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Unformatted text preview: CHAPTER 1 What’s so important about units ... ? INTRODUCTION
• Standards of measurement
• Conversion of units **
• Dimensions **
• Scientific notion **
• Estimates and creative thinking
• Fermi problems **
** Detailed notes on the web-site I’m 185
What do you think that means? Mars Climate Orbiter:
launched December 1998,
crashed September 23, 1999.
From the Sun Sentinel ... Different countries/continents may have different units
Europe : Meters, kilometers, grams, kilograms,
liters, o C ... USA: Feet, miles, pounds, tons, pints, o F ...
Just like coinage ... Euros and Dollars
But like coinage there must be accepted conversions and
agreements ... otherwise ... “... two spacecraft teams [in Colorado and California]
... unknowingly were exchanging some vital information
in different units on measurement.” Origin of standardized units:
LENGTH In 1736-1737 Pierre Louis Moreau de Maupertuis leads
an expedition to Lapland to measure the length of a
• degree along the meridian. Originally defined as the length of the king’s foot! In France ... “le pied du roi” (the foot of the king).
• Would vary from country to country and from one king to his successor.
In 1670 Gabriel Mouton (a French vicar) suggested a Originally, 1 meter ⇒ 110, 000, 000 of the pole to unit of length called a meter. equator quadrant of the Earth’s meridian passing close to
Dunkirk, Paris and Barcelona. Origin of standardized units:
• Origin of standardized units:
LENGTH A “fallback” unit of length ... a bar of platinum- iridium with two scratches, exactly 1 meter apart.
DISCUSSION PROBLEM 1.1:
Can you think of reasons why this may not be such a
good standard today? • Today 1 meter is the distance that light travels in
299, 792, 458th of a second. Origin of standardized units:
+4 Origin of standardized units:
TIME +3 +2 SECOND:
Originally defined as 160 th of a minute, which was 160
th of an hour, which was 124 th of a day. So, basically,
it was determined by the rate of rotation of the Earth. +1 0 1980 1981 1982 1983 But there’s a problem ... of course!
We now know that the length of the day is not constant!
It varies by a few-thousanths of a second each 24 hour
period. Origin of standardized units:
TIME Origin of standardized units:
TIME One solar day
⇒ 86400s Now defined as the time that passes during
9,192,631,770 atomic vibrations of a cesium atom.
(The “atomic clock” is shown above.)
By counting we can tell the difference between
9,192,631,770 and 9,192,631,771, i.e., 1 vibration in
9,192,631,770, or 0.000000011%.
This is a precision of almost 1s in 300 years! Because of the approximately 0.0025s longer time
between the solar day and the “atomic day” ...
... every 400 days the discrepancy is 1s. This is
“made up” by adding a second to the official year every
so often! Origin of standardized units: Origin of standardized units:
Is this metrication gone mad? ... KILOGRAM:
Mass of water contained in 1 liter. (1 liter is a cube
measuring 0.10m on each side.)
(Requires a specific temperature to be specified. ) Following the proposal of the metric system in 1792 the
hour was re-defined to a 10-hour day. But, of course, the
idea didn’t catch on! However, this watchmaker did hedge
his bets by including a 12-hour watch as well. Now, the mass of a block of platinum maintained in a
vault near Paris.
A new definition based on a specific number of atoms is
currently being sought. Most physical quantities have a “dimension”, i.e., some
ratio of length, time and mass.
** Do not confuse dimensions with units **
Length ⇒ [L]:
Time ⇒ [T]:
Mass ⇒ [M]: Dimensions of some physical quantities Area units
units ⇒ m, ft, km, mi
⇒ s, min, h
⇒ kg, lb, g 2 Area: [L] × [L] ⇒ [L] .
Possible units: m2, ft 2, etc.
Volume: [L] × [L] × [L] ⇒ [L]3.
Possible units: m3, ft 3, etc.
Speed: TABLE 1 distance [L]
⇒ [L][ T]−1.
Possible units: m/s , ft/s, mi/h , etc. speed
⇒ [L][ T]−2 .
[ T][ T]
Possible units: m/s2, ft/s2 , etc. [ L ]2 Volume [ L ]3 Velocity and speed [L]
[T ] Acceleration
[T ]2 [ M ][ L ]
Pressure Area [ L ][T ]2 Mass Density Volume [ L ]3 Energy
Energy Power Time [M] [ M ][ L ]2
[ M ][ L ]2
[T ]3 Example 1: What is the dimension of 2π l
g l ⇒ [L], g ⇒ [L][ T]−2 and 2π ⇒ dimensionless ∴ l
−2 ⇒ [ T ] ⇒ [ T ]
[L][ T] (Periodic time of a pendulum). DISCUSSION PROBLEM 1.2:
Can you think of any physical quantities that do not have
dimension? Example 2: What physical quantity could
× mass × (speed)2
Use Table 1 to convert to dimensions ...
[L ] [L ]
× mass × (speed)2 ⇒ [M] ×
[ T] [ T] ⇒ [M][L]2
[ T]2 In addition to the basic units, e.g., meter, kilogram and
second, there are sub-units, such as millimeters and You can download a Table of dimensions and a Table of nanoseconds. The prefixes milli - and nano- denote prefixes from the website (under handouts). multipliers of the basic units based on various powers of
ten. For example, 1 millimeter (1 mm) is 1 × 10 −3 m. • Conversion between units
• scientific notation Important common prefixes • significant figures
Abbreviation on web-page under “useful notes”. Power Prefix 10 −15 femto- f 10 −12 pico- p 10 − 9 nano- n 10 − 6 micro- µ 10 −3 milli- m 10 −2 centi- c 10 −1 deci- d 10 3 kilo- k 10 6 mega- M • how long it takes the average person to run 100m 10 9 giga- G • price of gold 1012 terra- T Think creatively: guess NO! Examples of things you should know:
• approximate age of the universe
• approximate age of the Earth
• approximate population of the US DISCUSSION PROBLEM 1.3:
Estimate how many quarts of milk (all kinds) are
consumed each day at breakfast in the U.S. ...
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