2. Lecture Outline
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
•
•
•
•
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
3. Equations
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Equations denote equality:
When A = B , it implies that :
1. The numerical values are the same,
2. The quantities are of the same type,
Example exam
question
3. The dimensions (and thus the units) are
the same.
There is Dimensional Homogeneity!
3
4. Dimensions vs Units
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
We shall normally use the dimensions of
mass,M, length L, and time,T.
The usual corresponding units
are kg, m, and s.
i.e., L/T m/s units of speed,
velocity
4
5. Equations & Equality
Dimensions &
Units
Equations denote Equality !
Equations &
Equality
The Dimensions of the LHS
Dimensional
Analysis
must be the same
Operating on
Dimensional
Quantities
as the Dimensions of the RHS
Example exam
question
(i.e. Equation is dimensionally homogeneous)
+
=
5
6. Equations & Equality
Dimensions &
Units
A familiar example:
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
s ut
1 2
at
2
The dimensions of the LHS are
denoted:
LHS: [ s ] = length = L
6
7. Equations & Equality
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
A familiar example:
What about the dimensions on the RHS?
s ut
1 2
at
2
Both terms on the RHS must have the same
dimensions if they are to be added in any
meaningful way.
7
8. Equations & Equality
Dimensions &
Units
A familiar example:
Equations &
Equality
Dimensional
Analysis
s ut
Operating on
Dimensional
Quantities
Example exam
question
ut
1 2
at
2
u t
a t
2
1 2
at
2
L
T
T
L 2
T
2
T
L
L
8
9. Equations & Equality
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
A familiar example:
s ut
1 2
at
2
The dimensions of both sides are: length, [L]
Example exam
question
The equation is dimensionally homogeneous
9
10. Equations & Equality
Dimensions &
Units
A different arrangement:
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
How do we analyse if the equation is dimensionally
homogeneous?
Example exam
question
Expand the equation, and analyse every term,
just as before!
s
ut
1 2
at
2
10
11. Dimensional Analysis
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
A preliminary requirement in dimensional
analysis…
…is the need to establish the units of the
various quantities in the equations.
Example exam
question
Some examples...
11
12. Dimensional Analysis - Angles
Dimensions &
Units
arc
Equations &
Equality
The usual unit for
an angle is the
radian
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
radius
The dimensions are:
arc
radius
L
L
dimensionless
12
13. Dimensional Analysis - Angles
Dimensions &
Units
Frequency, and angular speed may be
measured in units of…
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
radian s-1 (rad/s)
revolutions s -1 (Hz)
revolutions min -1 (RPM)
We use the S.I. unit to find its dimensions,
i.e. rad/s [T]-1
13
14. Dimensional Analysis - Forces
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
We can also general equations to find the
dimensions of a quantity.
For example, to find the dimensions of
Force (S.I. unit = Newton)
Operating on
Dimensional
Quantities
Example exam
question
Force has dimensions of :
M L T -2
14
15. Dimensional Analysis - Pressure
Dimensions &
Units
Pressure has units of Force per unit area:
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
The dimensions are:
[P] = [Force/Area] = MLT-2/L2
= ML-1T-2
This unit is called a Pascal
15
16. Dimensional Analysis - Work
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
Work has units of
Force times distance,
The dimensions are thus
[W] = [Force*Distance] = (M LT-2) L
= M L2 T-2
This unit is called a Joule – i.e. a Nm.
16
17. Dimensional Analysis - Power
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
Power has units of:
Force times velocity.
The dimensions are thus
[P] = [Force*Velocity] = (M LT-2) (LT-1)
= M L2 T-3
This unit is called a Watt – i.e. Nm/s.
17
18. Dimensional Analysis – Other Units
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
Other quantities widely used in
engineering:
Torque, Bending Moment, Shear Modulus,
Momentum, Stress, Strain, Stiffness,
Damping Coefficient, Moment of Inertia,
Dynamic Viscosity, Impulse, etc.
Practice Classes will provide an opportunity
to become familiar with many of these…
18
19. Derivatives of Dimensional Quantities
Dimensions &
Units
y
Equations &
Equality
x
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
x
The units of
dy
dx
Lt
dy
are those of
dx
These are the same as :
X
y
x
0
y
x
y
x
19
20. Derivatives of Dimensional Quantities
Equations &
Equality
Dimensional
Analysis
Force
Dimensions &
Units
Operating on
Dimensional
Quantities
The
dimensions of
dF/dT would
be?
Time
Example exam
question
dF
dT
F
T
MLT
T
2
MLT
3
20
21. Derivatives of Dimensional Quantities
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
dy
dx
dy
dx
Operating on
Dimensional
Quantities
Example exam
question
x
o
x
2
d y
2
dx
Lt
x
0
dy
dx
x
21
22. Derivatives of Dimensional Quantities
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
dy
dx
dy
dx
Operating on
Dimensional
Quantities
Example exam
question
x
o
x
2
d y
2
dx
Lt
x
0
dy
y
x
dx
x
22
23. Derivatives of Dimensional Quantities
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
dy
dx
dy
dx
Operating on
Dimensional
Quantities
Example exam
question
x
o
x
2
d y
2
dx
Lt
x
0
dy
y
x
dx
x
Dimensions
are:
y
x
x
23
24. Derivatives of Dimensional Quantities
Dimensions &
Units
Equations &
Equality
Dimensional
Analysis
dy
Remember by looking at
dx
the quantities being
dy
dx
Operating on
Dimensional
Quantities
Example exam
question
differentiated
x
o
x
2
d y
2
dx
Lt
x
0
dy
y
x
dx
x
Dimensions
are:
y
2
x
24
25. Dimensions of Integrals
Dimensions &
Units
Integrals are
areas
Equations &
Equality
Dimensional
Analysis
A
y
ydx
Operating on
Dimensional
Quantities
Example exam
question
x
Thus the dimensions of an integral are:
y x
25
26. Dimensions of Integrals
Dimensions &
Units
E.g., Impulse:
t2
Impulse
Equations &
Equality
Dimensional
Analysis
Fdt
t1
Force
Operating on
Dimensional
Quantities
Example exam
question
t1
Time
t2
ML
T
Units of Impulse = [F][t] =
T
2
= MLT-1
26
27. Past exam question
Dimensions &
Units
• Mid-semester Sem 1, 2013
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
27
29. Past exam question
Dimensions &
Units
• Sem 2, 2007
Equations &
Equality
Dimensional
Analysis
Operating on
Dimensional
Quantities
Example exam
question
29
31. 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].
31
