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

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

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

Equations denote Equality !

The Dimensions of the LHS

must be the same

as the Dimensions of the RHS

Dimensions & Units

(i.e. Equation is dimensionally homogeneous)

A familiar example:

s ut at 1

The dimensions of the LHS aredenoted:

LHS: [ s ] = length = L

Dimensions & Units

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

Dimensions & Units

What about the dimensions on the RHS?

22 taat21

A familiar example:

s ut at 1

Dimensions & Units

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

The equation is dimensionally homogeneous

A familiar example:

s ut at 1

Dimensions & Units

How do we analyse if the equation is dimensionally

homogeneous?

A different arrangement:

Dimensions & Units

𝑠=𝑡(𝑢+12𝑎𝑡 )

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

s ut at 1

A preliminary requirement in dimensional analysis…

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

Some examples...

Dimensions & Units

radius

The usual unit for an angle is the

radian

radius

Ldimensionless

The dimensions are:

Dimensional Analysis - Angles

Dimensions & Units

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

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

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

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

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

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

The units of are those ofdx

These are the same as : xy

Derivatives of Dimensional Quantities

Dimensions & Units

The dimensions of dF/dT would

Dimensions & Units

Dimensions are:

Dimensions & Units

xy Dimensions

Remember by looking at the quantities being

differentiated

Dimensions & Units

Thus the dimensions of an integral are: xy

Integrals are areas

Dimensions of Integrals

Dimensions & Units

Impulset

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

= MLT-1

Dimensions of Integrals

E.g., Impulse:Dimensions & Units

Past exam question

• Mid-semester Sem 1, 2013

Dimensions & Units

Past exam question

• Sem 2, 2007

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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