DIMENSIONS AND UNITS

A lesson from:

IOWA STATE UNIVERSITY

Engineering Fundamentals and Multidisciplinary Design


LESSON CONTENTS:

  1. Mechanical System Description
  2. Definitions: Dimensions and Units
  3. Fundamental and Derived Dimensions
  4. Derived Dimension Example
  5. Dimensional System
  6. Newton's Law
  7. SI Metric System
  8. Classes of Units
  9. Base Units
  10. Supplementary Units
  11. Example Problem
  12. Numbers and Errors
  13. Significant Figures
  14. Rules for Significant Figures
  15. Rounding Numbers
  16. Example: Rounding Numbers
  17. Rules for Addition and Subtraction
  18. Example: Addition and Subtraction
  19. Rules for Multiplication and Division
  20. Example: Use of Significant Figures

Credits for Lecture and Authoring

Lecture written by: Dr. James Hilliard

Authored for Presentation by: Mark Sobek and Dana Steffey

Revised by: Lex Jacobson

HTML Documentation by: Lex Jacobson

Last Updated: 8/8/95

Consider the following mechanical system:

System Description:


--In order to describe the above system, one needs to define a set of physical parameters.
 Mass             Velocity 

 Time             Momentum

 Length           Energy

 Etc.

Dimensions-- The set of physical parameters used to define the state of a system

Units-- A standard quantity that allows one to establish a value for a dimension.

Fundamental and Derived Dimensions:

Fundamental Dimension--A dimension that cannot be derived from other dimensions.

Derived Dimension--A dimension that can be defined by some mathematical combination of fundamental dimensions.

Example:

Let length(L) and time(T) be fundamental dimensions.
Let velocity(V) be another dimension used to describe a system state.
A relationship exists as follows:

V = L / T

Therefore, velocity is a derived dimension!

Dimensional System--The smallest set of fundamental dimensions that can be used for a complete study of a field of science.

Examples: (Systems used in Mechanics)

"Absolute System"
-Fundamental dimensions are length, mass, time.

"Gravitational System"
-Fundamental dimensions are length, force, time.

Note: Force and Mass are related by Newton's Law: F=(M)(A)

Where:

  • F = Force (N)
  • M = Mass (kg)
  • A = Acceleration (m/s^2)

The SI Metric System

During the past several years, many types of unit systems have been in use. Generally, they were variations of two types:

  1. British System

  2. MKS Metric

The new common system being adopted is the SI Metric System!

The SI system contains 3 classes of units:

  1. Base Units

  2. Supplementary Units

  3. Derived Units

Base Units:

Supplementary Units:

Example Problem:

The density of water at 50 degrees F is about 1.94 slug/ft^3. Determine the volume of a 50.0 ft. diameter cylindrical tank with a height of 35.0 ft. Then, calculate the mass of water contained if the tank is full. Express volume in cubic meters and mass in kilograms.

Numbers and Errors

Types of Errors:

Systematic (Bias)

Accidental (Statistical)

Note:
Precision, which is a measure of error, is properly conveyed numerically by using appropriate significant digits.

Significant Figures

Significant figures are figures (digits) used in writing a number, except those used to define only the decimal location such as:

2000, 200, 125000, ETC.

Or those zeros which do not have non-zero digits on their left,

0.0015, 0.01063, ETC.

Rules for Handling Significant Figures

Rounding Numbers

Multiplication & Division Rules

Addition & Subtraction Rules

Rounding Numbers

If the first digit discarded is <5, leave the last digit kept unchanged.

If the first digit discarded is > 5 or is = 5 followed by other non-zero digits, increase the last digit kept by 1.

If the first digit discarded is = 5 followed by all zeros, then increase the last digit kept by one if it is odd (not even).

Example
Rounding Numbers

Rules for Addition & Subtraction

Find the least precise number.

Round all other numbers to one place farther to the right.

Add (Subtract) the numbers.

Round the answer such that the last digit is no farther right than that of the least precise number.

Example
Rules for Addition & Subtraction

Rules for Multiplication & Division

The product or quotient shall contain no more significant digits than are contained in the number with the fewest significant digits initially.

Example
Proper Use of Significant Figures