PSY 324 Lecture Notes - Lecture 2: Kinematics, Johnson Matthey, Florian Cajori
21 Jun 2018
Department
Course
Professor
Chapter 2 Units, Dimensional Analysis, Problem Solving, and
Estimation
2.1 The Speed of light ................................................................................................... 1!
2.2 International System of Units ................................................................................ 1!
2.2.1 Standard Mass .................................................................................................... 2!
Example 2.1 The International Prototype Kilogram ................................................... 3!
Example 2.2 Mass of a Silicon Crystal ....................................................................... 4!
2.2.2 The Atomic Clock and the Definition of the Second ......................................... 5!
2.2.3 The Meter ........................................................................................................... 6!
Example 2.3 Light-Year .............................................................................................. 6!
2.2.4 Radians and Steradians ...................................................................................... 7!
2.2.5 Radiant Intensity ................................................................................................ 9!
2.3 Dimensions of Commonly Encountered Quantities ............................................. 9!
2.3.1 Dimensional Analysis ...................................................................................... 11!
Example 2.4 Period of a Pendulum ........................................................................... 11!
2.4 Significant Digits, Scientific Notation, and Rounding ....................................... 12!
2.4.1 Significant Digits ............................................................................................. 12!
2.4.2 Scientific Notation ........................................................................................... 13!
2.4.3 Rounding .......................................................................................................... 13!
2.5 Problem Solving .................................................................................................... 13!
2.5.1 General Approach to Problem Solving ............................................................ 14!
2.6 Order of Magnitude Estimates - Fermi Problems ............................................. 16!
2.6.1 Methodology for Estimation Problems ............................................................ 17!
Example 2.5 Lining Up Pennies ............................................................................... 17!
Example 2.6 Estimation of Mass of Water on Earth ................................................ 18!
find more resources at oneclass.com
find more resources at oneclass.com
2-1
Chapter 2 Units, Dimensional Analysis, Problem Solving, and
Estimation
But we must not forget that all things in the world are connected with one another
and depend on one another, and that we ourselves and all our thoughts are also a
part of nature. It is utterly beyond our power to measure the changes of things by
time. Quite the contrary, time is an abstraction, at which we arrive by means of
the change of things; made because we are not restricted to any one definite
measure, all being interconnected. A motion is termed uniform in which equal
increments of space described correspond to equal increments of space described
by some motion with which we form a comparison, as the rotation of the earth. A
motion may, with respect to another motion, be uniform. But the question whether
a motion is in itself uniform, is senseless. With just as little justice, also, may we
speak of an “absolute time” --- of a time independent of change. This absolute
time can be measured by comparison with no motion; it has therefore neither a
practical nor a scientific value; and no one is justified in saying that he knows
aught about it. It is an idle metaphysical conception.1
Ernst Mach
2.1 The Speed of light
When we observe and measure phenomena in the world, we try to assign numbers to the
physical quantities with as much accuracy as we can possibly obtain from our measuring
equipment. For example, we may want to determine the speed of light, which we can
calculate by dividing the distance a known ray of light propagates over its travel time,
speed of light =distance
time
. (2.1.1)
In 1983 the General Conference on Weights and Measures defined the speed of
light to be
c=299, 792, 458 meters/second
. (2.1.2)
This number was chosen to correspond to the most accurately measured value of
the speed of light and is well within the experimental uncertainty.
2.2 International System of Units
The system of units most commonly used throughout science and technology today is the
Système International (SI). It consists of seven base quantities and their corresponding
base units:
1 E. Mach, The Science of Mechanics, translated by Thomas J. McCormack, Open Court
Publishing Company, La Salle, Illinois, 1960, p. 273.
find more resources at oneclass.com
find more resources at oneclass.com
2-2
Base Quantity
Base Unit
Length
meter (m)
Mass
kilogram (kg)
Time
second (s)
Electric Current
ampere (A)
Temperature
kelvin (K)
Amount of Substance
mole (mol)
Luminous Intensity
candela (cd)
We shall refer to the dimension of the base quantity by the quantity itself, for example
dim length≡length≡L, dim mass ≡mass ≡M, dim time ≡time ≡T.
(2.2.1)
Mechanics is based on just the first three of these quantities, the MKS or meter-
kilogram-second system. An alternative metric system to this, still widely used, is the so-
called CGS system (centimeter-gram-second).
2.2.1 Standard Mass
The unit of mass, the kilogram (kg), remains the only base unit in the
International System of Units (SI) that is still defined in terms of a physical artifact,
known as the “International Prototype of the Standard Kilogram.” George Matthey (of
Johnson Matthey) made the prototype in 1879 in the form of a cylinder, 39 mm high and
39 mm in diameter, consisting of an alloy of 90 % platinum and 10 % iridium. The
international prototype is kept at the Bureau International des Poids et Mesures (BIPM) at
Sevres, France under conditions specified by the 1st Conférence Générale des Poids et
Mèsures (CGPM) in 1889 when it sanctioned the prototype and declared “This prototype
shall henceforth be considered to be the unit of mass.” It is stored at atmospheric pressure
in a specially designed triple bell-jar. The prototype is kept in a vault with six official
copies.
The 3rd Conférence Générale des Poids et Mesures CGPM (1901), in a declaration
intended to end the ambiguity in popular usage concerning the word “weight” confirmed
that:
The kilogram is the unit of mass; it is equal to the mass of the international
prototype of the kilogram.
There is a stainless steel one-kilogram standard that can travel for comparisons
with standard masses in other laboratories. In practice it is more common to quote a
conventional mass value (or weight-in-air, as measured with the effect of buoyancy), than
the standard mass. Standard mass is normally only used in specialized measurements
wherever suitable copies of the prototype are stored.
find more resources at oneclass.com
find more resources at oneclass.com
