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![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-6-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-9-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-13-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-15-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-16-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-17-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-26-320.jpg)
![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](https://image.slidesharecdn.com/eng1040lec02-131029001456-phpapp02/85/ENG1040-Lec02-31-320.jpg)
Uploaded byEvonne Lau
ENG1040 Lec02
1) The document discusses dimensions and units in engineering dynamics, explaining that equations represent equality where the dimensions and units of both sides must match. 2) It covers dimensional analysis, defining common engineering quantities like force, pressure, work, and power in terms of mass, length, and time dimensions. 3) Rules for operating on dimensional quantities are presented, showing that derivatives and integrals retain the dimensions of the original terms.
More Related Content
ENG1040 Lec02
- 1. Faculty of Engineering ENG1040 EngineeringDynamics Dimensions and Units Dr Lau Ee Von – Sunway Lecture 2 ENG1040 – Engineering Dynamics
- 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 Operatingon 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 DimensionalQuantities 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 DimensionalQuantities 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 DimensionalQuantities 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 DimensionalQuantities 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 DimensionalQuantities 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 DimensionalQuantities 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
- 28. Dimensions & Units Equations & Equality Dimensional Analysis Operatingon Dimensional Quantities Example exam question 28
- 29. Past exam question Dimensions& Units • Sem 2, 2007 Equations & Equality Dimensional Analysis Operating on Dimensional Quantities Example exam question 29
- 30. Dimensions & Units Equations & Equality Dimensional Analysis Operatingon Dimensional Quantities Example exam question 30
- 31. Conclusion • To ensurean 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