Units and Dimensions
Outline
• Terminology
– System, property (intensive, extensive)
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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
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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
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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 Temperature
0 °C = 32 °F = 273 K
100 °C = 212 °F = 373 K
-40 °C = -40 °F = 233 K
°C = (°F – 32) * 5/9
°F = (°C) * 9/5 + 32
K = °C + 273
1 °C change in temp. = 1.8 °F change in temp.
= 1 K change in temp.
1 °C ≠ 1.8 °F
1 °C ≠ 1 K
Conversion of Specific Heat
1 J/kg °C = 1 J/kg K = (1/1.8) J/kg °F
1 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 change
Note 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
Mass
Object
Hydrogen atom
8.5” x 11” notebook paper
1 teaspoon salt
Mass (kg)
1.67 x 10-27
4.5 x 10-3 (4.5 g)
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
Adult human being
Car (mid-size 4-door sedan)
4
62 (US: 81)
1,600 (~3,500 lbs)
18-wheeler truck
35,000
Airplane (Loaded Boeing 747)
400,000
Goods train (130 coal cars)
200 x 106
Largest cruise ship
220 x 106
Sun
2 x 1030
Length or Distance
Object or path
Radius of atom
Diameter of microbial cell
Diameter of human hair
Diameter of a 8 fl oz soda can
Height of a 2 liter coke bottle
Height of a 2-storey gable house
Length of an 18-wheel truck
Length of a football field
Average width of U.S.
Diameter of earth
Earth to moon
Earth to Pluto
Length or Distance (m)
10-10 (1 A)
2 x 10-7 (0.2 m)
10-4 (100 m)
0.06 (6 cm)
0.3 (30 cm)
8
20
91
4.8 x 106 (3,000 mi)
13 x 106 (7,918 mi)
384 x 106 (239 K mi)
4.2 to 7.5 x 109 (~3-5 B mi)
Time
Activity or Process
Time (s)
Light to travel 1 mile in vacuum
Duration of a camera flash
Human reaction time
Fastest car to go from 0 to 60 mph
Light to travel from Earth to Pluto
Sound to travel around earth
Life span of Mayfly
Fastest spacecraft: Earth to Pluto
Homo Sapiens on earth
Age of the universe
5 x 10-6
10-3
0.2
2
30,000 (5.5 hrs)
37,500 (10.5 hrs)
86,400 (1 day)
75 x 107 (23.5 yrs)
1.5 x 1012 (50,000 yrs)
4.4 x 1017 (13.8 B yrs)
Temperature
Object/Location/Process
Temperature (°C)
Liquid Helium
Freezer temperature
Refrigerator temperature
Melting of ice
Operating temperature of car engine
Boiling of water
Canning of soups
Frying
Baking
Melting of steel
Combustion temperature in car
Center of Sun
-269
-18
2
0
80
100
120-125
190
200
1500
2500
15 x 106
Velocity
Object or Action
Rate of growth of human hair
Velocity (m/s)
5 x 10-9
Walking
1.5
Sprinting
10
Top speed of fastest roller coaster
67
Fastest recorded ball (golf ball) on land
91
Car on highway
Airplane
Sound
Space shuttle (when booster rocket separates)
30 (67.5 mph)
250 (562.5 mph)
340 (765 mph)
1400 (3150 mph)
Escape velocity from Earth
11,200
Linear velocity of earth around sun
29,800
Lightning
4 x 105
Light
3 x 108
Density (At Room Temperature)
Material
Air
Apple
Printing paper
Cream
Commonly used plastics
Water
Skim milk
Glass
Aluminum
Steel
Mercury
Lead
Density (kg/m3)
1
700-900
800
980
700-1300
1000
1030
~2,600
2,700
7,800
13,534
11,340
Specific Heat
Material
Mercury
Lead
Sand
Steel
Dates
Aluminum
Air
Oil
Ice
Apple
Water
Hydrogen gas
Specific Heat (J/kg K)
140
160
290
420
840
900
1020
~2000
2010
3600
4180
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
tutelman.com
Bad precision
Bad accuracy
Good precision
Bad accuracy
Bad precision
Better accuracy
Good precision
Good accuracy
Calibration
• Calibrate against standards
• Calibrate within range of use
• Frequency of calibration
– Instrument or application dependent examples
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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