DIMENSIONS AND UNITS

A lesson from:

IOWA STATE UNIVERSITY

Engineering Fundamentals and Multidisciplinary Design

LESSON CONTENTS:

- Mechanical System Description
- Definitions: Dimensions and Units
- Fundamental and Derived Dimensions
- Derived Dimension Example
- Dimensional System
- Newton's Law
- SI Metric System
- Classes of Units
- Base Units
- Supplementary Units
- Example Problem
- Numbers and Errors
- Significant Figures
- Rules for Significant Figures
- Rounding Numbers
- Example: Rounding Numbers
- Rules for Addition and Subtraction
- Example: Addition and Subtraction
- Rules for Multiplication and Division
- Example: Use of Significant Figures

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:

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 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.

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:

*Therefore, velocity is a derived dimension!*

**"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)*

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

- British System
- MKS Metric

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

- Base Units
- Supplementary Units
- Derived Units

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.

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

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

Rounding Numbers

Multiplication & Division Rules

Addition & Subtraction Rules

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).

Rounding Numbers

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.

Rules for Addition & Subtraction

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

Proper Use of Significant Figures