units (systéme internationale) dimensionsi (mks) unitdefinition lengthmeters (m)distance traveled...
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UNITS (Systéme Internationale)
Dimension SI (mks) Unit Definition
Length meters (m) Distance traveled by light in 1/(299,792,458) s
Mass kilogram (kg) Mass of a specific platinum-iridium allow cylinder kept by Intl. Bureau of Weights and Measures at Sèvres, France
Time seconds (s) 9,192,631,700 oscillations of cesium atom
Standard Kilogram at Sèvres
Conversion Factors
• They are simply ratios that are equal to 1
e.g.
• These ratios make converting units easy!
e.g.
Conversion Factors
• The most holy of conversion factors:
1 inch = 2.54 centimeters
Dimensional Analysis
Dimensions & units can be treated algebraically.
Variable from Eq.
x m t v=(xf-xi)/t
a=(vf-vi)/t
dimension L M T L/T L/T2
Dimensional Analysis
Checking equations with dimensional analysis:
L (L/T)T=L
(L/T2)T2=L
• Each term must have same dimension• Two variables can not be added if dimensions are different• Multiplying variables is always fine• Numbers (e.g. 1/2 or ) are dimensionless
x f xi vit 1
2at 2
Example 1.1
Check the equation for dimensional consistency:
2
2
2
)/(1mc
cv
mcmgh
Here, m is a mass, g is an acceleration,c is a velocity, h is a length
Example 1.2
L3/(MT2)
Consider the equation:
Where m and M are masses, r is a radius andv is a velocity.What are the dimensions of G ?
mv2
rG
Mm
r2
Example 1.3
Given “x” has dimensions of distance, “u” has dimensions of velocity, “m” has dimensions of mass and “g” has dimensions of acceleration.
Is this equation dimensionally valid?
Yes
Is this equation dimensionally valid?
No
x (4 / 3)ut
1 (2gt 2 / x)
x vt
1 mgt 2
Units vs. Dimensions
• Dimensions: L, T, M, L/T …• Units: m, mm, cm, kg, g, mg, s, hr, years …• When equation is all algebra: check
dimensions• When numbers are inserted: check units• Units obey same rules as dimensions:
Never add terms with different units• Angles are dimensionless but have units
(degrees or radians)• In physics sin(Y) or cos(Y) never occur unless
Y is dimensionless
Example 1.3
Grandma traveled 27 minutes at 44 m/s.How many miles did Grandma travel?
44.3 miles
Prefixes
In addition to mks units, standard prefixes can be
used, e.g., cm, mm, m, nm
Example 1.4a
The above expression yields:
40m 11cm ?
a) 40.11 mb) 4011 cmc) A or Bd) Impossible to evaluate (dimensionally invalid)
Example 1.4b
The above expression yields:
1.5m 3.0kg ?
a) 4.5 m kgb) 4.5 g kmc) A or Bd) Impossible to evaluate (dimensionally invalid)
Example 1.4b
The above expression yields:
1.5m-3.0kgm/s?
a) -1.5 mb) -1.5 kg m2
c) -1.5 kgd) Impossible to evaluate (dimensionally invalid)