2.1 Physical quantities and units
Quantitative versus qualitative


Most observation in physics are quantitative
Descriptive observations (or qualitative) are usually imprecise
Figure 1: Qualitative
Observations
How do you
measure artistic
beauty?
2.1.1



Figure 2: Quantitative Observations.
What can be measured with the
instruments on an aeroplane?
Physical quantities
A physical quantity is one that can be measured and consists of a magnitude and unit.
It is a measurable property whose meaning is precisely defined so that everyone can have the same
understanding of the term.
The meaning of a physical quantity can be represented by :
Mass
A DEFINING EQUATION Density= Volume
A WORD DEFINITION The Density of a substance is the mass per unit volume of the substance
When quoting the measurement of a physical quantity it
is essential to state the unit as well as the numerical value
Physical quantities can be classified into two types:

Base / Fundamental quantities are the quantities
on the basis of which other quantities are
expressed. Fundamental quantities which cannot be
expressed in terms of any other physical quantity.
e.g. quantities like length, mass, time, temperature
are fundamental quantities

Derived quantities
The quantities that are
expressed in terms of base quantities are called
derived quantities
2.1.2
Base quantity is like the brick – the basic
building block of a house
Derived quantity is like the house that was
build up from a collection of bricks (basic
quantity)
S.I units
Unit
To measure a quantity, we always compare it with some reference standard. To say that a rope is 10
metres long is to say that it is 10 times as long as an object whose length is defined as 1 metre. Such a
standard is called a unit of the quantity.
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Therefore, unit of a physical quantity is defined as the established standard used for comparison of the
given physical quantity. The units in which the fundamental quantities are measured are called
fundamental units and the units used to measure derived quantities are called derived units.
Système International Units (SI system units) of fundamental quantities
In earlier days, many systems of units were followed to measure physical quantities. The British system of
foot−pound−second or fps system, the Gaussian system of cen metre −gram −second or cgs system, the
metre−kilogram−second or the mks system, were the three systems commonly followed. To bring
uniformity, the General Conference on Weights and Measures in the year 1960, accepted the SI system of
units.
Table 1: SI fundamental and supplementary quantities
 In the SI system of units there are seven
fundamental quantities and two
supplementary quantities.

It is an intriguing fact that some physical
quantities are more fundamental than
others and that the most fundamental
physical quantities can be defined only
in terms of the procedure used to
measure them.
Système International Units (SI system units) of derived quantities

SI system of units are coherent
system of units, in which the units of
derived quantities are obtained as
multiples or submultiples of certain
basic units.

A derived quantity has an equation
which links to other quantities.
Table 2: Derived quantities and their units
Units of Time, Length, and Mass:
The Second, Meter, and Kilogram
The Second

The SI unit for time, the second
(abbreviated s), has a long history.
For many years it was defined as
1/86,400 of a mean solar day. More
recently, a new standard was
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adopted to gain greater accuracy and to define the
second in terms of a non-varying, or constant,
physical phenomenon (because the solar day is
getting longer due to very gradual slowing of the
Earth’s rotation).

Cesium atoms can be made to vibrate in a very
steady way, and these vibrations can be readily
observed and counted. In 1967 the second was
redefined as the time required for 9,192,631,770 of
these vibrations.
Figure 3: An atomic clock such as this one uses the
vibrations of cesium atoms to keep time to a precision
of better than a microsecond per year
The Meter

The SI unit for length is the meter (abbreviated m); its definition
has also changed over time to become more accurate and
precise. The meter was first defined in 1791 as 1/10,000,000 of
the distance from the equator to the North Pole.

This measurement was improved in 1889 by redefining the
meter to be the distance between two engraved lines on a
platinum-iridium bar now kept near Paris.
Figure 4: 1791 definition of
meter
Figure 5: 1889 definition of meter


By 1960, it had become possible to define the meter even more accurately in terms of the wavelength
of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton
atoms.
In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light
travels in a vacuum in 1/ 299,792,458 of a second.
Figure 6: 1983 definition
of meter
The Kilogram

The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum-iridium
cylinder kept with the old meter standard at the International Bureau of Weights and Measures near
Paris.
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Dimensional analysis
In physics, the word dimension denotes the physical nature of a quantity. The distance between two
points, for example, can be measured in centimetres, metres, or kilometres, which are different ways of
expressing the dimension of length.







The symbols typically used to specify the Table 3: Dimensions of fundamental quantities
dimensions of length, mass, and time are L, M, and
T, respectively.
In physics, it’s often necessary either to derive a
mathematical expression or equation or to check its
correctness.
A useful procedure for doing this is called
dimensional analysis, which makes use of the fact
that dimensions can be treated as algebraic
quantities.
Such quantities can be added
Table 4: Dimensional formulae of some derived quantities
or subtracted only if they
have the same dimensions.
It follows that the terms on
the opposite sides of an
equation must have the
same dimensions.
If they don’t, the equation is
wrong.
If they do, the equation is
probably correct, except for
a possible constant factor.
Example problems on
dimensional analysis:
(1) Show that the expression
= + , is dimensionally
correct, where v and v0 represent
velocities, a is acceleration, and t
is a time interval.
(2) Determine whether the
equation x=vt2 is dimensionally
correct? If not, provide a correct
expression
Homogeneity of physical equations
A physical equation is homogeneous if quantities on BOTH sides of the equation have the same units.
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Note:
 Numbers have no units.
 Some constants have no unit e.g. π.
 A homogeneous equation may not be physically correct but a physically correct equation is always
homogeneous.
Table 5: Prefixes for power of ten
Prefixes and their symbols

Magnitudes of physical
quantity range from
very large to very
small. E.g. mass of sun
is 1030 kg and mass of
electron is 10-31 kg.

Prefix is used to
describe
these
magnitudes.

SI units are part of the
metric system. The
metric
system
is
convenient
for
scientific
and
engineering
calculations because
the
units
are
categorized by factors
of 10.
Convention for labelling tables and graphs

The symbol / unit is
indicated at the italics

Then fill in the data
with pure numbers

Then plot the graph
after labelling x axis
and y axis
v/ms-1
t/s
Estimation

On many occasions, physicists, other scientists, and engineers need to make estimates or
“guesstimates” for a particular quantity. What is the distance to a certain destination? What is the
approximate density of a given item? About how large a current will there be in a circuit? Many
estimates are based on formulae in which the input quantities are known only to a limited accuracy.
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As we develop problem-solving skills (that can be applied to a variety of fields through a study of
physics), we will also develop skills at estimating. These skills are developed through thinking more
quantitatively, and by being willing to take risks. As with any endeavour, experience helps, as well as
familiarity with units.
 Estimations allow us to rule out certain scenarios or unrealistic numbers. It also allows us to challenge
others and guide us in our approaches to our scientific world.
 In problem solving or calculations carried out in experiments you should always look at your answer
to see if it seems reasonable.
 When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures
since an estimate is not very precise.


Occasionally, students are asked to estimate the area under a graph. The usual method of counting
squares within the enclosed area is used.
Example problems on estimations
(1) Can you estimate the height of one of the buildings on your
campus, or in your neighbourhood? Estimate the height of a 39-story
building.
(2) Using mental math and your understanding of
fundamental units, approximate the area of a football
ground.
End of section problems
(1)
(2)
(3)
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(4)
The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h?
(5)
Tectonic plates are large segments of the Earth’s crust that move slowly. Suppose that one such
plate has an average speed of 4.0 cm/year.
(a) What distance does it move in 1 s at this speed?
(b) What is its speed in kilometres per million years?
(6)
How many heartbeats are there in a lifetime?
(7)
A generation is about one-third of a lifetime. Approximately how many generations have passed
since the year 0 AD?
(8)
How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime
of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10−22s .)
(9)
Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an
atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on
the order of 10−27 kg and the mass of a bacterium is on the order of 10−15 kg. )
(10)
Estimate the following:
Bibliography
1.
OpenStax
College,
College
Physics.
OpenStax
College.
21
June
2012.
<http://cnx.org/content/col11406/latest/>.
2.
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics [and Mastering
Physics TM], Pearson Education.
3.
Essentials of College Physics, Cengage Learning.
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