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)