cahier physique - physics workbook

Physics/Dr. Lankar Name: ________________ section: ______________: grade%: _______

LAB INTRODUCTION TO LOGARITHMIC SCALE
FITTING LARGE RANGE OF NUMBERS ON A PAPER

HOME TRY WITHOUT HINT

INTRODUCTION - DISCUSSION

Last time you built a scaled model of the solar system. The solar system was reduced 10 million times to fit in a poster.
If you want to include proxima centauri (closest star, 4.3 light years away) and Andromeda (the closest galaxy, 2.48 million light years away),
it would have been impossible to do so. Let's see why.
In the previous lab , in your poster, 75cm was the distance Sun-Pluto or 75cm was for 5, 910 million km (that was your scale).
Using the same same scale, find the distance in cm you need to place Proxima and Andromeda (on your poster, as an attempt)
follow step. Use 1 light year = 10⁷ million km ( or 10¹³ km )

distance Proxima from Sun = 4.3 light years = __________ million km (use scientific notation)
REPORT THIS NUMBER (in million km) IN THE 2nd COLUMN of TABLE BELOW
distance Andromeda from Sun = 2.48 million light years = 2.48 10⁶ light years = ___________ million km
REPORT THIS NUMBER (in million km) IN 2nd COLUMN of TABLE BELOW

Complete the table below. (fill the first column). Use proportion to find the missing distances in cm.
(cross multiply and divide, use first row)

distance from Sun in poster (cm)	distance from Sun in reality (in million km)	objects
75 cm	5,190	pluto
_____________	______________	Proxima
_____________	______________	Andromeda

Is it possible to fit these distances on a poster or even somewhere in the classroom ?
Convert to distance Sun-Proxima (poster) from cm to km ___________ (divide by 100,000)
Convert to miles ______________ miles ( divide by 1.5). Using your scaled model , the first star would be placed ________ miles away.

LOGARITHMIC FUNCTION LOG = A WAY TO EXTRACT THE EXPONENT ON A BASE 10

Conclusion: we need another way to scale a very large range of distances or numbers. (broad range of values)
We could use what we call a logarithmic scale. It is very simple. First we express the large distances as a power of ten,
(like 100 = 10² ) and we plot on our paper (poster) not the number itself (like 100) but the exponent (that is 2)

100 is 10² so the exponent to plot is __________.
1,000 is _____ so the exponent to plot is __________
1 million is _______, the exponent is ___________
1 billion is _________, the exponent is _____________.
1 trillion is _________, the exponent is ___________.
1 googol is __________, the exponent is __________
1 millionth is _________, the exponent is __________ (the ending th means 1 / .. )
1 billionth is _________, the exponent is __________

you get it ? The idea is to express the values (distances) as a power of 10 and, to plot on the paper only the exponent.

Say you want to express the distance Sun- Mercury 58 (in million of km) as a power of 10. How can you do that ?
You know that 58 = 10ⁿ (n is a whole numbeR) and that n has to be between 1 and 2 (58 is between 10 and 100), but how to find
the exact exponent ? The TI can do it for you. Use the function LOG. Try what it does.
LOG(100) = ___, LOG (1000) = ___, LOG (10,000) = _____, LOG (1000,000) = ___, LOG(1,000,000,000) = _____
LOG find the exponent to place on a base 10 !!
SO if you do LOG(58) you find _________, that means 58 = 10^1.76

Check if this is true, with your TI do 10^{^ 1.76} =________ (round to the nearest one)

Now, we can fit the whole solar system and more on this piece of paper. You are going to express
the distances in powers of ten, and extract the exponent using LOG.

Planet	average distance from sun in Km standard notation	distance in million of km standard notation	LOG (distance)	distance as a power of 10, in million of km
Mercury	58 000 000	= 58	=log(58)=1.76	=10^1.76
Vernus	108 000 000	=	=	=
Earth	150 000 000	=	=	=
Mars	229 000 000	=	=	=
Jupiter	777 000 000	=	=	=
Saturn	2 969 000 000	=	=	=
Uranus	3 000 000 000	=	=	=
Neptune	4 500 000 000	=	=	=
Pluto	5 910 000 000	=	=	=
Proxima	43,000,000,000,000	=4.3 10⁷	=	=
Andromeda	24800000000000000000	=2.48 10¹³	=	=

get a ruler and graph paper. Trace a line and make a mark every 2 squares.

The first mark is the Sun. Write Sun and note 0 (exponent 0), the second mark should be noted 1, the third, 2 ....
On that new scale plot the planets, the star and the galaxy using the 4th column of above table. So Mercury is 1.76 (1.8).
below the line write: distance from Sun in millions of kilometers.
You made a logarithmic scale to model part of the universe !!

USE YOU SKILL to build a time line for the history of the world

Consider ages of various events in earth's history

events	Age (million years)	log(age) exponent on base 10
Man emerges	1
Ape man fossils	5
Rise of small mammals	37
extinction of dinosaurs	67
rise of mamals	245
cambrian exposion, vertebrae	570
first plants	2500
EArth forms	4450

Get a ruler. Trace a again a 30cm line and trace marks every 1.5cm. In the middle call it 0 . The line becomes a number line.
to the right of 0, you have 1, 2, 3, 4, 5 ... To the left you have -1, -2, -3 , -4.
(these are the exponents of base 10 ). WRite Man emerges above the zero mark. Then plot all the events using 3rd column from table
Write below the line : millions of years ago.

Is that cool ?

Say now you want to plot more recent events that happened after the homo sapiens sapiens appears.

The stonehenge was built 5000 years ago. 5000 years ago is __________ millions years ago (divide by 1,000,000)
Find the log of that number ____________ and plot this event on your line. (negative whole numbeR)

The FRench revolution was about 200 years ago or ___________ million years ago. Find the log ____________
and plot the event on this number line.

Find another event and plot it.
event = __________ years ago = ___________ million years ago. log = _____________.

GOING FURTHER

The table below gives the size in cm of different animals. Use a log scale to plot the sizes in cm on a number line. (like before).
Fill the table before.

animals	virus	bacterium	human cell	ant	hummingbird	domestic cat	wolf (with tail)	Thresher shrk	giant squid	sequoia
size in cm	0.0000005	0.0002	0.002	0.8	12	60	200	600	2200	7500
Log (size) is exponnt on base 10	=	=	log(0.002)= -2.7	=	=	=	=	=	=	=

write below the line: size in cm

REFERENCE: Functions modeling change, Connally, Hughes-Hallet. Gleason, et al.

© haplosciences 2000-2009, Veronique Lankar, PhD