CHAPTER 1 INTRODUCTION • Standards of measurement • length • time • mass • Conversion of units * • Dimensional analysis • Scientific notation * • Prefixes • Estimates and creative thinking • Fermi problems * * Notes on the website UNITS are critical to all of science and engineering and everyday life: However, different countries/continents may have different systems of units Europe: Meters, kilometers, grams, kilograms, liters, ! C ... USA: Feet, miles, pounds, tons, pints, ! F ... Just like coinage ... Euros and Dollars But like coinage there must be accepted conversions and agreements ... otherwise ...
Transcript
CHAPTER 1
INTRODUCTION
• Standards of measurement • length • time • mass
• Conversion of units *
• Dimensional analysis
• Scientific notation *
• Prefixes
• Estimates and creative thinking • Fermi problems *
* Notes on the website
UNITS are critical to all of science and engineering and everyday life:
However, different countries/continents may have different systems of units
Europe: Meters, kilometers, grams, kilograms,
liters, !C ...
USA: Feet, miles, pounds, tons, pints, !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.”
Mars Climate Orbiter: launched December 1998, crashed September 23, 1999.
From the Sun Sentinel ...
Origin of standardized units:LENGTH
FOOT:• Introduced by the Holy Roman Emperor Charlemagne (~800 CE) the foot was defined as the length of the king’s foot! In France it was called ... “le pied du roi”.
• Would vary from country to country and from one king to his successor.
METER:In 1670 Gabriel Mouton (a French vicar) proposed a unit of length based on the size of the Earth rather than the human anatomy. In 1791-1793 the meter was defined.
In 1736-1737 Pierre Louis Moreau de Maupertuis led an expedition to Lapland to measure the length of a degree along the meridian.
In 1793 the meter was defined as 110,000,000 of the pole
to equator quadrant of the Earth’s meridian passing close to Dunkirk, Paris and Barcelona.
Origin of standardized units:
LENGTH
Between 1795 - 1927 a “fallback” unit of length was based on the length of metal bars. For example, in 1889 the meter was defined as the distance between two lines on a standard bar of an alloy of platinum with 10% iridium measured at the melting point of ice. The bar was kept at the International Bureau of Weights and Measures near Paris. The estimated uncertainty in
measurement was 0.1→ 0.2 µm ( 1 µm = 10−6 m).
Origin of standardized units:LENGTH
• Today 1 meter is the distance light travels in
1
299,792,458th of a second.
Origin of standardized units:TIME
SECOND:
Originally defined as 1
60 th of a minute, which was
1
60 th of an hour, which was 1
24th of a day. Basically,
it was determined by the rate of rotation of the Earth.
But there’s a problem ... of course!
Origin of standardized units:TIME
The length of the day ( 86,400 seconds) is not constant! It varies by a few-thousanths of a second each 24 hour period.
× 10−3s
1973 1980 1990 2000 2010
4
3
2
1
0
−1
Origin of standardized units:TIME
With the NBS-6 clock the second is defined as the time that passes during 9,192,631,770 atomic vibrations of a cesium atom. This translates to an accuracy of <1 s in 300,000 years.
NIST-F1, which began operation in 1999, has an accuracy of 1 s in 20 million years.
The NBS-6 “atomic clock” introduced in 1975
Origin of standardized units:TIME
Metrication gone mad? ...
Following the proposal of the metric system during the French Revolution, in 1792 the 24 hour day was re-defined to a 10-hour day. But, of course, the idea didn’t catch on! Some clocks produced in France during this period like this one combined a 10 hour clock (inner circle) and a 24 hour clock (outer circle).
Origin of standardized units:MASS
KILOGRAM:The mass of a block of a platinum-iridium alloy stored, since 1889, in a vault near Paris belonging to the International Bureau of Weights and Measures. 40 identical replicas were made a distributed to countries around the world.
A new definition based on fundamental natural constants is currently being sought.
Dimensional analysis
Most physical quantities have “dimension”, i.e., some ratio of length, time and mass.
** Do not confuse dimensions with units **
• Area: [L]× [L]⇒ [L]2. Possible units: m2, ft2, etc.
• Volume: [L]× [L]× [L]⇒ [L]3.Possible units: m3, ft3, etc.
• Speed:
distancetime
⇒[L][T]
⇒ [L][T]−1.
Possible units: m/s, ft/s, mi/h, etc.
• Acceleration:
speedtime
⇒[L][T][T]
⇒ [L][T]−2.
Possible units: m/s2, ft/s2, etc.
Dimension UnitLength ⇒ [L] m, ft, km, miTime ⇒ [T] s, min, hMass ⇒ [M] kg, lb, g
TABLE 1Dimensions of some physical quantities
Area [L ]2
Volume [L ]3
Velocity and speed [L ][T]−1
Acceleration [L ][T]−2
Frequency [T]−1
Force [M ][L][T]−2
Pressure Force
Area( ) [M ][L]−1[T]−2
Density Mass
Volume( ) [M ][L]−3
Energy [M ][L]2[T]−2
Power Energy
Time( ) [M ][L]2[T]−3
Momentum [M ][L][T]−1
Angular momentum [M ][L]2[T]−1
DISCUSSION PROBLEM 1.1:
Can you think of any physical quantities that do not have dimension?
Example 1: What is the dimension of 2πℓg
?
ℓ⇒ [L], g⇒ [L][T]−2 and 2π ⇒ dimensionless
∴ℓg⇒
[L][L][T]−2 ⇒ [T]2 ⇒ [T]
This is the periodic time of a simple pendulum.
Example 2: Use dimensional analysis (not units) to determine what physical quantity is described by
Force × speed?(Use Table 1 to get dimensions.)
Force × speed ⇒
[M][L][T]2
×[L][T]
=[M][L]2
[T]3
⇒ Power .Note: an equation must be dimensionally consistent, i.e., the quantities of the left hand side must have the same dimension as quantities on the right hand side.
Question 1.1: The force (F) acting on an object that is
being swung around in a circle at the end of a string,
depends on the mass (m) and speed (v) of the object and
the radius (r) of the circle. How are these quantities
related?
Put
Because an equation must be dimensionally consistent,
then using Table 1, we have
Equating exponents we find
Hence,
We will see later that this is the expression for a centripetal
force.
F ∝mavbrc.
[M][L][T]−2 ∝[M]a[L]b[T]−b[L]c.
a = 1, b + c = 1 and b = 2.
∴F ∝ mv2
r.
c = −1.
Question 1.2: The periodic time of the swing of a
compound pendulum is
where I is the moment of inertia of the pendulum, m is its
mass, g is the acceleration of gravity and d is the distance
from the pivot point to the center of mass. What is the
dimension of I?
T = 2π I
mgd,
Since T = 2π Imgd
,
I = T2
4π2mgd ⇒ [T]2[M][L][T]−2[L]
= [M][L]2.
TABLE 2
In addition to the basic units, e.g., meter, kilogram and second, there are
sub-units, such as millimeters and nanoseconds. The prefixes milli- and
nano- denote multipliers of the basic units based on various powers of ten.
For example, 1 millimeter (1 mm) is 1× 10−3m.
Important common prefixes
Power Prefix Abbreviation
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
103 kilo- k
106 mega- M
109 giga - G
1012 terra- T
DISCUSSION PROBLEM 1.3:
Estimate how many quarts of milk (all kinds) are consumed each day at breakfast in the U.S.
