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By the end of this section, you will be able to:
  • Find the dimensions of a mathematical expression involving physical quantities.
  • Determine whether an equation involving physical quantities is dimensionally consistent.

The dimension    of any physical quantity expresses its dependence on the base quantities as a product of symbols (or powers of symbols) representing the base quantities. [link] lists the base quantities and the symbols used for their dimension. For example, a measurement of length is said to have dimension L or L 1 , a measurement of mass has dimension M or M 1 , and a measurement of time has dimension T or T 1 . Like units, dimensions obey the rules of algebra. Thus, area is the product of two lengths and so has dimension L 2 , or length squared. Similarly, volume is the product of three lengths and has dimension L 3 , or length cubed. Speed has dimension length over time, L/T or LT –1 . Volumetric mass density has dimension M/L 3 or ML –3 , or mass over length cubed. In general, the dimension of any physical quantity can be written as L a M b T c I d Θ e N f J g for some powers a , b , c , d , e , f , and g . We can write the dimensions of a length in this form with a = 1 and the remaining six powers all set equal to zero: L 1 = L 1 M 0 T 0 I 0 Θ 0 N 0 J 0 . Any quantity with a dimension that can be written so that all seven powers are zero (that is, its dimension is L 0 M 0 T 0 I 0 Θ 0 N 0 J 0 ) is called dimensionless    (or sometimes “of dimension 1,” because anything raised to the zero power is one). Physicists often call dimensionless quantities pure numbers .

Base quantities and their dimensions
Base Quantity Symbol for Dimension
Length L
Mass M
Time T
Current I
Thermodynamic temperature Θ
Amount of substance N
Luminous intensity J

Physicists often use square brackets around the symbol for a physical quantity to represent the dimensions of that quantity. For example, if r is the radius of a cylinder and h is its height, then we write [ r ] = L and [ h ] = L to indicate the dimensions of the radius and height are both those of length, or L. Similarly, if we use the symbol A for the surface area of a cylinder and V for its volume, then [ A ] = L 2 and [ V ] = L 3 . If we use the symbol m for the mass of the cylinder and ρ for the density of the material from which the cylinder is made, then [ m ] = M and [ ρ ] = ML −3 .

The importance of the concept of dimension arises from the fact that any mathematical equation relating physical quantities must be dimensionally consistent    , which means the equation must obey the following rules:

  • Every term in an expression must have the same dimensions; it does not make sense to add or subtract quantities of differing dimension (think of the old saying: “You can’t add apples and oranges”). In particular, the expressions on each side of the equality in an equation must have the same dimensions.
  • The arguments of any of the standard mathematical functions such as trigonometric functions (such as sine and cosine), logarithms, or exponential functions that appear in the equation must be dimensionless. These functions require pure numbers as inputs and give pure numbers as outputs.

Questions & Answers

what is microbiology
Agebe Reply
What is a cell
Odelana Reply
what is cell
Mohammed
how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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