Engineering Statics

ENGR 2301

Chapter 1

Introduction And Measurement

What is Mechanics?

1 - 2

• Mechanics is the science which describes and predicts

the conditions of rest or motion of bodies under the

action of forces.

• Categories of Mechanics:

- Rigid bodies

- Statics

- Dynamics

- Deformable bodies

- Fluids

• Mechanics is an applied science - it is not an abstract

or pure science but does not have the empiricism

found in other engineering sciences.

• Mechanics is the foundation of most engineering sciences

and is an indispensable prerequisite to their study.

Fundamental Principles

1 - 3

• Parallelogram Law

• Principle of Transmissibility

• Newton’s First Law: If the resultant force on a

particle is zero, the particle will remain at rest

or continue to move in a straight line.

• Newton’s Third Law: The forces of action and

reaction between two particles have the same

magnitude and line of action with opposite

sense.

• Newton’s Second Law: A particle will have

an acceleration proportional to a nonzero

resultant applied force.

amF

• Newton’s Law of Gravitation: Two particles

are attracted with equal and opposite forces,

22,

R

GMgmgW

r

MmGF

Significant Figures

Scientific Notation

• Leading or trailing zeroes can make it hard to

determine number of significant figures: 2500, 0.000036

• Each of these has two significant figures

• Scientific notation writes these as a number from 1-10

multiplied by a power of 10, making the number of

significant figures much clearer:

2500 = 2.5 × 103

If we write 2.50x103, it has three significant figures

0.000036 = 3.6 x 10-5

Significant Figures

Round-off error:

The last digit in a calculated number may vary depending

on how it is calculated, due to rounding off of insignificant

digits

Example:

$2.21 + 8% tax = $2.3868, rounds to $2.39

$1.35 + 8% tax = $1.458, rounds to $1.46

Sum: $2.39 + $1.46 = $3.85

$2.21 + $1.35 = $3.56

$3.56 + 8% tax = $3.84

Numerical Accuracy

1 - 6

• The accuracy of a solution depends on 1) accuracy of the given

data, and 2) accuracy of the computations performed. The solution

cannot be more accurate than the less accurate of these two.

• As a general rule for engineering problems, the data are seldom

known with an accuracy greater than 0.2%. Therefore, it is usually

appropriate to record parameters beginning with “1” with four digits

and with three digits in all other cases, i.e., 40.2 lb and 15.58 lb.

• The use of hand calculators and computers generally makes the

accuracy of the computations much greater than the accuracy of the

data. Hence, the solution accuracy is usually limited by the data

accuracy.

Chapter 1: U.S. Customary Units

The base U.S. customary units are the units of length,

force and time.

These units are the foot (ft), the pound (lb) and the

second (s).

The second (s) is same as corresponding SI unit.

The foot is defined as 0.3048 m.

The pound (lb) is defined as the weight of a platinum

standard, called the standard pound, which is kept at

the National Institute of Standards and Technology,

outside Washington, the mass of which is 0.453 592

43 kg.

Chapter 1: U.S. Customary Units

Since weight of a body depends on upon the earth

gravitational attraction, which varies with location,

the U.S. customary units do not form an absolute

system of units.

The standard pound (lb) needs to be placed at sea

level and at a latitude of 45° to properly defined a

force of 1 lb.

On the other hand, SI system of units, the meter (m),

the kilogram (kg), and the second (s) may be used

anywhere on the earth. They may even be used on

another planet. They will always have same

significance. Hence, they are called absolute system

of units.

Chapter 1: U.S. Customary Units

The standard pound also serves as the unit of mass in commercial transactions in the United States, it can not be so used in engineering computations since it will not be consistent with Newton’s second law, F = ma.

So, the unit of mass was derived from basic U.S. system of units. This unit of mass is called the slug.

F = ma, therefore, 1 lb = (1 slug) (1 ft/s²). And

1 slug = (1 lb) ÷ (1 ft/s² ) = 1 lb · s²/ft

Since acceleration of gravity g is 32.2 ft/s², slug is a mass 32.2 times larger than the mass of standard pound (lb).

Chapter 1: Other U.S. Customary Units

Other U.S. customary units frequently used are:

mile (mi) = 5280 ft.

inch (in) = 1/12 ft

kilopound (kip) = force of 1000 lb

ton = mass of 2000 lb. Note: In engineering computation, this

must be converted into slugs.

Conversion into basic units of feet, pounds, seconds and slug is

often necessary in engineering computation. This is a very

involved process in U.S. system of units than in SI system of

units.

E.G., to convert velocity of 30mi/h into ft/s, following steps

are required:

v = (30 mi/hr) (5280 ft/1 mi)(1h/3600s) = 44 ft/s

Chapter 1: System of Units

International System Of Units (SI Units): The universal system used around the world except U.S.A. and a couple of other small countries. SI stands for System Universal, a French word translated in English.

Four fundamental units, called Kinetic Units are units of length, time, mass and force.

Three of these units (Length, Time and Mass) are defined arbitrarily and are referred to as basic units.

The fourth one, the force, is defined by equation F = ma and hence called derived unit.

Chapter 1: SI Units – Length and mass

Base unit of Length: The Meter: Originally defined as one ten-millionth of the distance from the equator to either pole, is now defined as 1 650 763.73 wavelengths of the orange-red light corresponding to a certain transition in an atom of krypton-86. This was changed once again in 1983 to: “The meter is the length of path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second.

Base unit of mass: The Kilogram originally defined as equal to mass of the 0.001 m³ of water is now defined as mass of a platinum-iridium standard kept at the International Bureau of Weights and Measures at Serves, near Paris, France.

A

C

B

earth

equator

Chapter 1: SI Units -- Time

Base unit of Time: The Second:

Originally defined as 1/86 400 of the

mean solar day, is now defined as the

duration of 9 192 631 770 cycles of

the radiation corresponding to the

transition between two levels of the

fundamental state of the cesium-133

atom.

Chapter 1: SI Units -- Force

Base unit of Force: The Newton(N): The unit of force is a derived unit. It is defined as the force which gives an acceleration of 1 m/s² to a mass of 1 kg.

As we know from Newton’s second fundamental law, F = ma

So, 1 N = (1 kg ) (1 m/s² ) = 1 kg · m/s²

Chapter 1: SI Units –Weight Weight of a body: It is the

force of gravity exerted on body.

Like any other force, should be expressed in Newtons, not in kg.

W = mg

I.E., W = ( 1 kg)( 9.81 m/s² )

I.E. W = 9.81 N

While standard kg also serves as the unit of Weight in commercial transactions, it can not be so used in engineering computations.

Chapter 1: SI Units – commonly used units

The most frequently used units are kilometer(km), millimeter(mm), megagram(Mg) which is known as metric ton, gram(g) and kilonewton(kN).

1 km = 1000 m 1mm = 0.001 m

1 Mg = 1000 kg 1 g = 0.001 kg

1 kN = 1000 N

3.82 km = 3820 m 47.2 mm = 0.0472 m

3.82 km = 3.82 x 10³ m

47.2 mm = 47.2 x 10³־ m

Chapter 1: SI Units – Derived units

There are many other units derived from the basic kinetic units (Length, Mass, Time and Force).

The most common derived units are units of Area and Volume.

The unit of Area is square meter (m²) which represents the area of a square of side 1 m.

The unit of Volume is the cubic meter (m³), equal to the volume of a cube of side 1 m.

The Volume of liquid is measured in cubic decimeter (dm³) is commonly referred as a liter (L).

Chapter 1: SI Units – Multiplication factors-Length

Multiple and sub-multiple of the units of Length:

1 dm = 0.1 m = 10¹־ m

1 cm = 0.01 m = 10²־ m

1 mm = 0.001 m = 10³־ m

1 km = 1 000 m = 10³ m

Multiple and sub-multiple of the units of Area:

1 dm² = (1 dm)² = (10¹־ m)² = 10²־ m²

1 cm² = (1 cm)² = (10²־ m)² = 10 -4m2

1 mm² = (1 mm)² = (10³־ m)² = 10 -6m 2

Multiple and sub-multiple of the units of Volume:

1 dm³ = (1 dm)³ = (10¹־ m)³ = 10³־·m³

1 cm³ = (1 cm)³ = (10²־ m)³ = 10 -6m 3

1 mm³ = (1 mm)³ = (10³־ m)³ = 10 -9m 3

Chapter 1: SI Units – Multiplication factors conventions-

In order to avoid exceedingly small or large numerical values, many sub-units are defined and used.

When more than four digits are used on either side of the decimal point -- as in

427 200 m or 0.002 16 m – spaces, never commas, should be used to separate the digits into groups of three. This is to avoid confusion with the comma used in place of a decimal point, which is the convention in many countries.

Example for use of multiple and sub-multiple of the units of Length:

You write 427.2 km instead of 427 200 m

You write 2.16 mm instead of 0.002 16 m.

Chapter 1: SI Units – Multiplication factors-Length

Multiple and sub-multiple of the units of Length:

1 dm = 0.1 m = 10¹־ m

1 cm = 0.01 m = 10²־ m

1 mm = 0.001 m = 10³־ m

1 km = 1 000 m = 10³ m

Multiple and sub-multiple of the units of Area:

1 dm² = (1 dm)² = (10¹־ m)² = 10²־ m²

1 cm² = (1 cm)² = (10²־ m)² = 10 -4m2

1 mm² = (1 mm)² = (10³־ m)² = 10 -6m 2

Multiple and sub-multiple of the units of Volume:

1 dm³ = (1 dm)³ = (10¹־ m)³ = 10³־·m³

1 cm³ = (1 cm)³ = (10²־ m)³ = 10 -6m 3

1 mm³ = (1 mm)³ = (10³־ m)³ = 10 -9m 3

Chapter 1: SI Units – Two ideas

However, there have been two ideas as to which metric units

should be preferred in science. Scientists working in

laboratories, dealing with small quantities and distances,

preferred to measure distance in centimeters and mass in

grams. Scientists and engineers working in larger contexts

preferred larger units: meters for distance and kilograms for

mass. Everyone agreed that units of other quantities such as

force, pressure, work, power, and so on should be related in a

simple way to the basic units, but which basic units should be

used?

The result was two clustering of metric units in science and

engineering. One cluster, based on the centimeter, the gram,

and the second, is called the CGS system. The other, based on

the meter, kilogram, and second, is called the MKS system

Chapter 1: SI units vs. US units

The beauty of the metric system lies in its simplicity and

consistency.

Despite the advantages of the metric system, English units are still in wide use, therefore we must be able to work with all kinds of units.

Unlike the English system, which uses a hodge-podge of units to express the same physical quantity (for example length) with no consistent conversions between them.

the metric system uses a single unit of measure modified by a prefix to change the measurement scale. For example, the English system uses inches, yards, and miles to measure distances of varying scales, which have no consistent conversion factors between them. In contrast, the metric system uses a single unit, the meter, which, with appropriate prefix modifications produce roughly comparable scales: centimeters, meters, and kilometers. Moreover, the metric system uses the same set of prefixes for scaling regardless of the physical quantity under consideration.

Chapter 1: SI Units vs. US unit

Hence mass can be measured in centigrams, grams, and kilograms.

it is not true that the US remains the last holdout.

While the rest of the world is pretty much standardized on the metric system of measurements, when it comes to mandatory use, the United States has company in its foot dragging. Great Britain, Liberia and Burma are right there along with the United States.

Some international organizations have threatened to restrict U.S. imports that do not conform to metric standards and rather than trying to maintain dual inventories for domestic and foreign markets, a number of U.S. corporations have chosen to go metric.

Chapter 1: SI Units vs. US units

You will be seeing more and more of your

customers in the US using the Metric system in

their purchases and writing more original

specifications in Metric.

Scientists have adopted the metric system to

simplify their calculations and promote

communication across national boundaries

Some Motor vehicles, farm machinery, and

computer equipment are now manufactured to

metric specifications.

Chapter 1: Conversion of units

There are many instances when conversion from U.S. system to SI system or vice versa is required. Since the unit of time is the same in both the system, only two kinetic base unit need to be converted.

1 ft = 0.3048 m and 1 lb = 4.448 N

This makes 1 slug = 14.59 kg.

Since all other kinetic units and conversion factors can be derived from these base units.

E.G. 1 mi = 5280 ft = 5280·(0.3048 m) =1609 m = 1.609km

It is important to solve many problems involving conversion of these units to understand these concepts.

Based of the base units, we may need to change the units of a

given quantity using the chain-link conversion.

For example, since there are 60 seconds in one minute,

Conversion between one system of units and another can

therefore be easily figured out as shown.

The first equation above is often called the “Conversion

Factor”.

Changing units

ss

xx

ands

s

120)min1

60(min)2()1(min)2(min2

,min1

601

60

min1