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Chapter 10.2: Gravity

Please remember to photocopy 4 pages onto one sheet by going A3A4 and using back to back on the photocopier.

Questions to make you think

1. An apple falls from a tree due to the force of gravity acting on the tree. However the earth also experiences a force

of gravity due to the existence of the apple. Which (the apple or the planet) experiences the larger force (or are

they both the same)?

. Why do two ob!ects with different masses fall at the same rate (given that from "ewton#s law the force

between two ob!ects is proportional to the product of their masses$ e.g. the force between the %arth and a

bloc& is 1''' times greater than the force between the %arth and a pebble)?

Student Notes

Newtons Law of Gravitation

states that any two point masses in the universe attract each other with a force that is directly proportional to the

product of their masses$ and inversely proportional to the suare of the distance between them.

(&nown as *big #) is a constant+ its value is ,.- x 1'11" m&g(you don#t need to &now this).

Gravity and !ei"ht

/t turns out that weight and mass are related by the following formula (nobody is uite sure why)+

Weight(as we have seen from the chapter onForce) is merely a shorthand way of sayingforce due to gravity.

0herefore we could rewrite W mg$ as 2g mg 3where 2grepresents force due to gravity4

5ut we have !ust seen above that

1

d

mGmFg =

We can euate both euations to get

1

d

mGm

mg"ow on the left hand side one of these masses (m1) represents the mass of the %arth (or any other planet).

mrepresents the mass of a second ob!ect$ while on the right hand side m also represents the mass of this ob!ect.

6o cancelling this m on both sides (corresponding to the mass of the ob!ect) we get

0his lets us calculate acceleration due to gravity (g) at any distance from a given planet.

/t also allows us to calculate the answer to the following common exam uestion+

Yo mammas so fat that objects 10 m away accelerate towards her at a rate of 1 m s-2. What is yo mammas mass

"ote that these three formulae are in page 7, of the log tables.

1

1

d

mGmFg =

W mg

d

G!g =

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Leavin" Cert #hysi$s Sy%%a&us

Content 'epth of (reatment )$tivities S(S

ravity "ewton#s law of universal

gravitation.

1

d

mGm

gF =

8ompare gravitational

forces between %arth and

6un and between %arth and

9oon.

6olar 6ystem.

Weight mg*ariation of "+ and hen$e !+

with distan$e from $entre of

,arth

)ppropriate $a%$u%ations. -!ei"ht%essness and

artifi$ia% "ravity.

:alue of acceleration due to

gravity on other bodies in space$

e.g. 9oon.

8alculation of weight on

different planets.

;resence of atmosphere.

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,/tra Credit

Newtons Law of Gravitation

6tates that any two point masses in the universe attract each other with a force that is directly proportional to the

product of their masses$ and inversely proportional to the suare of the distance between them.

9athematically+ 2 (91 9)$

And 2

1

d

;utting this together

1

d

mGmFg =

Where is a constant$ as mentioned above.

0his means that each of us is *attracted# to every one of our colleagues$ and the bigger they are$ the greater is our

attractionf course the fact that "ewton spent over ' @ of his research on the very unscientific pursuit of alchemy$ or that he

actually invo&ed deity to nudge the planets bac& in line whenever his law wasn#t providing a full explanation doesn#t

tend to ma&e it into the textboo&s.

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He also got into some nasty disputes with some of his fellow scientists$ and wasn#t above cogging results from others

and claiming them as his own. 5ut that#s another story.

#hysi$s and )stro%o"y

We can use this law to test one of the fundamental concepts in astrology= how strong is the (gravitational) influence of

a planet (or the moon) on a newborn baby compared to the influence of a nearby building?

6tic& the numbers into the formula above and see for yourselfver the following years the men became bitter enemies$

to the extent that when "ewton replaced Hoo&e as president of the Doyal 6ociety in %ngland$ the painting of Hoo&e

which had pride of place in the building mysteriously went *missing# shortly after "ewton replaced him.

>ne reason they fell out was because Hoo&e was a mechanist. He believed that it was the scientist#s !ob to engage with

nature#s causes F the cause of light$ of gravity and so on F and how these causes produce effects that we can detectwith instruments. "ewton$ on the other hand$ saw physics as concerned with the mathematical expression of effects.

2or "ewton the true causes of nature$ being divine$ were beyond the business of science.

0he truth is$ "ewton could describe gravity$ but he didn#t &now how it wor&ed. Gravity must be caused by an agent

acting constantly according to certain laws$ he admitted. G5ut whether this agent be material or immaterial$ / have

left to the consideration of my readers.

After "ewton died scientists went to a lot of trouble to promote him as the person to whom all other scientists should

aspire. 6o they downplayed his interest in both astrology and alchemy. 0hey would also have been uncomfortable with

the role which he believed od played in maintaining the motion of the heavens. 6ometimes they went a little

overboard. >ne :ictorian biographer noted that in addition to his intellectual contribution$ "ewton#s life was *one

continual source of labour$ patience$ charity$ generosity$ piety$ goodness and all other virtues$ without a mixture of any

vice whatsoever#.Hmmm. . .

"evertheless$ as / go on to explain in another chapter (8ircular 9otion) "ewton#s big achievement was to show that

the motion of the heavens did actually follow rules of ;hysics and so could be predicted. /t was this which later led

%instein to comment that Gthe most incomprehensible fact about the world is that it is comprehensible.

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!hats in a word what is"ravity

0he reason apples fall to the ground is because of *gravity#.

Bes the term *gravity# is what we use to label the phenomenon$ but does tell us anything new?

After all$ /#m sure the word preceded /ssac "ewton$ so why did he feel that there was anything left to explain?

Anytime / mention this in class / am invariably met with a set of puIIled faces$ so in an effort to illuminate the

problem / draw a picture of the planet %arth$ with an apple tree on top$ and an apple hanging out of it.

"ow everyone &nows that when the apple falls it will fall *down# to %arth. What happens next is very interesting=

6tudents don#t see that there is anything to explain about why an apple falls to the ground F it#s !ust the way it#s meant

to be (har&ing bac& to Aristotelian notions).

"ow draw an upsidedown apple tree underneath the %arth and have an apple in midair. Which way does it fall? An

upsidedown stic& figure might help them appreciate what#s going on. /t#s still pretty amaIing though. "ow

complicate things !ust a little by saying the man has !ust spilt a cup of tea F what will happen? Bou may find that our

language causes confusion here. Joes the apple *fall up to the ground#? What does this tell us about the words *fall#

and *up#?

0he apple which *falls up# may now need a little explanation. /t#s at this stage that the resident nerd declares

confidently *0hat#s easy$ gravity pulls the apple to earth#$ and unfortunately$ as with most nerds$ he#s right. 5ut wait.

>ne simple uestion can now cause the rest of your class plan to go out the window (there#s a moral in there

somewhere). GWhat is gravity? "ow at this stage a series of hands may go up because everyone &nows what gravityis= it#s what causes ob!ects to be pulled to %arth. "ow it#s time to explain what a tautology is.

5ut bac& to the apple.

Hopefully by now those of you still reading this will &now where /#m heading. Knowing aboutthe word *gravity# and

&nowing what gravity isare two rather different concepts.

/ personally &now very little about what gravity is. / could try to sound impressive throwing out terms li&e

gravitatio$al field% gravito$s% su&erstri$g theory% relativity% s&ace-timeand the li&e$ but that would be little better than

what / was correcting the students over.

/nstead what / prefer to do is to highlight the fact that "ewton himself was very uncomfortable with tal&ing about

what gravity actually was. What he did was to show mathematically some of rules which govern how gravity wor&s.He came up with a formula which would predict the force of attraction between any two masses$ and because

everybody thought that "ewton was so brilliant they decided to call his formula a Eaw$ which sounds much more

impressive (well almost everybody thought he was brilliant. how and why "ewton was promoted as (one of?)

science#s first secular saints is another story).

Which brings us nicely bac& to the apple. "ewton has many claims to fame$ but perhaps one of his greatest

achievements was acting as a signpost to all other scientists$ showing them how science shouldbe carried out. 0here

should be no reference to divine intervention$ every claim should be amenable to being verified by experiment. 0his

was why he was so uncomfortable tal&ing about whygravity wor&ed. How did the apple '$owthat the earth was

beside it? 6omehow this information must have been transferred across the space between them$ but did it ta&e time or

was it instantaneous? /n his monumental wor&(ri$ci&iahe went to great lengths to point out that his treatment of

motion would be a mathematical treatment+

#he words )attractio$*% )im&ulse* or a$y )&ro&e$sity* towards a ce$tre% however% + em&loy i$differe$tly a$d

i$tercha$geably% co$sideri$g these forces $ot &hysically but merely mathematically. #he reader should he$ce beware

lest he thi$' by words of this sort + a$ywhere defi$e a s&ecies or mode of actio$% or a &hysical cause of reaso$.

/n a separate letter to a friend he writes+

You sometimes s&ea' of gravity as esse$tial as i$here$t to matter. (ray do $ot ascribe this $otio$ to me% for the cause

of gravity is what + do $ot &rete$d to '$ow a$d therefore would ta'e more time to co$sider it.

...+f o$e as's why o$e thi$g draws a$other - +t is a$swered by ,a certai$ drawi$g$ess it hath,.

As for the LEaw of ravityM$ later Eord Kelvin flatly stated N;resent science has no right to attempt to explain

gravitation. We &now nothing about it. We simply &now ">0H/" about itN$

5ut "ewton#s gravitational law did have one thing going for it F it wor&ed.

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,/am Questions

Dadius of the earth ,., O 1',m Acceleration due to gravity at the earth#s surface .P1 m sQ

Jistance from the centre of the earth to the centre of the moon .PC O 1' Pm

1. R''CSR''7SR''PSR''P >ESR'1'S

6tate "ewton#s law of universal gravitation.

. ive two factors which affect the siIe of the gravitational force between two bodies. R'', >ES

. What is the relationship between the acceleration due to gravity g and the distance from the centre of the earth?

R''S

C. R''PS

(i) 0he international space station (/66) moves in a circular orbit around the euator at a height of C'' &m.

What type of force is reuired to &eep the /66 in orbit?

(ii) What is the direction of this force?

7. 8alculate the acceleration due to gravity on the moon.

0he radius of the moon is 1.- O 1',m and the mass of the moon is - O 1' &g. R''P >ES

,. 8alculate the acceleration due to gravity at a point C'' &m above the surface of the earth.0he radius of the earth is ,.C O 1',m and the mass of the earth is ,.' O 1'C&g. R''PS

-. 8alculate the acceleration due to gravity at a height above the surface of the earth$ which is twice the radius of the

earth. R'1'S

P. An astronaut in the /66 appears weightless. %xplain why. R''PS

. Jescribe the variation in the weight of the astronauts as they travel to the moon. R'1'S

1'. Why is the acceleration due to gravity on the moon less than the acceleration due to gravity on the earth?

R'' >ES

11. 0he earth is covered with a layer of air called the atmosphere.

What holds this layer of air close to the earth? R''7 >ES

1. 0he earth is surrounded by a layer of air$ called its atmosphere.

%xplain why the moon does not have an atmosphere. R'', >ESR'1'S

1. At what height above the earth#s surface will the astronauts experience weightlessness?

Assume the mass of the earth is P1 times the mass of the moon. R'1'S

1C. 0he moon orbits the earth every -. days. What is its velocity$ expressed in metres per second?

17. What speed are you travelling at right now? Tustify all assumptions.

-

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,/am So%utions

1. "ewton#s Eaw of ravitation states that any two point masses in the universe attract each other with a force that is

directly proportional to the product of their masses$ and inversely proportional to the suare of the distance

between them.

2. 0he mass of the ob!ects and the distance between them.

. g is proportional to 1Ud

C.

3i4 ravity(ii) 0owards the centre of the orbit U inwards U towards the earth

d

G!g =

1

d

mGm

7.

mg

g (,.- O 1'F11)( - O 1') U (1.- O 1',)

g 1., m s

,. d

G!g =

1

d

mGm

mg

note that d represents the distance from the point to the ce$treof the earth.

g (,.,- O 1'F11)( ,.' O 1'C) U (C'' ''' V ,.C O 1',) g P., m s

-. "ote that d above surface is d from earth#s centre

d

G!g =

)A( d

G!g$ew =

where d ,., O 1',m

gnew 1.' m s

P. He is in a state of freefall (the force of gravity cannot be felt).

5. Weight decreases as the astronaut moves away from the earth and gains (a lesser than normal) weight as sheUhe

approaches the moon

10. 0he earth has a greater mass than the moon.

11. ravity.

1. 0he gravitational force is too wea& to sustain an atmosphere.

1. ravitational pull of earth gravitational pull of moon

1d

mGm.

d

mGmm

1)P1(d

d

!

!

!

.==

1?

d

d=

d% dmand d%V dm .PC O 1'Pm

1' dm .PC O 1'P

dm .PC O 1'-

d% .7, O 1'P

Height above the earth (.7, O 1'P) F (,., O 1',) . O 1'Pm

P

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1C. v=distancetravelled

time =

2r

T

3.8410

2 ( 8)

27.3242460

v=

v 1'. m s1