Units and Dimensions

Outline• Terminology

– System, property (intensive, extensive) • Dimension versus unit• Primary and secondary dimensions• SI units• Conversion of temperature• Conversion of specific heat• Order of magnitude• Significant digits/figures and round-off• Resolution (least count), precision, accuracy

Terminology• System: Region defined by a boundary• Property (of a material): An attribute that is

characteristic of that material. It may vary with temperature or direction; eg. density, elasticity

• Property (of a system): A characteristic that defines the equilibrium state of a system. It does not depend on how that state was attained.

• Intensive property (of a system): Does not depend on system size; eg. pressure, density, temperature

• Extensive property (of a system): Extensive property depends on system size; eg. volume, energy

Terminology (contd.)• Density: Mass per unit volume

– SI Units: kg/m3

• Specific heat: Energy required to raise the temperature of unit mass of material by one unit– SI Units: J/kg K

• Velocity: Distance traveled per unit time– SI Units: m/s

• Energy: Capacity to do work– SI Units: J (or N m)

• Work: Transference of energy– SI Units: J (or N m)

• Power: Rate of doing work– SI Units: J/s or Watts (W); Note: 1 hp = 745.7 W

Dimension versus Unit

• Dimension– A physical entity that can be measured

• Unit– Quantitative magnitude of a dimension

• Example: TIME is measured in SECONDS– TIME is the DIMENSION– SECONDS is the UNIT

Primary and Secondary Dimensions

• Primary dimensions– Mass (M)– Length (L)– Time (T)– Temperature (K)

• Secondary dimensions– Combination of primary dimensions

• Example: Velocity = Distance/Time– Thus, its dimensions are L/T OR LT-1

Secondary Dimensions• Force (Units: N)

o Force = (mass) (acceleration) = (mass) (velocity)/(time) Dimensions: M LT-1 / T = MLT-2

• Energy or work done (Units: Joules)o Work = (force) (distance) Dimensions: (MLT-2) (L) = ML2T-2

• Power (Units: J/s or Watts)o Power = (work)/(time) Dimensions: (ML2T-2)/(T) = ML2T-3

• Specific heat (Units: J/kg K)o Energy reqd to inc. temp. of 1 kg of material by 1 °C Dimensions: (ML2T-2)/(M K) = L2T-2K-1

SI Units

• Mass: kg (g is NOT SI unit)• Length: m (cm is NOT SI unit)• Time: s• Temperature: K• Electric current: A• Amount of a substance: mol• Luminous intensity: Cd

Conversion of Temperature0 °C = 32 °F = 273 K100 °C = 212 °F = 373 K-40 °C = -40 °F = 233 K°C = (°F – 32) * 5/9°F = (°C) * 9/5 + 32K = °C + 2731 °C change in temp. = 1.8 °F change in temp.

= 1 K change in temp.1 °C ≠ 1.8 °F1 °C ≠ 1 K

Conversion of Specific Heat

1 J/kg °C = 1 J/kg K = (1/1.8) J/kg °F1 J/kg °F = 1.8 J/kg °C = 1.8 J/kg K

Note 1: There is NO factor of 32 to account for as we are NOT converting temperature, but ONLY temperature changeNote 2: Recall that specific heat was the energy required to change the temperature of unit mass of a substance by one unit

Order of Magnitude

• An order of magnitude is used to describe/determine the approximate range of any quantity– It is usually expressed in powers of 10

• Example– The diameter of a mustard seed is ~0.1" while that

of a basketball is ~ 9.39". Thus, a basketball is 2 orders of magnitude (or approximately 100 times) larger in diameter than a mustard seed.

Significant Digits/Figures• A measure of how accurately a measurement can be made• Example: An object weighs 5.2 kg on a household weighing

machine and 5.274 kg on an analytical weighing machine– Significant digits with household weighing machine: 1– Significant digits with analytical weighing machine: 3

• Number of digits used should be in accordance with its ability to measure at that level of accuracy– If the volume of the object was 0.7 m3, the density is calculated to

be 7.429 kg/m3 using the household weighing machine and 7.534 kg/m3 using the analytical machine

• If accuracy in volume measurement is only to 1 significant digit, density value should not be reported with more than 1 significant digit

• Round-off 7.429 kg/m3 to 7.4 kg/m3

MassObject Mass (kg)

Hydrogen atom 1.67 x 10-27

8.5” x 11” notebook paper 4.5 x 10-3 (4.5 g)1 teaspoon salt 6 x 10-3 (6 g)Large box of cereal 0.5 (500 g)8 fl oz soda can with soda 0.6 (600 g)1 gallon of milk in plastic jug 4Adult human being 62 (US: 81)Car (mid-size 4-door sedan) 1,600 (~3,500 lbs)18-wheeler truck 35,000Airplane (Loaded Boeing 747) 400,000Goods train (130 coal cars) 200 x 106

Largest cruise ship 220 x 106

Sun 2 x 1030

Length or DistanceObject or path Length or Distance (m)

Radius of atom 10-10 (1 A)Diameter of microbial cell 2 x 10-7 (0.2 m)Diameter of human hair 10-4 (100 m)Diameter of a 8 fl oz soda can 0.06 (6 cm)Height of a 2 liter coke bottle 0.3 (30 cm)Height of a 2-storey gable house 8Length of an 18-wheel truck 20Length of a football field 91Average width of U.S. 4.8 x 106 (3,000 mi)Diameter of earth 13 x 106 (7,918 mi)Earth to moon 384 x 106 (239 K mi)Earth to Pluto 4.2 to 7.5 x 109 (~3-5 B mi)

TimeActivity or Process Time (s)

Light to travel 1 mile in vacuum 5 x 10-6

Duration of a camera flash 10-3

Human reaction time 0.2Fastest car to go from 0 to 60 mph 2Light to travel from Earth to Pluto 30,000 (5.5 hrs)Sound to travel around earth 37,500 (10.5 hrs)Life span of Mayfly 86,400 (1 day)Fastest spacecraft: Earth to Pluto 75 x 107 (23.5 yrs)Homo Sapiens on earth 1.5 x 1012 (50,000 yrs)Age of the universe 4.4 x 1017 (13.8 B yrs)

TemperatureObject/Location/Process Temperature (°C)

Liquid Helium -269Freezer temperature -18Refrigerator temperature 2Melting of ice 0Operating temperature of car engine 80Boiling of water 100Canning of soups 120-125Frying 190Baking 200Melting of steel 1500Combustion temperature in car 2500Center of Sun 15 x 106

VelocityObject or Action Velocity (m/s)

Rate of growth of human hair 5 x 10-9

Walking 1.5Sprinting 10Top speed of fastest roller coaster 67Fastest recorded ball (golf ball) on land 91Car on highway 30 (67.5 mph)Airplane 250 (562.5 mph)Sound 340 (765 mph)Space shuttle (when booster rocket separates) 1400 (3150 mph)Escape velocity from Earth 11,200Linear velocity of earth around sun 29,800Lightning 4 x 105

Light 3 x 108

Density (At Room Temperature)Material Density (kg/m3)

Air 1Apple 700-900Printing paper 800Cream 980Commonly used plastics 700-1300Water 1000Skim milk 1030Glass ~2,600Aluminum 2,700Steel 7,800Mercury 13,534Lead 11,340

Specific HeatMaterial Specific Heat (J/kg K)

Mercury 140Lead 160Sand 290Steel 420Dates 840Aluminum 900Air 1020Oil ~2000Ice 2010Apple 3600Water 4180Hydrogen gas 14,267

Resolution, Precision, and Accuracy

• Resolution (Least count): Smallest increment in reading of instrument– Interval between consecutive markings (analog)– Interval between closest possible consecutive

readings (digital)• Precision: Repeatability of reading

– Can be a higher value than resolution• Accuracy: Correctness of reading

– Precision does not always equate to accuracy• Quality of instrument and calibration are factors

Resolution, Precision, and Accuracy (contd.)

• Resolution versus precision– Resolution of an analog stopwatch is 0.1 s– Resolution of a digital stopwatch is 0.01 s– Precision of both are 0.1 s (human reaction time)

• Precision versus accuracy

Bad precisionBad accuracy

Good precisionBad accuracy

Bad precisionBetter accuracy

Good precisionGood accuracy

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Calibration

• Calibrate against standards• Calibrate within range of use• Frequency of calibration

– Instrument or application dependent examples• Each time instrument is turned on• Each time instrument is moved• When range of use of instrument changes• When external factors such as temperature, pressure

or humidity changes

Other Factors Affecting Readings

• Ambient conditions– Temperature, humidity

• External equipment– Electronic noise/interference

• Good contact/proximity of instrument/probe with measurement location

• True non-invasive nature of probe– Are flow or heat transfer affected by probe?– Does the probe create channeling or shielding?

Summary• Primary dimensions

– Mass (M), Length (L), Time (T), Temperature (K)• SI Units

– Mass (Kg), Length (m), Time (s), Temperature (K), Electric current (A), Amount of a substance (mol), Luminous intensity (Cd)

1 °C change in temp. = 1.8 °F change in temp.= 1 K change in temp.

• 1 J/kg °C = 1 J/kg K = (1/1.8) J/kg °F• Resolution (least count), precision (repeatability),

accuracy (correctness), calibration