Physical Quantities Handout.........

7/27/2019 Physical Quantities Handout.........

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Working with physical quantities, dimensions & units I.J. Walker 1 of 5

Department of GeographyUniversity of Victoria

WORKING WITH PHYSICAL QUANTITIES, DIMENSIONS & SI UNITS

IN PHYSICAL GEOGRAPHY

by Dr. Ian J. Walker

When working with data in earth science problems, it is obviously useful to knowthe numerical value of the quantities that you are working with but it is also important tounderstand the units and physical dimensions of the variable(s). This handout willintroduce you to working with physical quantities, SI units and dimensional analysis; aninvaluable skill when working with and manipulating mathematical equations ingeomorphology, hydrology, climatology or other physical geography applications.

What are quantities and dimensions?

A quantity is a measurable value of size, number, weight or amount ofsomething. All physical quantities on Earth have dimensions that can be expressed interms and combinations of 5 basic dimensions: mass (M), length (L), time (T), electrical

current (I), and temperature (). These 5 dimensions are considered basic becausethey are easy to measure in experiments.

Dimensions are not the same as units. Rather, units express the system ofmeasurement for the various dimensions. For example, speed can measured in units ofmetres per second (m s-1) or kilometers per hour (km hr-1) but the dimensions of speedare always a length (L) divided by time (T), or simply L T-1. Similarly, the dimensions ofarea are L x L or L2 and the units can be expressed in m2. This is a useful means ofworking with physical quantities in that it enables you will to determine the "dimensions"involved and the appropriate units of the quantity, especially in equations involvingmany variables and parameters.

What are SI units?

SI stands for Systme Internationale or the International System convention ofmeasurement. The basis of measurement of SI units is the metric system in which allunits are based on a factor of ten. It is the only measurement system based on astandard length derived from the size of the earth. The metre, the basic unit of

measurement of the metric system, is defined as 1 x 10

-7

times the distance from theequator of the earth to either pole. Because of the difficulty in actually measuring andreproducing this unit, it was later defined more precisely by a relation to a wavelength ofvisible light (e.g., 1 m = 1650763.73 wavelengths of orange light in the spectrum ofkrypton 86!?!). More recently (1983), it was defined further as 1 metre = the distancelight travels in a vacuum in a fraction of a second (e.g., 3.34 x 10 -9 seconds) veryprecise indeed! The seven basic units of the SI system are listed in Table 1.


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Table 1: Fundamental quantities, dimensions and units of the SI system

Quantity Unit Symbol

length (L) meter m

mass (M) kilogram kg

time (T) second s

electric current (I) ampere A

thermodynamic temperature () kelvin K

amount of a substance mole mol

luminous intensity candela cd

All other SI units are eitherderived from or related to these fundamental units.Derived quantities (Table 2) are defined in terms of relations between the seven basequantities. For instance, area is the square of length (LxL) and volume is the cube oflength (L3).

Table 2: Derived SI Units

SI derived unit

Derived quantity Name Symbol

area square meter m2

volume cubic meter m3

speed, velocity meter per second m s-1

acceleration meter per second squared m s-2

mass density kilogram per cubic meter kg m-3

specific volume cubic meter per kilogram m3 kg-1

amount-of-substance concentration mole per cubic meter mol m-3

luminance candela per square meter cd m-2

For convenience purposes, several SI derived units have been given specialnames and symbols, as shown in Table 3.


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Table 3: SI derived units with special names

SI derived unit

Quantity Name Symbol Expression interms of other

units

Expression interms of base

SI units

frequency hertz Hz - s-1

force newton N - kg m s-2

pressure, stress,momentum flux

pascal Pa N m-2 kg m-1 s-2

work, energy joule J N m kg m2 s-2

power watt W J s-1 kg m2 s-3

charge coulomb C - s A

electric potential volt V W A-1 kg m2 s-3 A-1

electric resistance ohm V A-1 kg m2 s-3 A-2

Celsiustemperature

degree Celsius C - K

Some SI units include and are derived from other SI units as shown in Table 4.

Table 4. SI units whose names and symbols include SI derived units with specialnames and symbols

SI derived unit

Derived quantity Name Symbol

dynamic viscosity pascal second Pa s

moment of force newton meter N m

momentum newton second N s

surface tension newton per meter N m-1

heat flux density, irradiance, stream

power

watt per square meter W m-2

heat capacity, entropy joule per kelvin J K-1

specific energy joule per kilogram J kg-1

Useful links:

http://www.physics.uoguelph.ca/tutorials/tutorials.htm http://physics.nist.gov/cuu/Units/units.html


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What is dimensional analysis?

Dimensional analysis is a way of managing the dimensions of physical quantitiesin algebraic equations. A 'physically correct' equation must have the equality ofdimension. In other words, each term in the equation must have the same dimensions

to dimensionally correct. Ideally, all equations should be presented in dimensionlessform (i.e., units cancel out) to be dimensionally correct. However, many equationsproduce values that have dimensions (and units) to provide a 'real' value.

For example, consider the dimensions on each side of the following equation forcalculating pressure (P):

ghP =

where is density (kg m-3), g is gravitaitonal acceleration (m s-2), and h is height (m).What are the dimensions of the equation?

LEFT SIDE: Recall that the units of pressure arepascals (Table 3) and havedimensions of M L-1 T-2 (or respective units of kg m-1 s-2).

RIGHT SIDE: [M L-3 L T-2 L]1/2 or M1/2 L-1/2 T-1 (or respective units of kg1/2 m-1/2 s-1).

Thus, the dimensions of the equation are not balanced. Consider the formula fordetermining force (F) using Newton's 2

ndlaw of motion:

F = m a

where F is force (N or kg m s-2

), m is mass (kg), and a is acceleration (m s-2

).

LEFT SIDE: M L T-2

RIGHT SIDE: M L T-2

Thus, the dimensions of the force equation are balanced.

Aside from being able to detect whether there are algebraic (i.e., dimensional)errors in an equation, there are practical advantages to using dimensionlessexpressions of physical quantities. For instance, dimensionless parameters can be auseful way to express quantitative data gathered under slightly different initialconditions. For example, Reynolds Number (Re) expresses the ratio of inertial toviscous force (i.e., relative amount of turbulence) in fluid flow and is described by:

Re = u L /

where u is velocity (m s-1), L is a length term (m), and is kinematic viscosity (m2 s-1).

Note that the dimensions on the right-hand side cancel out [(L T -1 L) / (L2 T-1) =0]. This produces a value that can be used as a basis for comparison even though anyone of the state variables (e.g., velocity) may vary between cases. For example,Reynolds number is often used for scaling fluid dynamics between differentenvironments (i.e., different lengths scales between field conditions and a wind tunnelsimulation). As long as Re can be matched between cases, dynamic similitude orproper dimensional scaling of all forces involved is said to exist.


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Summary

SI units provide a standardized and internationally recognized convention formeasurement and expression of quantitative data. Furthermore, the metric (base 10)system allows for more convenient use and conversion of units used to measure

physical quantities and dimensions in science (compared to the Imperial or EnglishSystem for example).

Macro-scale modelling of earth systems is often too complicated for analyticalsolution because of the inherent complexity and multivariate nature of natural systems.This can be overcome in part by clearly defining and measuring only a 'sub-system' ofinterest (e.g., sediment transport system in fluvial environments). However, there oftenexist external variables and controls that must be subsumed as 'parameters'.

In addition to theoretical considerations, studying earth systems often requiresempirical observation and controlled experimentation. Dimensional analysis is a usefultechnique for obtaining quantitative data on relations between important variables and

for understanding responses of the system (or relations) to change. Remember that aphysically correct equation must show equality of dimension, or a balance in thedimensions expressed by all quantities in the relation.

Basic terms & definitions

Mass: the quantity of matter in a body that gives it inertia measured by resistance toapplied force or acceleration (e.g., gravity)

Weight: force exerted on a mass by earths gravitational field w = mg MLT-1

Force: something that imposes a change in motion of a material body F=ma MLT2

Inertia: the property of a body that keeps it at rest or in motion OR the difficulty ofsetting a body in motion

Momentum: the difficulty of stopping a body already in motion = mv MLT-1

Mass density: measure of fluids inertial resistance to acceleration = m/V ML-3

Specific density (specific weight): weight per unit volume = mg / V = g(m/V) = g

ML-3T-2

Pressure: force per unit area P=ma/A ML-1T-2

Work: force applied over a distance m a d ML2T-2

Discharge: volume of fluid moved per unit time, typically measured using the velocityarea method Q = V A L T-1 L2 = L3 T-1

Power: rate at which work is performed mad/t Fv ML2T-3

Energy: any phenomenon capable of conversion to work

Kinetic energy: the energy a mass contains by virtue of its motion mv2ML2T-2

Potential energy: the energy a mass contains by virtue of its position mgd =

wz ML2T-2

Physical Quantities Handout.........

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