# Dimensions vs. Units ```UNITS (Syst&eacute;me 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 platinumiridium allow cylinder kept by
Intl. Bureau of Weights and
Measures at S&egrave;vres, France
Time
seconds (s)
9,192,631,700 oscillations of
cesium atom
Standard Kilogram
at S&egrave;vres
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 &amp; 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:
1
x f  xi  vi t  at 2
2
(L/T2)T2=L
L
(L/T)T=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 p) are dimensionless
Example 1.1
Check the equation for dimensional consistency:
2
mc
2
mgh 

mc
2
1  (v / c )
Here, m is a mass, g is an acceleration,
c is a velocity, h is a length
Example 1.2
Consider the equation:
v2
Mm
m G 2
r
r
Where m and M are masses, r is a radius and
v is a velocity.
What are the dimensions of G ?
L3/(MT2)
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?
(4 / 3)ut
x
1  (2gt 2 / x)
Yes
Is this equation dimensionally valid?
vt
x
1  mgt 2
No
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
• 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
standard prefixes can be used,
e.g., cm, mm, mm, nm
Example 1.4a
40 m  11cm  ?
The above expression yields:
a)
b)
c)
d)
40.11 m
4011 cm
A or B
Impossible to evaluate (dimensionally invalid)
Example 1.4b
1.5 m  3.0 kg  ?
The above expression yields:
a)
b)
c)
d)
4.5 m kg
4.5 g km
A or B
Impossible to evaluate (dimensionally invalid)
Example 1.4b
1.5 m-3.0 kg m/s  ?
The above expression yields:
a)
b)
c)
d)
-1.5 m
-1.5 kg m2
-1.5 kg
Impossible to evaluate (dimensionally invalid)
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