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Definitions of Physical Quantities

A physical quantity is a physical property that can be quantified by measurement.

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Types of physical quantities

- Scalar quantities

are quantities that have magnitude only; they are independent of direction.

(time, temperature, mass, density, energy,…)

- Vector quantities

are quantities that have magnitude and direction. (displacement, velocity, moment, force, ….)

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Symbols for physical quantities

Usually, the symbols for physical quantities are chosen to be a single letter of the Latin or Greek alphabet, and are often printed in italic type (F, t, m, ……).Often, the symbols are modified by subscripts and superscripts, to specify what they refer to - for instance Ek is usually used to denote kinetic energy and cp heat capacity at constant pressure.

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Physical Quantities

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Vector presentation

- When vectors are written, they are represented by a single letter in bold type or with an arrow above the letter, such as or . Some examples of vectors are displacement (e.g. 120 cm at 30°) and velocity (e.g. 12 meters per second north). The only basic SI unit that is a vector is the meter. All others are scalars. Derived quantities can be vector or scalar, but every vector quantity must involve meters in its definition and unit.

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Vector notation

- Vectors are distinguished from scalars by writing them in special ways. A widely used convention is to denote a vector quantity in bold type, such as A, and that is the convention that will be used. you may also encounter the notation Ū orḛ.
- The magnitude of a vector A is written as |A|.

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Representation of vectors in Cartesian coordinates

The vector can be represented by its components (magnitude)in (x,y,z) directions. The direction of the vector can be represented by a unit vector u = i + j + k,

so:

v = vxi + vyj + vzk

where the units can be anything

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Representation of vectors in Cartesian coordinates (cont.)

- F = Fxi + Fyj + k

Where:

Fx = F cosα

Fy= F cosβ

Fz= F cosγ

α, β and γ are the angles the force vector (F) making with x, y and z axis respectively. And

F = √ Fx2 + Fy2 + Fz2

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(adding, subtracting, multiplying ..)

- you have to consider both

the magnitude and the direction.

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Vectors Addition

Adding two vectors in Cartesian form v1and v2

v3 = v1+ v2

v3 =(vx1i + vy1j + vz1 k)+(vx2 i + vy2 j + vz2k)

v3 =(vx1 + vx2) i + (vy1 + vy2) j + (vz1 + vz2) k

That is, the components of a sum are the sums of the components.

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Units of physical quantities

- Most physical quantities Q include a unit. Neither the name of a physical quantity, nor the symbol used to denote it, implies a particular choice of unit.

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Systems of Units

- The Imperial units or the Imperial system is a collection of English units, first defined in the Weights and Measures Act of 1824, later refined (until 1959) and reduced. The units were introduced in the United Kingdom and its colonies, including Commonwealth countries, but excluding the then already independent United States.
- The United States customary system (also called American system or, more rarely, "English units") is the most commonly used system of measurement in the United States. It is similar but not identical to the British Imperial units.

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Systems of Units (cont.)

- The most widely used system of units and measures around the world is the Systeme International d'Unites(SI), the modern form of the metric system. This originated in France, where in 1790 the French Academy of Science was commissioned to design a new system of units (the International System of Unit).

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International system of units (SI)(cont.)

- Associated with basic units are a variety of supplementary derived units.
- Derived units:

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International system of units (SI) (cont.)

- Derived units with special names and symbol:

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International system of units (SI)(cont.)

- In the International system of unit, a unit is chosen for a particular purpose and larger and smaller are obtained by applying a prefix to this unit and multiplying or dividing by 10 or power of 10.
- Larger units, obtained by multiplying by 1000, etc., are called ‘multiples’, smaller units, obtained by multiplying by 0.1, etc. are termed ‘sub-multiplies.

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International system of units (SI)(cont.)

- Units outside the SI that accepted for use with the SI:

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International system of units (SI)(cont.)

- Other units outside the SI that are currently accepted for use with the SI:

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SI unit rules and style conventions

- Unit symbols are placed after the numerical value, leaving a space between the value and symbol. e.g. 5 V not 5V.
- Only one prefix can be applied to a unit at a time. e.g. 1000 kilonewton must not express 1kilokilonewton but as 1 meganewton, (1000 kN≠ 1 kkN = 1 MN).
- The correct use of upper and lower case letters (capital and small letters) is important. i.e. m- meter; but M- mega; k- kilo; K- kelvin.
- Symbols must not made plural by adding ‘s’ since ‘s’ is symbol of second. e.g. 10 kg not 10 kgs.

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SI unit rules and style conventions (cont.)

- When a prefix is attached to a unit there should be no space between the prefix and the unit; i.e. mm- millimeter, kW- kilowatt… etc.
- When a complex unit is formed by multiplying two units together, however, the symbol should be separated by a space; e.g. N m- newton meter (the unit of torque).
- Abbreviations such as sec, cc, or mps are avoided and only standard unit symbols, prefix symbols, unit names, and prefix names are used (s, cm3, m/s).

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SI unit rules and style conventions (cont.)

- Unit symbols are generally written in lower case letters, except when the name of the unit is derived from a proper name. (Note that the name of a unit which is derived from a proper name is written out in full, such as ampere or hertz, the name is not capitalized. The only exception to this is Celsius. Note that "degree Celsius" conforms to this rule because the "d" is lowercase.

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Dimensional analysis

- Technique used in the physical and engineering to reduce physical properties such as acceleration, velocity, energy and others to their fundamental dimensions of length(L), mass (M), and time (T). This technique facilitate the study of interrelationships of systems (or models of systems) and their properties. Acceleration, for example, is expressed as length per unit of time squared (LT-2); whether the units of length in the English or the metric system is immaterial.

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Physical quantities: Time

Time (t)

The fundamental unit of time suggested by SI system is

the second, since 1967 defined as the second of

international Atomic Time, based on the radiation

emitted by a Caesium-133 atom in the ground state.

Based on the second as base unit, the following units

are in use:

- minute (1 min) = 60 s
- hour (1h) = 60 min = 3.6 ks
- Julian day (1 day) = 24 h = 86.4 ks

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Physical quantities: Displacement, Area & Volume

Displacement and distance (L)

Displacement is measured as meters (m), or sometimes more

conveniently as kilometers, millimeters or centimeters. Distance

(scalar) is the magnitude of the displacement (vector).

Area (A)

Any flat, curved, or irregular expanse of a surface. It is measured

in meter square (m2).

Volume (V)

The volume of any solid, liquid, gas, plasma, or vacuum is how

much three-dimensional space it occupies, often quantified

numerically. It is measured in cubic meter (m3) or liter (l).

- 1 l = 1000 cm3 = 0.001 m3

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Physical quantities: Angle

Angle (θ)

- Angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other.
- The degree and the radian are by far the most common. The turn (or full circle, revolution, rotation, or cycle) is one full circle.
- 1 turn = 360° = 2π (rad)

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Physical quantities: Linear & Angular Velocities

Velocity (v) and speed

Velocity is measured as meters per second (m/s), or sometimes more

conveniently as kilometers per hour (km/h).

Useful conversions are m/s = 3.6 km/h, and 0.278 m/s = 1 km/h). The

speed is the magnitude of the velocity vector.

Angular velocity (ω)

Angular velocity is a measure of the angular displacement per unit

time.

The angular velocity can be measured in revolution per minute (rpm).

ω = 2 π N/60

- Where: N = revolution per minute (rpm)

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Physical quantities: Angular & Linear Velocity

Angular and linear velocity

The linear velocity of a particle is related to

angular velocity by :

v = ω r

v = (2 π N/60) r

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Physical quantities: Linear Acceleration

Acceleration (a)

Acceleration or deceleration is the rate of change of speed. It is measured as meters

per second per second or m/s2. If the speed increases from u m/s (initial velocity) to v

m/s (final velocity) during t seconds (time), then the average acceleration a m/s2 is

given by

a = (v-u)/t m/s2

Acceleration due to gravity (g)

In physics, gravitational acceleration is the acceleration on an object caused by

gravity, a conventional standard value of exactly 9.80665 m/s2 (g = 9.81 m/s2)

* Car acceleration and deceleration indication

Another way to indicate the car acceleration is the time taken by the car form 0

velocity to reach 100 km/h, or (0- 60 mph).

* A convenient way to measure braking action is to equate (compare) vehicle deceleration to the gravity acceleration constant [g]. Example: declaration = 0.3 g.

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Physical quantities: Angular acceleration

Angular acceleration (α)

Angular acceleration is the rate of change of angular

velocity over time. In SI units, it is measured in radians

per second squared (rad/s2).

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Physical quantities: Mass & Weight

Mass (m) and weight (w)

A 'body' contains a certain amount of stuff or

matter called mass (m). The unit of mass is the

kilogram (kg).The pull of earth -the force of

gravity- acting on this mass is the weight (w)of

the body. The unit of weight is the newton (N).

w = mg

Where: g is the acceleration of falling body due

to gravity in meter per second square (m/s2).

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Physical quantities: Mass moment of inertia

Mass moment of inertia (I):

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, or the angular mass, (SI units kg m2), is a measures of an object’s resistance to changes to its rotation.

I = ∫ r2 dm

Where:

m is the mass,

r is the perpendicular distance to the axis of rotation.

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Physical quantities: Density

Density (ρ)Density is the mass ofa substance per unit volume (kg/m3). The density of water is, for practical purposes, 1000 kg/m3 or 1 kg/l. (The litre (l) is 10-3 m3.)

- The density of solids and liquids are usually stated in g/cm3, while gases are usually in kg/m3.

Relative density or specific gravity

- relative density of the substance = (mass of a substance / mass of an equal volume of water)
- This ratiois called the relative density of the substance, and represents how many times it is heavier or lighter than the same volume of water. Note that relative density has no units. Example of substances relative densities (oxygen= 0.0014, steel=8.0, lead=11.4, mercury = 13.6).

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Physical quantities: Force

Force (F)

The force is a measurable influence tending to cause

movement of body (its intensity). The unit of force is newton,

1 newton is the value of a force which if exerted upon a mass

(m) of one kilogram gives it an acceleration (a) of 1 m/s2.

F = m a, 1 N = (1 kg) (1 m/s2) = 1 kg m/s2

There are different types of forces; external force, internal

force, friction force, inertia force and reaction force.

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Physical quantities: Pressure

Pressure (p)Pressure is the force per unit area; the unit is N/m2 or the Pascal (Pa). Larger practical units are kN/m2 (kPa) and MN/m2 (MPa). Note that

1 MN/m2 (MPa) = 1N/mm2

A pressure of 7 MPa means that each mm2 subject to the pressure has a force of 7 N acting on it, and the total force on the surface will be the product of the pressure and the area.

Atmospheric pressureAir has weight. The atmosphere above the earth produces a pressure at sea level of approximately 1 bar, where 1 bar = 105 N/m2 or 105 Pa. Standard atmospheric pressure (atm) is 1.01325 bar.

1 atm = 760 mm Hg (Torr) (mercury column) = 10333 mm H2O (water column)

Gauge and absolute pressureThe ordinary pressure gauge gives readings measured above atmospheric pressure. To obtain the absolute pressure, that is the pressure measured above a perfect vacuum, atmospheric pressure must be added to the gauge reading:

absolute pressure = gauge pressure + atmospheric pressure

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Physical quantities: Torque

Torque (T)When a force (F) acts on a body pivoted on a fixed axis, the product of the force perpendicular to the radius, and the radius at (r) which it acts, is termed the turning moment of the force or torque. Torque (T) is measured in newton meters (N m).

T = F r

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Physical quantities: Work & Power

Work (W)

Work is done when a force overcomes resistance and causes movement. Work is

measured by the product of the force (f) and the distance moved (s) in the direction of

the force, the unit being the joule (J):

W = F s

If the force causes no movement, then no work is done, or if the force in the apposite

direction of the movement, then the work is negative.

Power (P)

Power is the rate of doing work. The unit, the watt, is a rate of working of 1 joule per

second (1 J/s)

power = work done per second = [W / t] {N m /s}, {J/s}, (W)

The relation between power and torque:

If a tangential force is applied to a shaft, then the work done per one revolution is:

W = F s = F (2 π r) = (2 π) T

Since the power is

P = W/t = (2 π / t) T, then: P = ωT, P = (2 π N/60) T

Using these formulae, the power can be calculated from the torque and speed of a

shaft.

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Physical quantities: Temperature

Temperature (t)

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Quantitatively, temperature is measured with thermometers.The International System of Units (SI) defines a scale and unit for the thermodynamic temperature by using the kelvin temperature. The unit symbol of the kelvin is K. While the Kelvin scale is the principal temperature scale for use in science and engineering, much of the world uses the Celsius scale (°C) for most temperature measurements.

1 K = 1°C = 1.8 °F = 1.8 °R

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Physical quantities: Energy

Energy (E)In all such energy transformation processes, the total energy remains the same. Energy may not be created nor destroyed.

- Any form of energy can be transformed into another form. When energy is in a form other than heat, it may be transformed with good or even perfect efficiency, to any other type of energy.
- Measurement of energy in the SI unit is the joule. In addition to the joule, other units of energy include the kilowatt hour (kW h) and the British thermal unit (Btu). These are both larger units of energy. One kW h is equivalent to exactly 3.6 million joules, and one Btu is equivalent to about 1055 joules.
- There are different types of energies; kinetic, potential, mechanical, thermal, chemical, electric and nuclear energy.

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