ENG1040 Lec02

Faculty of Engineering

ENG1040 – Engineering Dynamics

ENG1040Engineering Dynamics

Dimensions and Units

Dr Lau Ee Von – Sunway

Lecture 2

Lecture Outline

• Dimensions and units – a definition• Equations and equality – why is this

important ?• Dimensional Analysis• Operating on dimensional quantities

• What happens when I integrate ?• What happens when I differentiate ?

• Example exam question

2

Dimensions & Units

Equations & Equality

Dimensional Analysis

Operating on Dimensional Quantities

Example exam question

When A = B , it implies that :

Equations

Equations denote equality:

3. The dimensions (and thus the units) are the same.

There is Dimensional Homogeneity!

1. The numerical values are the same,

2. The quantities are of the same type,

Dimensions & Units





3

The usual corresponding units are kg, m, and s.

We shall normally use the dimensions of mass,M, length L, and time,T.

Dimensions vs Units

i.e., L/T m/s units of speed, velocity

In dynamics, everything can be described using these dimensions alone.

Dimensions & Units





4

Equations denote Equality !

The Dimensions of the LHS

must be the same

as the Dimensions of the RHS

+ =


Dimensions & Units





5

(i.e. Equation is dimensionally homogeneous)

A familiar example:

s ut at 1

22

The dimensions of the LHS aredenoted:

LHS: [ s ] = length = L


Dimensions & Units





6

Both terms on the RHS must have the same dimensions if they are to be added in any

meaningful way.

A familiar example:

s ut at 1

22


Dimensions & Units





7

What about the dimensions on the RHS?

tuut

22 taat21

L

L

A familiar example:

s ut at 1

22


22T

T

L

TT

L

Dimensions & Units





8

The dimensions of both sides are: length, [L]

The equation is dimensionally homogeneous

A familiar example:

s ut at 1

22


Dimensions & Units





9

How do we analyse if the equation is dimensionally

homogeneous?

A different arrangement:


Dimensions & Units





10

𝑠=𝑡(𝑢+12𝑎𝑡 )

Expand the equation, and analyse every term, just as before!

s ut at 1

22

A preliminary requirement in dimensional analysis…

…is the need to establish the units of the various quantities in the equations.

Some examples...


Dimensions & Units





11

arc

radius

The usual unit for an angle is the

radian

arc

radius

L

Ldimensionless

The dimensions are:

Dimensional Analysis - Angles

Dimensions & Units





12

Frequency, and angular speed may be measured in units of…

radian s-1 (rad/s)

revolutions s -1 (Hz)

revolutions min -1 (RPM)

Dimensional Analysis - Angles

Dimensions & Units





13

We use the S.I. unit to find its dimensions, i.e. rad/s [T]-1

We know that,

Force has dimensions of : M L T -2

Dimensional Analysis - Forces

Dimensions & Units





14

We can also general equations to find the dimensions of a quantity.

For example, to find the dimensions of Force (S.I. unit = Newton)

Pressure has units of Force per unit area:

The dimensions are:[P] = [Force/Area] = MLT-2/L2

= ML-1T-2

Dimensional Analysis - Pressure

This unit is called a Pascal

Dimensions & Units





15

Work has units of Force times distance,

The dimensions are thus[W] = [Force*Distance] = (M LT-2) L

This unit is called a Joule – i.e. a Nm.

= M L2 T-2

Dimensional Analysis - Work

Dimensions & Units





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Power has units of: Force times velocity.

The dimensions are thus[P] = [Force*Velocity] = (M LT-2) (LT-1)

This unit is called a Watt – i.e. Nm/s.

= M L2 T-3

Dimensional Analysis - Power

Dimensions & Units





17

Dimensional Analysis – Other Units

Other quantities widely used in engineering:

Practice Classes will provide an opportunity to become familiar with many of these…

Torque, Bending Moment, Shear Modulus, Momentum, Stress, Strain, Stiffness,

Damping Coefficient, Moment of Inertia, Dynamic Viscosity, Impulse, etc.

Dimensions & Units





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The units of are those ofdx

dy

x

yLt

dx

dyX

0

xy

These are the same as : xy

x

y

x

Derivatives of Dimensional Quantities

Dimensions & Units





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Fo

rce

Time

The dimensions of dF/dT would

be?

TF

dT

dF

3

2

MLTT

MLT


Dimensions & Units





20

xdx

dy

Ltdx

ydx

02

2

dx

dy

xo

dx

dy

x


Dimensions & Units





21

xdx

dy

Ltdx

ydx

02

2

xy

dx

dy

xo

dx

dy

x


Dimensions & Units





22

xdx

dy

Ltdx

ydx

02

2

xy

Dimensions are:

xxy

dx

dy

xo

dx

dy

x


Dimensions & Units





23

xdx

dy

Ltdx

ydx

02

2

xy Dimensions

are:

dx

dy

xo

dx

dy

x


2xy

Remember by looking at the quantities being

differentiated

Dimensions & Units





24

Thus the dimensions of an integral are: xy

ydxA

Integrals are areas

y

x

Dimensions of Integrals

Dimensions & Units





25

Force

Time

2

1

Impulset

t

Fdt

Units of Impulse = [F][t] = TT

ML2

t1 t2

= MLT-1

Dimensions of Integrals

E.g., Impulse:Dimensions & Units





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Past exam question

• Mid-semester Sem 1, 2013

27

Dimensions & Units





28

Dimensions & Units





Past exam question

• Sem 2, 2007

29

Dimensions & Units





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Dimensions & Units





Conclusion

• To ensure an equation is dimensionally homogeneous, ensure that the units are the same (SI or Imperial) for every term on both LHS and RHS

• All engineering dynamics equations can be described by the 3 dimensions, mass [M], time [T] and length [L].

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Education

ENG1040 Lec02