Physics

Measuring Gravity: Thinking outside the box

Sir Isaac Newton told us how important gravity is, but left some gaps in the story. Today scientists are measuring the gravitational forces on individual atoms in an effort to plug those gaps.

In this lesson you will investigate the following:

• What is “big G”?

• How do we measure it?

• What else determines the strength of gravity?

• What is “little g”?

So stay grounded as we go back in time to follow the story of a universal constant.

This is a print version of an interactive online lesson. To sign up for the real thing or for curriculum details about the lesson go to www.cosmosforschools.com

Introduction: Gravity (P1)

Over 300 years ago the famous English physicist, Sir Isaac Newton, had the incredible insight that gravity, which we’re so familiarwith on Earth, is the same force that holds the solar system together.

Suddenly the orbits of the planets made sense, but a mystery still remained. To calculate the gravitational attraction between twoobjects Newton needed to know the overall strength of gravity. This is set by the universal gravitational constant “G” – commonlyknown as “big G” – and Newton wasn't able to calculate it. Part of the problem is that gravity is very weak – compare the electricforce, which is 10 times stronger (that’s a bigger difference than between the size of an atom and the whole universe!). Newton'swork began a story to discover the value of G that continues to the present day.

For around a century little progress was made. Then British scientist Henry Cavendish got a measurement from an experimentusing 160 kg lead balls. It was astoundingly accurate.

Since then scientists have continued in their attempts to measure G. Even today, despite all of our technological advances (we livein an age where we can manipulate individual electrons and measure things that take trillionths of a second), different experimentsproduce significantly different results.

Experiments have always used masses that you might measure in kilograms, but in the latest attempt scientists used individualatoms. They launched atoms of rubidium (a metal) up a tube with a laser, tracking their motion as it was altered by heavy metalblocks around the tube.

So, is it settled? Do we know big G? No, the scientists got a figure of 6.67191 x 10 m kg s , too far from the current “official”value, 6.67384 x 10 . While the new method marks an astounding change in tactics, the hunt goes on.

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Read the full Cosmos magazine article here.

Reflecting upon why an apple falls in a straight line perpendicular to the ground, Newton had his epiphany about gravityand its application across the cosmos. Or so the story goes.

Question 1

Isolate: Gravity is a very weak force. If you were setting up an experiment to measure the gravitational attraction between twoobjects you would need to be sure that no other forces were interfering, or at least that their influence was minimised.

What are some of the other forces you would have to consider? What sorts of steps could you take to ensure that these had noinfluence?

Gather: Gravity (P1)

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Newton said that much as an apple is attracted to the Earth and falls towards it, so too does the Moon. But clearly the Moon hasn'tcrashed into Earth and it doesn't look like it will. So how can it be falling?

A force, like gravity, acting on a mass makes it accelerate. But acceleration is change in velocity, which has two components – direction and speed. So you can accelerate a mass by:

1. changing its speed travelling in a straight line (e.g. apples falling), and/or

2. changing the direction it is moving, away from a straight line (e.g. the Moon in orbit).

Acceleration

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Newton had made incredible progress in understanding that the Moon was continually falling to Earth, but never reaching it. But hecouldn't measure big G because he didn't have all the information he needed. The Earth was the only object with a gravitational

effect strong enough to measure but he didn't know its mass – one of the values he needed for his formula. Still, he had otherinformation:

The Earth's radius is 6.37 x 10 m (this came from the ancient Greek astronomer, Eratosthenes, 240 BC)

The distance from the Earth's centre to the Moon is 3.84 x 10 m (this from another Greek, Hipparchus, 190 BC)

All objects on Earth fall with the same acceleration – 9.8 m/s (famous Italian astronomer Galileo demonstrated this in the late16 century)

In addition, Newton had discovered the formula for the acceleration of an object moving in a circle:

where a = acceleration, r = radius of orbit and T = time for one revolution, called the period.

He couldn't calculate the gravitational force acting on the Moon, but he could work out its acceleration. Acceleration is directlyproportional to the force that causes it (this is Newton's second law of motion), so he still had something to work with.

6

8

2

th

a =4 rπ2

T 2

The Moon's period, T, is 27 days, 7 hours and 43 minutes. What is its acceleration?

Note: We have to use standard units, i.e. metres and seconds, to get an answer in m/s

Calculation

First convert the period T into seconds:

Substituting into the equation:

The Moon's acceleration – worked example

2

T = ((27 × 24 + 7) × 60 + 43) × 60 = 2, 360, 580 seconds

r = 3.84 × m108

a =4 rπ2

T 2=

4 × × 3.84 ×3.142 108

2, 360, 5802

= 0.00272 m/s2

Question 1

Calculate: As well as the moon there are many man-made satellites in orbit around the Earth. Some of these are "geostationary",meaning they circle the Earth once every 24 hours, moving in the same direction as the Earth's rotation. They stay fixed over thesame point on the planet.

Geostationary satellites orbit at an altitude of 36,000 km.

What is the acceleration of a geostationary satellite?

Calculate to three significant figures. You may be best to write your calculation on paper, photograph it, then upload as an image.

Hint: What is altitude?

Question 2

Annotate: Put in the missing distances and accelerations from the calculations above to complete the diagram.

Note: The diagram isn't to scale – in reality the distance from the satellite to the Moon is vastly more than from the satellite to the Earth's

surface

Looking at the diagram it is clear that acceleration drops drastically as you move away from the Earth. How much?

This can be explored by comparing the ratios:

for objects orbiting the planet at different distances.

Inverse square law

andlarger distance

smaller distancelarger acceleration

smaller acceleration

Question 3

Contrast: The table below is set out to show the relationship between distance and acceleration for two cases illustrated in thediagram:

1. apple (on the Earth's surface) compared to a geostationary satellite, and

2. apple (on the Earth's surface) compared to the Moon.

To help show the relationship we've included a third column for the squares of the number of times the distance is increased.

Some of the values have been filled in. Fill in the rest (to two significant figures).

Case 1. How may times thedistance increased...

2. How many times theaccelerationdecreased...

3. Square the valuefrom step 1...

Apple vs. satellite r / r =sat e 6.7 a / a =e sat (r / r ) =sat e2

Apple vs. moon r / r =m e a / a =e m 3600 (r / r ) =m e2

The relationship should be clear – the values from steps 2 and 3 should be equal or very close to equal. That means that when thedistance is increased by a factor, the acceleration is decreased by the square of that factor. For example:

If you double the distance of the satellite (x2), you reduce its acceleration by four ( )

If you triple the distance (x3), the acceleration decreases nine times ( ).

This is called an inverse square relationship. It occurs whenever something is spreading out through space, like gravity from theEarth, light from the Sun or sound from a loudspeaker.

=122

14

=132

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Process: Gravity (P1)

Although he didn't know the value of G and so couldn't apply it, Newton had it right with his general equation for gravity...

General equation for gravity

The equation says that for any two objects the force of gravity acting on each one, pulling it towards the other, is:

1. Proportional to the masses of both objects

2. Inversely proportional to the distance between the objects

3. A constant value – big G – is required to set the scale so the values on each side of the equation agree. As Prof. Brian Cox saysin the video, "it sets the overall strength of gravity".

Note:

The force is the same on both objects

Even though gravity reduces rapidly as objects move apart, it never goes away entirely – it extends across the cosmos.

We can use Newton's equation to work out the gravitational attraction between all sorts of objects – for example, on the rubidiumatoms in the experiment discussed in the Cosmos article.

Let's say that a 10 kg metal block was used to attract the falling atoms, one of which was 1 cm away. The mass of a rubidium atomis 1.4 x 10 kg.

What is the gravitational attraction between the atom and the block?

Solution:

Note: Masses must be in kilograms and distances in metres for input into the equation. And we've rounded G to 6.7 x 10 m kg s

That is a very small force!

Attracting atoms – worked example

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F = G = 6.7 × = 9.4 × NmRbmblock

r2 10−11(1.4× )(1× )10−25 101

(1× )10−22 10−31

Question 1

In contrast, what is the gravitational attraction between the Sun and Mars?

Use these values:

Mass of Mars = 6.4 x 10 kg

Mass of the Sun = 2.0 x 10 kg

Distance Mars–Sun = 2.3 x 10 m

G = 6.7 x 10 m kg s

Use the text tool to show your calculations, or write on paper, photo, and upload.

Attracting planets – The Sun and Mars

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Question 2

Draw: The sketchpad below has two diagrams: A) the Sun and Mars, and B) a rubidium atom and a 10 kg metal block.

Draw arrows to represent the gravitational forces acting on all four bodies. To the best you are able with the space available, makethe size and length of the arrows represent the strength of the forces.

Question 3

Apply: Is the 10 kg block attracted towards the rubidium atom?

No

Yes

Question 4

Consider: If Mars were twice the mass that it is, what effectwould this have on the gravitational force between it and theSun?

It will not change the force

Double it

Increase it four times

Reduce it by a quarter

Question 5

Calculate: If the rubidium atom was moved from 1cm to 3 cm away from the 10 kg block, the force between theatom and block would:

decrease to one ninth the value

stay the same

decrease to one third the value

increase to nine times the value

increase to three times the value

Question 6

Consider: If you wanted to double the gravitational force actingon an apple hanging from a tree you could:

quarter the height of the apple above the ground

double the mass of the apple

double the mass of the Earth

halve the height of the apple above the ground

For a hundred years after Newton discovered that big G existed, no one knew how to measure it. The force of gravity between thesorts of objects that we can easily weigh and handle is very small, and overwhelmed by the huge downward gravitational force fromthe Earth.

Then Henry Cavendish came on the scene.

Measuring big G

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Cavendish's setup was big – the large balls were each 158 kg and 30 cm in diameter. That's the weight of two men in something thesize of a medicine ball. When he ran the experiment the ends of the rod moved just 4.1 mm, showing a force equal to the weight ofa large grain of sand.

Before the experiment Cavendish carefully tested the apparatus so he knew the force needed to twist the wire. So after it heknew the masses of the two attracting bodies, the distance between them, and the force of attraction. Plug those into Newton'sequation and it just leaves G to calculate.

Cavendish got 6.75 x 10 m kg s , compared to the currently accepted 6.67 x 10 . Not bad for such a small value detected withfairly basic equipment!

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Question 7

Think: Cavendish placed his setup in a wooden box with walls over half a metre thick, built inside a closed shed. He usedtelescopes through holes in the walls to observe the experiment.

What factors was this design intended to mitigate? How well does it deal with the factors you raised in the Introduction question?

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Question 8

Observe and describe: In your own words, describe what happened in the video when the lead bricks were placed beside the steelballs on the styrofoam bar.

Question 9

Imagine: Big G determines the strength of gravity.

Imagine a world similar to ours but where big G is greater or less than it actually is. Write a short story describing a visit to thisworld.

Hint: You could think about how plants, animals and birds have evolved. What about the forces of geology? Or sports?

Apply: Gravity (P2)

Experiment: Measuring "little g"

"Little g" is the acceleration on the Earth's surface due to gravity. Unlike big G, little g is not a universal constant but a property ofthe Earth due to its mass and size.

In fact, little g varies between 9.76 to 9.83 ms depending on where you are. The image above shows how its value changes acrossthe surface of the Earth. Red indicates slightly higher values and blue indicates slightly lower values.

Unlike measuring big G, measuring little g is relatively straightforward. It can be calculated by dropping an object from a height (h),measuring the time it takes for it to hit the ground (t) and using this simple formula:

Background

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g =2h

t2

Materials and method

Question 1

Use the above information to design an experiment to measure little g. Identify your materials, experimental setup and procedure.

Hint: You may wish to upload and annotate a photo of your experimental setup using the Sketchpad.

Results

Question 2

Conduct your experiment and record your measurements below and then calculate your value for the acceleration due to gravity.

Question 3

Calculate: Use your value for little g to calculate the mass of the Earth using this formula:

where G = 6.67 x 10 m kg s and R = radius of the Earth at 6.38 x 10 m.

Hint: Use the equation editor located in the Text widget.

M =g × R2

G

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Did you know?

Gravimeters are special machines that can measure little g very accurately. They are so sensitive that they can be fitted to planesand detect variations of little g from the air. This technique is used in mineral exploration. If there is a high concentration of a densemineral (such as iron ore) underground then the gravity reading is higher than the surrounding area.

Gravimeters have even been used in archaeology to find hidden chambers in ancient pyramids!

Question 4

Other than discovering subterranean minerals and hidden chambers in ancient pyramids, what uses can you think of for extremelyaccurate gravimeters?

Career: Gravity (P2)

What goes through your head when you compete in a sporting event? Motivational pep talks and well-planned strategies,maybe – but almost certainly not physics concepts! Yet that's exactly what Cameron McEvoy thinks about as he swims.

Cameron grew up on the Gold Coast and began swimming whenhis brother joined the local swim squad. Since then, he’s goneon to swim for Australia at the Olympic Games, theCommonwealth Games and other swimming championships —all before he turned 20!

Besides his promising swimming career, Cameron is also a hugephysics buff. After high school he spent a year focusing onqualifying for the Olympics team and decided to learn moreabout physics at the same time. He opened up the textbook hewas given by a professor while training in Manchester, andquickly fell in love with the topic. Cameron has his hands fullwith swim training and university, but always makes time toread about his favourite subject. One of his favourite topics isparticle physics, but “anything and everything to do withquantum mechanics is fascinating!” After all, physics is key tounderstanding the underlying principles of the universe, hesays.

It’s also helped him in other ways. In the pool, Cameron useswhat he knows about forces and hydrodynamics to optimise theway he moves through the water. It's also reassuring, he says,knowing that the laws of physics will get him from point A to B inthe shortest time possible as long as he obeys them. And itclearly works – Cameron recently won the Pan Pacific 100 mfreestyle championships, beating Michael Phelps and otherOlympic and world champions.

Cameron may be an Olympic athlete, but he insists he's just atypical teenager who loves surfing and hanging out with friends.He also enjoys life at Griffith University, where he studiesphysics and maths, and is looking forward to embarking on aresearch project over the summer.

Question 1

Investigate: Cameron states that he uses his understanding of physics to improve his swimming technique. Name three or moreother activities that can be improved with a better understanding of physics.

Cosmos Lessons team

Lesson authors: Dan O'KeeffeProfile author: Yi-Di NgEducation editor: Jim RountreeEducation director: Daniel Pikler

Image credits: Kate Patterson / Medipics and prose, MirelaTufman, and iStock.Video credits: Frank Noschese, Scarlett Sims, Sixty Symbols,BBC and YouTube.