Units and measurements

Motivation

One of the core values of science is its consistency, and in order to achieve such amazing consistency, hence precision and accuracy, we need a unified measuring qu

Definitions

• Quantities that can be measured are known as physical quantities
     Examples: Length, mass, time, weight, electric current, velocity, force, etc.
• When stating a physical quantity, two items must be mentioned:
     The first is the numerical value
     The second is the unit
     Examples:
     1.75m. Numerical value: 1.75, unit: m.
     59.3kg. Numerical value: 59.3, unit: kg
     38ms^-1. Numerical value: 38, unit: ms^-1
• To measure a physical quantity, a standard size of that quantity is required, it is also known as the unit for that physical quanitity
• Scientists have agreed to use a common system of units called the S.I. The advantage of the S.I. system is that a physicsl quantity has only one unit. Prefixes are used for multiples or submultiples of the unit

  Prefix	Multiplying factor	Symbol
  yocto	10-24	y
  zepto	10-21	z
  atto	10-18	a
  femto	10-15	f
  pico	10-12	p
  nano	10-9	n
  micro	10-6	μ
  mili	10-3	m
  centi	10-2	c
  deci	10-1	d
  deca	101	da
  hecto	102	h
  kilo	103	k
  mega	106	M
  giga	109	G
  tera	1012	T
  peta	1015	P
  exa	1018	E
  zetta	1021	Z
  yotta	1024	Y

  Prefixes in bold are the most commonly used

• The units for the basic physical quantities are known as base units
  

  Basic quantities	Base units	Symbols
  Length	metre	m
  Mass	kilogram	kg
  Time	seconds	s
  Electric current	ampere	A
  Temperature	kelvin	K
  Amount of substance	mole	mol
  Luminous intensity	candela	cd

• Physical quantities other than the basic quantities are known as derived quantities. Units for such quantities are known as derived units

Derived quanitties	Derived units
Area	m2
Volume	m3
Density	kg m-3
Velocity	m s-1
Acceleration	m s-2
Force	N or kg m s-2
Pressure	Pa or kg m-1 s-2
Energy	J or kg m2 s-2
Power	W or kg m2 s-3
Electric charge	C or A s
Voltage	V or kg m2 s-3 A-1
Resistance	Ω or kg m2 s-3 A-2
Frequency	s-1

You cannot add terms of different units unless you convert them to one UNIFIED unit.

When you multiply or divide quanities, you must divide BOTH the numerical value and the units

• The dimensions of a physical quantity is the relationship between the physical quantity and the basic quantities
  It can also be written as [physical quantity]
     Examples:
     [Area] = [length × breadth] = L × L = L². Unit for area: m²
     [Velocity] = [displacement ÷ time] = L ÷ T = LT^-1. Unit for velocity: ms^-1
     [Acceleration]= [change in veloctiy ÷ time] = LT^-1 ÷ T = LT^-2. Unit for acceleration: ms^-2
• Units and dimensions are used to check the homogeneity of physical equations. When all of the terms of an equation has the same unit, we say that they are dimensionally homogeneous. An equation which is not homogenous must be wrong. However, the reverse is not true.    Examples:
     1. Incorrect coefficient(s)
     1.1
     s = ut + 1/2 at² (Correct equation)
     s = ut + 1 at² (Incorrect equation)
     s = 2 ut + 1 at² (Incorrect equation)
     1.2
     F = ma (Correct equation)
     F = 2 ma (Incorrect equation)
     2. Missing or extra term(s)
     2.1
     s = ut + 1/2 at² (Correct equation)
     s = 1/2 at² (Incorrect equation)
     s = ut (Incorrect equation)
     s = ut + vt + 1/2 at² (Incorrect equation)
     2.2
     F = ma (Correct equation)
     F = ma + pV (Incorrect equation)
  

Revision

DO NOT SCROLL UP WHILE ATTEMPTING THE QUESTIONS, OR ELSE THE PURPOSE OF THIS REVISION WILL BE NULLIFIED.

Also, take your pen(cil) and paper out and write, not type it out or keep it in your mind, it won't work.

What are physical quantities? Name four of them.

What must be mentioned when stating a physical quantity? Give examples and explain.

Name four most commonly used SI prefixes and their multiplying factor

List some of the basic quantities and their corresponding basic units

What does the phrase "dimensions of a physical quantity" mean?

Can a homogenous physical equation be wrong? If so, explain and give examples.

Questions

• It is known that the force exerton on two point mass on each other is expressed by this relation:

   F = k m₁ m₂ ÷ r².
   What are the dimensions and unit of k?

• It is known that the force exerted on two point charge on each other is expressed by this relation:

   F = k q₁ q₂ ÷ r².
   What are the dimensions and unit of k?

• We have the following relation:

   f = 2 π m^1/2 ÷ a^1/2
   Verify the homogenity of this equation.