Namas ChandraIntroduction to Mechanical engineering Chapter 8-1

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CHAPTER 8

Dimensions, Units, and Error

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Objectives

Understand the difference between a unit and dimensionsConvert quantities between the different systems of measurementUnderstand the types of error in a measurement Express large numbers in scientific notation

Sections

8.1 Introduction8.2 Dimensions and Units8.3 Conversion between Different Units8.4 Errors and Accuracy8.5 Significant Digits8.6 Summary

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Basic Units and Dimensions

Unit is the basic quantity in a given system of measurement yard, meter, inch

Some units can be a combination of different basic units Example: Unit of force depends on the choice of the units

of mass and acceleration Dimension describes the quantity in terms of Mass, Length, Time

and Temperature without specifying any unit.Example:

-22

-2

La LT

T

F m a MLT

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Basic Units and Dimensions…2

One of the fundamental tenet for any physically valid equation is that dimensions on either side of the equation should be identical.Now F=ma=kx. Now to get the units of stiffness k,

-2

-2MLTk = = MT

L

In fact dimensional analysis is used to check any proposed equation is valid or not.

Groups of quantities can be combined in a manner that the group becomes dimensionless and they oftentimes represent very important fundamental quantities.

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8.2 Systems and Measurement

There are three major systems of measurement They are

BGS (British Gravitational System)EES (English Engineering System)SI System (The current worldwide standard)

BGS Here force (not mass), feet (for length) and second (time) are the

fundamental units.

Since, ,mass is derived unit. 2 -1FM FT L

a

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Systems of Measurement…2

The unit of mass is a slug

Since weight is a force acting on mass of 1 slug

A body with a mass of 1 slug has a weight of 32.174 lb.

2lb sec1 slug = 1

ft

2

ftW = 1 slug 32 = 32.174 lb

sec

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Systems and Measurement…3

EES Here the unit of mass is defined such that weight is 1 lbf Mass is 1 lbm

For this system use

SI SI is the metric system. Fundamental units are grams (weight),

cm ( length), and seconds (time).

2

ft1 lbf = 1lbm 32.174

sec

c 2c

ma ftF = with g = 32.174

g sec

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Conversion between Units

We are faced with converting a quantity from one system to another. Use the logic that units should cancel on both sidesExample:

6

3

3

mile m m30 30 0.447 13.411

hour sec sec

m 10 m 1sec13.411

sec 1m 10 millisec

m13.411 10

millisec

mile mile 5280 ft 0.3048 m 1 hour1 1

hour hour 1 mile 1 ft 3600 sec

m0.447

sec

Conversion of FPS SI

Force 1 lb = 4.4482 NMass 1 slug = 14.5938 kgLength 1 ft = 0.3048 m

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Comparison of Units in BGS, EES, and SI

Quantity BGS EE SIMass Slug lbm kilogram

Length ft ft meter (m)

Time sec (s) sec (s) sec (s)Area

Velocity Ft/s Ft/s M/sAceleration

Density Force Lbf Lbf Newton(N)

Pressure Pascal (Pa)

Energy Ft x Lbf Ft x Lbf Joule

Volume

Power Ft x Lbf/s Ft x Lbf/s Watt (W)

2ft2m

2ft/s 2m/s3Slug/ft 3lbm/ft 3Kg/m

2Lbf/ft

3m3ft

2ft

2ft/s

2Lbf/ft

3ft

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Order of 10 in metric system

Mult. Factor Prefix SI Symbol tera T giga G mega M kilo k

hecto h deka da deci d centi c milli m micro nano n pico p

femto f atto a

12

9

6

3

2

1

1

2

3

6

9

12

15

10

10

10

10

10

10

10

10

10

10

10

10

10

μ

1810

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Errors and Accuracy

Every quantity is measure in some form or another Accuracy is defined as the difference between true value and

measured value All measurements contain error Error has two parts

Bias error-inherent in the measurement systems. E.g. graduation in a tape

Precision error- Error that arises due to lack of repeatability. Arises due to operator or environmental conditions

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CHAPTER 8..concludes

Dimensions, Units, and Error