Unit Analysis

Unit Analysis

“Measurement units can be manipulated in a similar way to variables in algebraic relations.”

SWTJC STEM – ENGR 1201

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Unit Analysis

The basis for this analysis is embodied in three rules:

1. Dimensional Consistency Rule2. Algebraic Manipulation Rule3. Transcendental Function Rule

Rule #1 – Dimensional Consistency Rule

In a unit relation, all terms must be dimensionally consistent. This means that each term must have the same units or be reducible to the same units.


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Rule 1 Dimensional Consistency

Dimension refers to “what is being measured”. For instance, when measuring the length of a table, “length” is the dimension. The unit could be feet, meters, or a variety of other “length” units.

Reducible refers to rewriting the units in fundamental units. Units are either fundamental or derived. Derived units are combinations of eight fundamental units.

Refer to Derived Units Charts on “Useful Links”

The Dimensional Consistency Rule simply reinforces the common sense idea that you can only add and subtract identical things. You cannot mix apples and oranges!


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Adding Apples and Oranges

?


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Applying Consistency Rule

x = a + b - c (Terms are separated by addition or subtraction)

terms

Consistency means terms “x”, “a”, “b”, and “c” must have the same dimension, i.e. units, in this case length/meters. If “x” is length (meters), all other terms must be length (meters)!

(meters) = (meters) + (meters) – (meters)

(length) = (length) + (length) – (length)

Example

Note that (meters) - (meters) = (meters) not zero!16 meters - 12 meters = 4 meters!


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Rule 2 Algebraic Manipulation

Rule #2 – Algebraic Manipulation Rule

Unit relations that are multiplied and/or divided can be treated like variables; i.e., canceled, raised to powers, etc.

During algebraic manipulation of a relation, dimensional consistency must be maintained. When finished, if dimensional inconsistency is noted, then either an algebraic manipulation error occurred or the original formulation of the relation was faulty.

Working through the units is a great way to check your algebra!


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Distance Formula Example

Distance formula: d = v0 . t + (1/2) . a . twhere d (m), v0 (m/s), t (s), and a (m/s2)m = (m/s) . s + (none) . (m/s2) . sm = m + m/sA problem? The relation is inconsistent!Formula is incorrect!

Distance formula: d = v0 . t + (1/2) . a . t2

m = (m/s) . s + (none) . (m/s2) . s2

m = m + mThe formula is consistent!

Examplesdimensionless constant

(no units!)


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Particle Energy Example

2

2


1E mgh mv

2m m

where E [J], m [kg], g [ ], h [m], v [ ]ss

Assume the relation is consistant.


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2

2

2

Solve for v.

1mv E mgh

2

mv 2E 2mgh

2E 2mgh Ev 2( gh)

m m

Ev 2( gh)

m


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2

Check for consistency.

Ev 2( gh)

m

kgm Nm m

ms kg s

2

mm

skg

2

2

2 2 2

2 2 2

m

s

m m m m m

s ss Consistent

s s

Rule #3 – Transcendental Function Rule

Transcendental functions (trig, exponential, etc.) and their arguments cannot have dimensions (units).

Examples of transcendental functions includes:

sin(x), cos(x), tan(x), arcsin(x)

ex

log(x), ln(x)


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Rule 3 Transcendental Function


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Transcendental Function Examples

Consider the relation A = sin(a t + b).

Neither (a t + b) nor A can have a unit.

Note that a and t can have units provided they cancel. Variable b cannot!

Suppose a = 6 Hz, t = 10 s, and b = 5 (no units). Hz is the derived unit Hertz and is reducible to fundamental units 1/s.

Then A = sin(6 1/s 10 s + 5) = sin(60 + 5) = sin(65) = 0.906

No units!


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Richter Scale Example

Earthquake intensity is measured on the Richter Scale.

MR = log(A) where is a seismic amplitude factor.

The famous San Francisco earthquake of 1906 was MR = 7.8 on the Richter scale.

Does A have units?

No! According to Rule 3, Transcendental Function

Does MR have units?

No! Ditto.


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Bernoulli Example

2

3 2

23 3 2

Bornoulli's Law

1P v g h constant

2where

kg m mP [Pa], [ ], v [ ], g [ ], h [m]

sm s1

Note: is a dimensionless constant.2

kg m kg mPa ( ) m constant

sm m s


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Bernoulli Example

2 3

N kg

m m

2

1

m

2 3

kg

s m

1

m

2m

s constant

kg m

2

2s

m 1 2 2

2 2 2

kg kgconstant

m s m skg kg kg

cons Consistant m s m s m s

tent


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Coherent Systems of Units

A system of units is coherent if all units use the numerical factor of one. For example, the SI system is coherent, so the unit m/s implies (1 meter) / (1 second).

Both the SI and USCS systems are coherent.

This means that when you use SI or USCS units in a relation (formula), no numeric factors will be needed.

Unless otherwise indicated, change all units to SI and USCS base or derived units before plugging in a formula.


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Base Units SI (Metric)

SI - Systeme International or Metric System

Fundamental Dimension

1. Length

2. Mass

3. Time

4. Temperature

5. Electric current

6. Molecular substance

7. Luminous intensity

Base Unit

meter (m)

kilogram (kg)

second (s)

kelvin (K)

ampere (A)

mole (mol)

candela (cd)

Note: Force and charge are not fundamental units.


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Derived Units in SI


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Base Units USCS

USCS - United States Customary System

Fundamental Dimension

1. Length

2. Force

3. Time

4. Temperature

5. Electric current

6. Molecular substance

7. Luminous intensity

Base Unit

foot (ft)

pound (lb)

second (s)

rankine (R)

ampere (A)

mole (mol)

candela (cd)

Note: Mass and charge are not fundamental dimensions.


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Derived Units in USCS

What is the kinetic energy of a 20 ton ship moving 5 knots?


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Coherent System Example

Ke = W · v2 / (2 · g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2)

What is the system of units?

Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s

Is USCS coherent? Yes.

What is the relation for kinetic energy?

Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s

What is first step?


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Ke = W · v2 / (2 · g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2)

Convert to base units.

W in tons and v in knots.

20 tons · 2000 lbs/ton = 40,000 lbs = 4 · 104 lbs

What’s not in base units?

5 knots · 1.688 (ft/s)/knot = 8.44 ft/s

Substituting,

Ke = 4 · 104 lb · (8.44 ft/s) 2 / (2 · 32.2 ft/s2)


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Ke = 4 · 104 · 71.2336 / 64.4 lb · ft 2 /s2 · s2/ft

Ke = 4.42 · 104 lb·ft Ans

1


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Torricelli Example

h

d3

22

2 32 2 2

2 2

Relations : FromTorricelli's Principle,

Q A 2gh A 2gh

where

m mQ( ), A(m ), g( ), h(m)

s sSolution:

(a) Verify units of Q from the formula

m m m mQ A 2gh m m m m , same as Q.

s ss s(b) Calculating the ar

2 24 2

4 22

33

ea A of a circular opening,

d 2 cm 0.02 m (Must work in units given above!)

d 0.02A 3.14 10 m

4 4

Q A 2gh

mQ 3.14 10 m 2 9.81 1.2m

s

mQ 1.52 10 Ans

s

Unit Analysis

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