When was the last time you watched a young child playing that favourite game of throwing the toy out of the pram? You know what happens.

The child throws the soft toy out of the pram. The parent puts it back. The child throws it out… and so on. Sometimes the child pulls up on the side of the pram to see where the toy has gone.

Which way do they look? Downwards of course!

From a very early age, children become aware of the fact that things fall downwards when you let go of them. Much later on in school science lessons, they are introduced to the idea that things don't just fall but that they fall because of the force of gravity – but what is this gravity?

Gravity is a force that exists between all objects with mass in the universe. The soft toy falls to the ground because of the gravitational attraction of the Earth. Sometimes we refer to this force as the pull of the Earth.

The fascinating thing about the gravitational force is that it allows one object to pull on another without being in contact with it. If you think that this is a curious state of affairs, you'd be absolutely right!

For example: If you hold a bag of sugar in the palm of your hand, it is tempting to think that it is just the heaviness of the sugar that you can feel pushing down on your hand. Not so! In fact it is the gravitational force of the Earth that is acting on the sugar and pulling down through your hand.

Even though the bag of sugar and the surface of the Earth might be separated by a distance of more than one metre, the Earth is still able to exert its pulling force. This feature of the gravitational force is in contrast with those everyday pushes and pulls where the force acts through direct contact with the object.

For example: When you push on a door to open it, it doesn't start moving until contact is made and the force is exerted. We register this difference by saying that the gravitational force is not a contact force but acts at a distance.

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The strength of the gravitational force

Newtons Law of Gravitation
Earth and Space

The strength of the gravitational force

Physics Narrative for 11-14

The gravitational force depends on mass

It is truly mind-bending that the gravitational force acts between any two objects with mass in the universe.

For example, if you meet a pupil walking down the corridor, there is a gravitational force acting between the pair of you.

You exert a gravitational force on the pupil and the pupil exerts an equal gravitational force on you. However, it is obviously not the case that as the pupil approaches, you can feel the pull of the pupil and are drawn towards them.

The size of the gravitational force depends on the mass of the objects involved. The greater the mass, the greater the gravitational force. Furthermore, if the gravitational force is to be detectable on a human scale, then one of the objects must be as massive as a planet.

For example, at the surface of the Earth:

  • The pull of the Earth on a 1 kilogram mass is approximately 10 newton.
  • The pull of the Earth on a 2 kilogram mass is approximately 20 newton.

Here the gravitational force acts in a direction towards the centre of the Earth. In other words a 1 kilogram mass at the surface of the Earth experiences a gravitational force of about 10 newton pulling it towards the centre of the Earth.

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Action at a distance and gravitational fields

Newtons Law of Gravitation
Earth and Space

Action at a distance and gravitational fields

Physics Narrative for 11-14

How gravity alters from one planet to the next and how it varies with distance

Physicists use the idea of a field when dealing with forces, such as gravity, that act at a distance with no contact between the objects involved. The field occupies the space where the gravitational force acts, so at the surface of the Earth we live our lives in a gravitational field.

At any position, the strength of the gravitational field, or the gravitational field strength, is given by the gravitational force per kilogram.

At the surface of the Earth, for example, the strength of the Earth's gravitational field is 9.8 newton / kilogram.

This is so close to 10 newton / kilogram that this approximate value is often used.

Check this value out by hanging a 1 kilogram mass on a spring balance or newton meter. The reading shows about 10 newton.

The Moon's gravitational field

The strength of the Moon's gravitational field is only one sixth of the strength of the Earth's gravitational field. This is because the Moon has a much smaller mass.

At the surface of the Moon, the strength of the Moon's gravitational field is 1.7 newton / kilogram(An approximate value of 2 newton / kilogram is often used).

This means that if you go to the Moon, you are still pulled towards the surface by a significant force but you can jump a lot higher.

Alternatively, if you were able to land on a bigger planet such as Jupiter or Saturn, you would experience a much bigger gravitational force.

How field strength varies with distance: the inverse square law

You may have noticed that the gravitational field strengths given above are quoted for the surface of the Earth or Moon. The gravitational field strength decreases as the separation from the planet increases.

As you move away from the surface of the planet this dilution of gravity follows a definite inverse square relationship, where distances are measured from the centre of the Earth (or Moon).

Write the radius of the Earth as R, so that the surface of the Earth is a distance R from the centre.

Then:

  • At a distance R from the centre of the Earth gravitational field strength is 10 newton kilogram-1.
  • At a distance 2R from the centre of the Earth gravitational field strength is 2.5 newton kilogram-1.
  • At a distance 3R from the centre of the Earth gravitational field strength is 1.1 newton kilogram-1.

The rule is that if you:

Double the distance from the centre of the Earth, the force will be four times weaker.

Triple the distance from the centre of the Earth, the force will be nine times weaker.

In general terms, we can say that:

Gravitational field strength is inversely proportional to the square of the distance from the centre of the Earth. You can also write this as: gravitational field strength is proportional to 1r 2, where r is the distance from the centre of the Earth.

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Newton and universal gravity

Newtons Law of Gravitation
Earth and Space

Newton and universal gravity

Physics Narrative for 11-14

Newton's law

Newton's great achievement was to identify gravity as a force pulling objects (such as the infamous apple) to the ground and that this is the same force that keeps the Moon in orbit around the Earth, the Earth in orbit around the Sun, and so on. In other words, gravity is not some local force restricted to acting on the Earth but is a universal force which acts throughout the solar system and the universe.

Moreover, what Newton managed to do was to establish that the force depended directly on the masses of the objects involved and inversely on the square of the distance between them (as previously outlined). This he expressed mathematically in his universal law of gravitation, which states that:

The force which acts on mass m due to another mass M, a distance r away, is directed towards that second mass M and has a strength F which is given by:

forcegravity = G  ×  M  ×  mseparation2

Where G is the universal gravitation constant.

Notice that the gravitational force of one mass on another pulls from the centre of the first towards the centre of the second (and vice versa). Strictly speaking, Newton's equation applies to objects with spherically symmetric distributions of mass where r is the distance between their centres.

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Calculating gravitational forces

Newtons Law of Gravitation
Earth and Space

Calculating gravitational forces

Physics Narrative for 11-14

Find out how to calculate gravitational forces

Suppose you want to calculate the size of the gravitational force acting between you and your colleague as you approach each other (one metre apart) in the corridor. We can do this quite simply by using Newton's equation: forcegravity = G  ×  M  ×  mseparation2 .

Suppose: your mass, m, is 60 kilogram; the mass of your colleague, M, is 70 kg; your centre-to-centre separation, r, is 1 m; and G is 6.67 × 10-11 newton square metre kilogram-2.

Inserting these values into the equation to get 6.67 × 10-11 newton square metre kilogram-2  ×  60 kilogram  ×  70  kilogram1 metre2. You can work out this force and you'll get 2.8 × 10-7 newton.

In other words you exert a gravitational pulling force of 0.28 millionths of a newton on your colleague! The force exists but it is too small to notice in practice.

From the numbers it is clear that because the value of G is so small, the magnitude of the gravitational force will be very small, unless one or other of the objects has a very large mass.

You can use Newton's equation to check out the empirical observation that a 1 kilogram mass experiences a gravitational pull of about 10 N at the surface of the Earth. This is calculating the gravitational pull at the Earth's surface

forcegravity = G  ×  M  ×  mseparation2

Where: mass, m, is 1 kilogram; mass of the Earth, M, is 6.0 × 1024 kilogram; the radius of the Earth (separation of masses), r, is 6.4 × 106 m; and G is 6.67 × 10-11 newton square metrekilogram-2}.

Inserting these values into the equation, and work this out to get a force of 9.8 newton .

As expected, the pull of the Earth on a 1 kilogram mass at its surface is about 10 N .

Up next

But what is gravity?

Newtons Law of Gravitation
Earth and Space

But what is gravity?

Physics Narrative for 11-14

What is gravity?

You, or one of your pupils, might pose the question: Yes, but what is gravity? There is certainly a difference between being able to describe the gravitational force acting in a given situation and being able to account for the origins of that force in more fundamental terms.

Henry: Why do things in this room fall?

Teacher: They move closer to the middle of the planet.

Clare: Why do they move closer to the centre of the planet?

Teacher: Because gravity pulls them.

Bo: Why does gravity pull them?

Teacher: Gravity pulls all objects with mass.

Rajit: Why does gravity pull all masses?

Thinking only about teaching physics in school, it is legitimate to state that gravity just exists and is a phenomenon of the natural world. Somehow we have to break out of the But why? cycle. Posing the question: What is gravity? is rather like asking Why does the universe exist? Both are interesting questions and physicists are currently engaged in basic research in an attempt to uncover the fundamental nature of gravity and the universe, but you will need extensive study to understand the answers they are likely to give.

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Accounting for gravity

Newtons Law of Gravitation
Earth and Space

Accounting for gravity

Physics Narrative for 11-14

A brief explanation of gravity

This gives a very brief description of how physicists try and explain gravity, rather than just saying it is there.

The modern explanation for the gravitational force is based on Albert Einstein's general theory of relativity. According to Einstein, Newton's notion that gravity is due to a force that acts instantly, and at a distance, between objects with mass is wrong. There is no gravitational force. Rather, in Einstein's view, the gravitational force is a consequence of the geometry of space and time.

The essential idea is that space is rather like an elastic medium that can be distorted or curved and that this curvature is caused by the energy and momentum of matter and radiation.

So, in the case of the solar system, the Sun (which has a great deal of energy) causes a significant geometric distortion. A planet moving in this region of distorted (curved) space-time reacts to the distortion by moving differently to the way it would move if the Sun had been absent and the space undistorted. This is why the planets orbit the Sun, even in the absence of the invisible hand of the gravitational force reaching out to grasp them.

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Falling

Newtons Law of Gravitation
Earth and Space

Falling - physics narrative

Physics Narrative for 11-14

Falling at the same speed

One of the consequences of living in a gravitational field, where the force on an object depends on its mass, is that all objects fall at the same rate providing that there are no other forces acting.

A much-celebrated example of this occurred in 1971 when Apollo 15 astronaut Dave Scott dropped a hammer and a feather on the surface of the Moon.

The Moon's lack of atmosphere provided the ideal conditions to confirm what Galileo had anticipated centuries before. The hammer and feather fell together to hit the Moon's surface at the same instant and Dave Scott was able to report:

How about that, this proves that Mr. Galileo was correct in his findings.

Here on the surface of the Earth there is another force acting, which is air resistance. This has a significant effect on light objects (such as pieces of tissue paper or feathers) and balances out the gravitational force at relatively low speeds. The result is that most people, including most children, believe that heavier objects fall faster, which is true if air resistance has an effect and not true if this effect is removed or minimised (as it is on the Moon).

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How do we know that objects of different mass fall at the same rate?

Firstly, it is easy to repeat the experiment that Galileo is supposed to have carried out in dropping a musket ball and a cannon ball off the Leaning Tower of Pisa.

For a spectacular demonstration of this you might try dropping a melon and a peach simultaneously from the school roof or upstairs window.

For a less spectacular demonstration, try dropping a bunch of keys and an A4 sheet of paper. Naturally the keys get there first. Now repeat this process but this time, scrumple the piece of paper. Half of the class, at least, will not believe their eyes.

An approach through dialogue

Galileo used an alternative approach to argue the case, by using a thought experiment that involves two characters. Simplicio is characterised as being rather simple or dumb, whilst Salviatti is a bit of a know-all.

Salviatti asks Simplicio to consider two balls, one heavier than the other and asks:

Salviatti: Which will fall faster?

Simplicio states that it will be the heavier one. Salviatti then poses the following conundrum.

Salviatti: But if we tie them together, the larger stone will be slowed down by the smaller and the smaller stone speeded up by the larger, will it not?

Simplicio: Yes.

Salviatti: But what this means is that their joint speed is less than the speed of the larger stone on its own. Yet tied together they have an even bigger mass than the larger stone which would mean they should go faster. Should they not?

This conversation shows that the idea that heavier things fall faster is simply logically inconsistent and the conundrum exposes the weakness of this theory.

If all things fall at the same rate there is no inconsistency.

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Why do things with different masses fall at the same rate?

Newtons Law of Gravitation
Earth and Space

Why do things with different masses fall at the same rate?

Physics Narrative for 11-14

A simple explanation of why all objects fall at the same rate

If one object has twice the mass of another the Earth will pull it with twice the force:

Box of mass 2 kilogram: Pull of Earth is 20 newton

Box of mass 4 kilogram: Pull of Earth is 40 newton

Since the larger box has twice the force pulling on it (and this is what you feel when you hold it in your hand), it is tempting to predict that it will fall more quickly. But, the larger box has twice as much mass to set into motion, so it will accelerate at the same rate as a lighter object.

A force of 20 newton on a 2 kilogram mass has the same effect as a force of 40 newton on a 4 kilogram mass.

In fact, we can use Newton's second law of motion (see the SPT: Forces topic) to calculate the acceleration in each of these cases.

This is quite unlike the case for horizontal motion, where you can vary the force exerted and the mass independently.

Using the relationship to reinforce the understanding

acceleration = forcemass

For the smaller mass, the force is 20 N and the mass is 2 kg, so:

2 kg10 m s-2 = 20 N

For the smaller mass, the force is 40 N and the mass is 4 kg, so:

4 kg10 m s-2 = 40 N

Both objects fall with an acceleration of about 10 metre second-2.

This is often referred to as the acceleration due to gravity and is the value obtained if the air resistance force acting on the falling object is negligible.

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Distinguishing between gravity force and mass

Newtons Law of Gravitation
Earth and Space

Distinguishing between gravity force and mass

Physics Narrative for 11-14

The distinction between the gravity force on an object and the mass of an object is one which many pupils find confusing

Teacher: In everyday speech the weight of something is the force to support it against the pull of gravity. If there is no gravitational field then there is no need to support it. The unit of force is the newton, whether it is the supporting force or the pull of gravity.

Teacher: Now, who can tell me about mass?

Jim: The mass of an object is a measure of the amount of stuff or matter in the object. The unit of mass is the kilogram.

Teacher: That's along the right lines, but we can be more exact. The mass is a measure of how difficult it is to accelerate an object – that is to change its motion. It is connected to the amount of matter, but that's measured in moles. But you're right about the unit of mass.

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Deeper into mass

Newtons Law of Gravitation
Earth and Space

Deeper into mass

Physics Narrative for 11-14

Mass is a subtle idea – here's an account of some of its subtleties

The definition of mass in terms of the amount of stuff is the one that is commonly used in school science. Strictly speaking, however, the amount of stuff or matter should be measured in moles.

For a physicist, the mass of an object is a measure of its reluctance to be accelerated by a given force. Thus a 1 kilogram mass has a certain resistance to being accelerated by a given force, and a 2 kilogram has double that reluctance.

We sometimes express this in terms of inertia, stating that a bigger mass has a greater inertia or reluctance to be accelerated. You might consider the inertia of a juggernaut lorry or a super-tanker ship. Both of these require huge forces to set them moving (to accelerate them from rest). Equally, both need huge forces to bring them to a halt once they are moving (to decelerate them). Both have a large mass and a large inertia.

The relationship between mass and reluctance to accelerate can be seen in Newton's second law of motion:

acceleration = forcemass

Which is sometimes (although less helpfully) written as:

force = mass × acceleration

This equation allows you to predict that an object with a big mass will undergo a small acceleration if a given force is acting on it and vice versa.

So, although the physics definition of mass is in terms of resistance to acceleration, you can see that this measure is directly linked to the amount of stuff idea in that, the more matter or stuff there is in an object, the harder it is to set into motion, or stop.

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Measuring mass and gravity force

Newtons Law of Gravitation
Earth and Space

Measuring mass and gravity force

Physics Narrative for 11-14

Two different measurements

Remember that mass does not vary. If you measured the mass of an object here on Earth and on the Moon, you would find it was exactly the same. This is in line with common sense. If you take an object to the Moon, it is the same object, it looks exactly the same, and will have exactly the same reluctance to change its motion.

The simplest way to measure mass is not to compare the response of different objects to a force (so to gauge the mass in terms of their inertia or reluctance to accelerate). It is easier to compare two things with a beam balance than to measure their accelerations. If the masses are the same, the gravitational force on each will be the same and the beam will balance.

This beam will balance anywhere – even on the Moon where the gravitational field is much less, but is the same for both sides of the balance. At one time, this kind of balance was used by greengrocers, so when they gave you 1 kilogram of potatoes, they were comparing the mass of the potatoes with a standard 1 kilogram mass. The unit of mass, 1 kilogram is defined in terms of a standard 1 kg mass that is held in Sevres, just outside Paris. The mass of other objects is measured by comparing them to this mass.

Nowadays, many balances are based on the idea of measuring the force of the object on the balance. The object is placed on the balance and gravity pulls it down, compressing something in the balance and giving a reading. Such balances should be calibrated in newtons as this is the unit of force.

Of course, any set of bathroom scales that you are likely to use at home will be calibrated in kilograms (and stones and pounds). In day-to-day life we take our weight in kilograms. In scientific contexts we measure force in newtons – here the supporting force provided by a balance. This is a good example of a situation where everyday and scientific ways of talking and thinking differ from one another.

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Weightlessness

Weight
Earth and Space

Weightlessness

Physics Narrative for 11-14

Feeling weightless

An interesting challenge for teaching in this area is prompted by those pictures of astronauts in spacecraft, orbiting the Earth, who float around inside the craft and appear to be weightless.

A common term for this experience is that the astronauts are in a zero-g environment. Yet this cannot be so as the strength of the gravitational field at that height of orbit is only about 12 % less than that at the Earth's surface! So why do the astronauts appear to be weightless?

This is because there is something we all experience which is the sensation of weight. This is not the same as the force of gravity. The sensation of weight is the feeling we get from the Earth pushing up on us stopping us from falling. It is the feeling that we take for granted as the floor pushes up on our feet.

If you removed the floor instantly, you would not feel this sensation, and instead, would feel weightless. Such an effect happens if you are unfortunate enough to be in a lift and the cable breaks or if you sky-dive out of an aeroplane.

In a lift

If the lift cable breaks, both you and the lift will drop towards the ground at the same rate and you will feel no push up from the floor, so you will experience weightlessness.

In the case of the orbiting spacecraft, both the astronaut and the spacecraft are accelerating towards the Earth at the same rate. The floor is falling away from the astronauts as they fall towards the Earth, so they feel weightless. See episode 02 to find out why orbiting involves falling.

This is exactly like being in the falling lift and there is no sensation of weight. So a push on the floor of the spacecraft sends you moving up towards the ceiling just as it would do in the lift. This state is often called microgravity or weightlessness as it appears as if there is no gravity.

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Gravity, weight, weightlessness and falling

Weight
Earth and Space

Gravity, weight, weightlessness and falling

Physics Narrative for 11-14

The force that shapes the behaviour of the universe

Here you've learnt about gravity as a force, linking it to experiencing weight and weightlessness. Knowing about the forces acting on objects as a way of modelling their interaction is essential to predicting their motions, whether forced or natural.

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