Physical Quantities and Measurement

Let's talk about physical quantities and what it takes to measure them!

• Physical Quantities and their classifications
• Fundamental and derived quantities 
• Measurement of length, mass, weight, time and electric charge
• Dimensional analysis

By the end of this lesson, students should be able to:

• Explain what is meant by a unit of measurement
• Discuss the significance of units in physical measurements
• State the different systems of measurement in physics
• List the Fundamental Units
• Distinguish between fundamental and derived quantities
• Distinguish between a fundamental unit and a derived unit
• Determine the dimension of at least 5 physical quantities 
• Determine the units of a physical quaintly given the dimensions

Topic: Physical quantities and Measurement of length, mass, weight, time and electric charge
Sub – Topics:
·         Physical quantities and their classifications
·         Fundamental and derived quantities
·         Measurement of physical quantities

·         Dimensional analysis

PHYSICAL QUANTITIES AND MEASUREMENTS

A quantity refers to any measurable property of a substance. For a clear indication of the property measured, measurements of quantity are specified by the units which are often written in their symbols. The international system of unit abbreviated SI after the French Systeme Internationale is a modernized version of quantity measurements. The SI unit was established by agreement in 1960.

Quantities could either be basic (fundamental) or derived. A quantity is said to be derived if its measurement depends on other measurements. For simplicity, SI uses seven basic units (fundamental). The seven basic quantities, their units and symbols are:

Quantity	Unit	Symbol
Length	Meter	M
Mass	Kilogram	Kg
Electric current	Amperes	A
Temperature	Kelvin	K
Amount of substance	Moles	mol.
Luminous intensity	Candelas	Cd

Physical quantities can also be classified as scalar and vector quantities. Scalar quantities are quantities with only magnitude e.g. length, distance, speed, temperature, heat, pressure, mass, amount of substance, density etc. while vector quantities or simply vectors are quantities with both magnitude and direction e.g. position, displacement, velocity, acceleration, force, temperature gradient, electric field, magnetic field, gravitational field etc.

UNITS AND DIMENSIONS

UNITS   

A unit has to be defined before any kind of measurement can be made. You are aware that different systems of units have been used in the past. However, the system which has gained universal acceptance is the System International d' Unites usually called S. I. Units (adopted in 1960). The S.1 Unit is based on the metre as the unit of length; the kilogram (kg) as the Unit of Mass; the second (s) as the Unit of Time, the ampere (A) the unit of current and the Kelvin (K) the unit of temperature. Other Units are given in Tables below

S/N	QUANTITY	FORMULA	S.I UNIT
1.	Area	l×b	m2
2.	Volume	l×b ×h	m3
3.	Density	M/V	Kg/m3
4.	Specific gravity	(Density of substance)/(Density of water)	No units
5.	Frequency	(no of Vibrations)/time	Hertz
6.	Angle	Arc/radius	No units
7.	Velocity	Displacement/time	m/sec
8.	Speed	Distance/time	m/sec
9.	Areal velocity	Area/time	M2sec-1
10.	Acceleration	(Change in velocity)/time	m/sec2

CLASSWORK

State the importance of unit in physics

ASSIGNMENT

Distinguish between dimension and dimensionless quantities

System	Length	Mass	Time
FPS	Foot	Pound	Second
CGS	Centimetre	Gram	Second
MKS	Meter	Kilogram	Second

C.G.S system: in this system, the unit of length is centimetre; the unit is mass of gram, and the unit of time is second.

F.P.S System: In the F.P.S system, the unit of length is foot, the unit of mass is pound and the unit of time is second.

M.K.S system: in this system, the unit of length is meter, the unit of times is second.

Coherent System of units: a coherent system of units is a system based on a certain set of basic (or fundamental) units from which all derived units may be obtained by simple multiplication or division without introducing any numerical factors. S.I system of units is a coherent system of units for all types of physical quantities.

CONVERSION OF UNITS

From the SI which is also known as the modernized MKS (meter, kilogram, second) system of unit; other systems CGS (centimetre, grams, second) or Gaussian and the British system are commonly used in scientific work. There exist relationships between these systems of units. For example;

        1 centimeter (cm) in CGS = 10^-2meter (m) in MKS (or SI)

        1 slug (= 32.17 pound) in British system = 1459kg in MKS

Bearing the relationship in mind, one can therefore convert a measurement from one system of units to another.

EXAMPLE

Convert the speed of a car moving at 50mile per hour into km/hr and m/s.

SOLUTION

Since 1 mi = 1.609km

50 mi/hr = (50 × 1.609km/1hr) = 80.45km/hr

Also, 1km = 1000m and 1hr = 3600s

80.45km/hr = 80 × 1000m/3600s =22.35m/s

CLASSWORK

Convert the speed of a car moving at the of 18km/hr to m/s

ASSIGNMENT

List 20 physical quantities different from those discussed in class

FUNDAMENTAL AND DERIVED QUANTITIES AND THEIR UNITS

FUNDAMENTAL QUANTITIES AND THEIR UNITS

You may have observed that there are two types of physical quantities. One group of such quantities keep occurring again and again. They are independent of others and cannot be defined in terms of other quantities.

You may also have noticed that most other quantities depend on them. This group of quantities are called fundamental quantities.

• Fundamental quantities are the basic quantities upon which other quantities depend • Fundamental units are the basic units upon which other units depend. They are the units of the fundamental quantities.

• Table below shows the fundamental quantities and their units.

Examples of Fundamental Quantities and their Units

Quantity	Unit	Unit abbreviation
Length	Metre	m
Time	Second	s
Mass	Kilogram	kg
Electric Current	Ampere	A
Temperature	Kelvin	K
Luminous Intensity	Candela	cd
Amount of substance	Mole	mol

The unit abbreviation is a capital letter when it refers to the name of a person e.g. Ampere (A) or Kelvin (K).

DERIVED QUANTITIES AND THEIR UNITS

At this point you may have realized that all the quantities with the exception of the seven fundamental quantities belong to this group.

• Derived quantities are those quantities derived by some simple combination of the fundamental quantities.

• Derived units are the units of the derived quantities.

• Example of derived quantities, their derivation and their units are summarized in Table below.

Examples of Derived Quantities and their Units

Derived Quantity	Derivation	Derived Unit
Area (A)	Length x breadth	m2
Volume (V)	Length x breadth x height	m3
Density	Mass/volume	kgm-3
Velocity (V)	Displacement/time	ms-1
Acceleration (a)	change in velocity/time	ms-2
Force (F)	Mass x acceleration	Newton; N
Energy or Work (W)	Force x distance	Joule, J (Nm)
Power (P)	Work/time	Watt, W (Js-1)
Momentum	Mass x velocity	kg.m.s-1, Ns
Pressure (P)	Force/area	Nm-2 or Pascal, Pa
Frequency (f)	number of oscillation/time	Per second or s-1 (Hertz, HZ)
Electric charge		Coulomb (c)
Electric potential difference	Work/charge	Volt (V)
Elector motive force	Work/charge	Volt (V)
Electric resistance	Electric potential difference/current	Ohm  ( Ω )
Electric capacitance	Charge/Volt	Farad (F)

• Supplementary Units

You will notice that the Units of plane angle, the radian and of solid angle, the steradian are not classified as fundamental or derived units. They are sometimes called supplementary units.

• For many measurements, you will realize that the units may be too big or too small and so multiples and submultiples of the basic units are used.

These are formed by using the following prefixes shown in Table below.

Multiple and Submultiples of Units with their Prefixes and Symbols

Submultiples	Example	Prefix	Symbol
0.1 or 10-1	decimetre = 10-1 m	Deci	d
0.01 or 10-2	centimetre = 10-2 m	Centi	c
0.001 or 10-3	millimetre = 10-3 m	Milli	m
0.000001 or 10-6	microfarad = 10-6 f	micro	µ
0.000000001 or   10-9	nanosecond = 10-9 s	nano	n
0.000000000001 or 10-12	picoampere = 10-12 A	Pico	p
Multiples	Example	Prefix	Symbol
101	decametre = 101 m	deca	da
102	hectormetre= 102m	hector	h
1000 or 103	kilometre = 103m	kilo	k
1,000,000 or 106	megawatts = 106W	mega	M
1,000,000,000,000 or 1012	Terabytes = 1012B	tera	T

CLASSWORK

Distinguish between:

a.   Fundamental and derived quantities; and

b.   Fundamental and derived units.

ASSIGNMENT

List in tabular form 10 other derived quantities and state their derivation and S.I Units

DIMENSIONS

We often use the term dimension in Physics to describe the relationship between a physical quantity and the fundamental quantities expressed in terms of the symbols M, L, T of the fundamental quantities mass, length and time respectively. You should note that physical quantities can either be dimensional or dimensionless.

• Dimensionless Quantities do not depend on the system of unit in which they are measured. That simply means that they have no units. Examples are angles and their trigonometric ratios (ratio of two lengths); relative density (or specific gravity), (ratio of two densities); the efficiency of a machine (ratio of two quantities of work).

• Dimensional Quantities on the other hand depend on the magnitude of the fundamental units in which they are measured and are different in different unit systems. If you now consider that the fundamental quantities are denoted by the symbols M, L, and T, you will be able to determine (derive) the dimensions of Physical quantities by your careful study of these illustrations.

i)             Area (length x breadth) has dimension of L x L = L2

ii)           Volume (length x breath x depth) has dimensions of L x L x L = L3

iii)          Velocity (distance ÷ time) has dimension of L  ÷ T = LT-1

iv)         Density (mass ÷ volume) has dimensions of M ÷ L2 = ML-2

v)           Acceleration (velocity ÷ time) has dimensions of L ÷ T2 = LT-2 

vi)         Momentum (mass x velocity) has dimensions of  (M x L) ÷ T = MLT^-1

vii)        Pressure (force ÷ area) = (mass x acceleration) ÷ area has dimension of MLT^-2 ÷ L² =MT^-2L^-1

DIMENSIONAL ANALYSIS

This refers to the expression of physical quantities in their dimensions. That is in terms of length, mass and time. Dimensional symbols are therefore L, M and T for length, mass and time respectively.

For example, a quantity expressed as [L]² can be interpreted as Length × length referring to the quantity area. Similarly, speed, which has units of length/time, will have dimensions L/T or LT^-1.

Summing (adding or subtracting) quantities together implies that the quantities have the same dimension. Similarly, irrespective of the system of units used, all mathematical expression and equation must be dimensionally correct. That is, quantities on both sides of an equation must have the same dimensions.

Analyzing dimensionally is important in checking the correctness of the form of the equation. Dimensional analysis is also a way to verify the dependence of physical quantities on one another.

USES OF DIMENSIONAL ANALYSIS

To check the accuracy of a given relation

To derive a relationship between different physical quantities

To convert a physical quantity from one system to another system

EXAMPLE 2

The equation of motion governing an object given an initial velocity, thus covering a distance x with a uniform acceleration a, over time t is expressed as

X = ut + 1/2at²

Show that this expression is dimensionally correct.

SOLUTION

Dimensionally: x = L (length); u = L/T (velocity = distance/time)

        a = L/T² (velocity/time = distance/time/time); t = T

Hence the expression becomes

        [L] = [L/T] [T] + ½ [L/T²][T]²

=> [L] = [L] + ½[L] = 3/2[L]

Since both sides of the equation have the same dimension, except for the constant factor 3/2, therefore, the equation is dimensionally correct.

CLASSWORK

1.   Determine the dimensions of force, work, surface tension, (force per unit length) and power (rate of doing work).

2.   Show that the equation V² – u² = 2ax is dimensionally correct [Ans… L²T²]

ASSIGNMENT

1.   The specific heat capacity for a particular solid at a temperature close to 0k is given by c =aT³. What is the unit in SI for the constant a when T is the absolute temperature? [m²s^-2K^-4]

2.   The velocity (V) of sound in a medium is determined to depend on the young’s modulus Υ, the density, ρ, of the medium and the wavelength λ. Use dimension analysis to derive a formula for the speed of sound in the medium. [v =  Υ^1/2 ρ^-1/2]