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Dimensional Analysis and Conversions

Chapter 1 Circuits

ENGR 1375

Units of Measurement

• Units are one of THE most important things to be aware of when working in engineering and technology.

• Incorrect or inconsistent units produce meaningless results. – Any numerical result is incorrect if it does not include units!!

• Don’t get too focused on obtaining the numerical result that you omit the units!

• Units can be VERY helpful in problem solving

– Dimensional Analysis

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Units of Measurement- Cont’ • Typical conversion factors:

– 1 in = 2.54 cm – 1 m = 39.37 in – 1 mile = 5280 ft – 1 ft·lb = 1.356 J (newton·meter) – 1 lbm = 0.4534 kg – 1 Slug = 14.6 Kg

• Get other conversions online or out of books • Another way to write conversion factors- IMPORTANT

– 2.54 cm/in, or (1/2.54) in/cm or 0.3937 in/cm – 5280 ft/mile or (1/5280) miles/ft or .00019 miles/ft – 0.4534 kg/lbm or (1/.4534) 2.205 lbm/kg

• Conversion Example: Convert 27 inches to cm 27 in · 2.54 cm/in = 68.58 cm • Conversion Example: Convert 75 kg to lbm 75 kg · 2.205 lbm/kg = 165.4 lbm

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Units of Measurement - Cont’ • For temperatures, there is no conversion factor, you

must us a formula – °F = (9/5)· °C + 32 – °C = (5/9) (°F- 32) – °K = °C + 273.15

• Caution! there are units that are equivalent that don’t seem to make sense – N·m is the same as Joule – Newton (force) is the same as kg·m/sec2

– Joule (energy) is the same as N·m – Joule is also the same as kg·m2/sec2

• These are mostly used for unit analysis or error checking

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Dimensional Analysis

• Always carry the units along with calculations • Example: d = v·t (let’s keep units in SI)

If v is 45 m/s and time is 10 sec, d = 45 m/s · 10 s = 450 m – You can cancel s’s and get 450 m

• Note that the answer units come out correctly • Another example: F = m·a (let’s again keep units in SI)

If m is 5 kg and a is 15 m/s2 F = 5 kg · 15 m/s2 = 75 kg·m/s2

Is this unit kg·m/s2 correct? Shouldn’t it be force (N) ?! – It is! Recall that N is equivalent to kg·m/s2 Some units may have an equivalent (N is the same as kg·m/s2).

• In a similar way, Energy = force (N) x distance (m) So the units here are N · m, which are the same as Joules (J)

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Dimensional Analysis- Cont’ • Unit analysis can be used for error checking during

calculations • Say you multiply velocity by time to get distance, where

velocity is 60 miles/hr and time is 30 minutes, let’s multiply d = v · t = 60 miles/hr · 30 min = 1800 (miles·min/hr) • These units do not make sense; something is not right! • To fix this, you can first convert minutes to hours • 30 min is 1/2 hour d = v · t = 60 miles/hr · ½ hr = 30 miles • This makes more sense, so remember that you can use

dimensional analysis anytime to check your calculations!

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More Conversion Examples

250 cm x

1 in

2.54 cm x

1

1 ft

12 in x

1 yd

3 ft = 2.73 yd

• Use this method if you have multiple conversions • Example, Convert 250 cm to yds • The conversion factors you will use are 2.54 cm/in, 12 in/ft and 3 ft/yd

• Convert 0.24 m to centimeters (Ex 1.18)

0.24 m x

100 cm

1 m 1 = 24 cm

• What conversion factor did we use?

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Conversion Examples- Cont

• Determine the number of minutes in ½ day (ex 1.19)

½ day x

24 hr

day x

1

60 min

hr = 720 min

Dimensional Analysis/Units Conversion – Things to Keep in Mind • A helpful tip: Always develop the analysis as completely

as possible before inputting numerical values – helps to minimize errors.

• Before inputting numerical values in your analysis, check the following:

1. Each quantity has the proper unit of measurement as defined by the equation.

2. The proper magnitude of each quantity in the defining equation is substituted.

3. Each quantity is in the same system of units (or as defined by the equation).

4. The magnitude of the result is of a reasonable nature when compared to the level of the substituted quantities.

5. The proper unit of measurement is applied to the result. Page 9