Mechanics of Materials/Physical Units and Standards

2. Physical Units and Standards

Infants know when they want "more" sleep, more milk, more cleaning, and more carrying. They might not attach units to their wants -- hours for sleep, milliliters for milk, diapers per day for cleaning, calories/joules burned by those carrying them -- but they have inherent methods of communicating their needs. As we grow, we become aware of our height (feet, meters) and weight/mass (pounds, kilograms). Indeed, we begin to quantify what we see in our surroundings, form comparisons and attach units to these comparisons. Adults will think about the size of their homes (square feet), their salaries (dollars/year), and time spent in school (years). While comparison may be the thief of joy, we can acknowledge that physical units and standards are necessary when considering the currency of compensation.

We involuntarily think and make comparisons when we make purchases or get paid with money, when we specify how much food we will consume and measure quantities when we cook, when we travel/commute over distances, and when we subject ourselves to professional or personal scrutiny regarding our own health. Some of the same metrics/physical units that we use in our everyday life are relevant to mechanical structures and material properties. In fact, when we think of ourselves as mechanical machines, we become aware that our brains are somehow balancing and automatically keeping track of units, loads, mechanics, and materials without a pencil or calculator.

Fun Questions

1 Question: Which of the listed items below is a unit of mass? slug newton metre joule 2 Question: How much power does a person spend approximately "at rest?" 10 watts 100 watts 1 kW 10 kW

2.1 When Units Get Mixed Up

In the classic science fiction movie Back to the Future, the DeLorean requires 1.21 Jigowatts of power for its flux capacitor to enable time travel. While the pronunciation of "Jigowatt" might be appropriate, the correct spelling would be "gigawatt." The prefix "giga" representing 1 billion or 10⁹. In mechanical engineering, we often use giga when discussing elastic moduli (e.g., the elastic modulus of aluminum is approximately 70 gigapascals (GPa)). We will often use the prefix "mega" representing 1 million or 10⁶ when discussing the strength of materials (e.g., the strength of different metal alloys on the order of 100s of MPa). We can avoid mix-ups by keeping in mind that elastic moduli are often in terms of GPa, while units of strength and stress are often MPa.

Artist's rendition of the Mars Climate Orbiter.

A costly mix-up with units occurred in 1998 with the Mars Climate Orbiter. This orbiter was to monitor dust and water vapor, take daily pictures, and help plot an evolutionary map of climate change. Unfortunately, the orbiter did not make it into orbit, as there was a "metric mix-up." The ground software by Lockheed Martin sent calculated trajectories in imperial/US customary units (pound-seconds), but NASA was expecting the results in metric units (Newton-seconds). The thrusters provided an inappropriate amount of force, and the orbiter was lost in the Martian atmosphere. While these types of robotic missions are low cost, the cost for the spacecraft development and launch exceeded $250 million.

There is a reason why many textbooks in physics and engineering dedicate a section to units and physical standards. Units add meaning and help us conceptualize the quantity, size, and magnitude of the quantities of interest. While it is doubtful that one would lose credit for not including units in their scratchwork, keeping track of units prevents embarrassing and costly mistakes. As a courtesy to others, using an appropriate prefix also helps convey understanding of the appropriate measurements, just like the use of an appropriate number of significant figures.

2.2 Units as a Tool for Solving Problems

We have already emphasized how units help avoid costly mistakes when solving engineering problems. Keeping track of units or manipulating units can also help us solve problems.

Example Problem to Demonstrate How Units Might Help Us Solve a Problem

At this point, we have not discussed the concept of axial stress, but we might keep track of units to solve a problem involving the calculation of stress. For example, Jumbo the Elephant asks you to calculate the axial stress in MPa for a bar with an axial load of 10 kN axial load to a bar with a cross-section of 1 cm x 5 cm. Go ahead and attempt to solve this question before clicking on the "Answer" below.

Answer

We recall that the units associated with MPa are those of pressure meaning force/area. We can then think in terms of the metric system and the units of relevance to our problem: N and m. We can write N/m² and then perform the calculations shown in the animated GIF.

Animated GIF showing how to use units to calculate stress

We do not need to know that axial stress equals force divided by area. We can guess at the correct answer of 20 MPa by arranging the units appropriately.

Another Example Problem to Demonstrate How Units Might Help Us Solve a Problem

This question explores how to use units to solve a question relating mechanical, rotational power to electrical power.

Question Relating Rotational Power to Electrical Power and Current

You are a new engineer working at Company XYZ, and your supervisor asks you to calculate how much electrical current you will need to power a motor connected to 120 V outlet, which delivers 1 N m of torque at a speed of 100 rpm. 1.21 GW. 9 A. 900 mA. 90 mA

2.3 Sources of Physical Units

There are many sources of physical units, which describe material properties or methods for converting between units. Textbooks and websites will often share common material properties. A commercial website such as MatWeb will have both overviews for classes of materials (e.g., an overview of silicone materials) and data for specific grades of materials. The Wikipedia page on Young's modulus also lists approximate values for common materials. Table 2.1 shows some sources for material properties. A mechanical engineer should memorize the following: the elastic/Young's modulus of aluminum (70 GPa), the elastic modulus of steel (~3 X that of aluminum), the density of aluminum (2,700 kg/m³), the density of steel (7,900 kg/m³), and the dimensionless Poisson's ratio of many metal alloys (~0.3).

2.1 Table for Sources of Mechanical Material Properties

Type of Mechanical Property	Websites
Density	MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box
Elastic Modulus	MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box
Yield Strength	MatWeb, Wikipedia, Engineering Tool Box
Ultimate Tensile Strength	MatWeb, Wikipedia, Engineering Tool Box
Poisson's Ratio	MatWeb, Wikipedia, Wolfram Alpha, Engineering Tool Box

For conversion between units, we can use multiplication and fractions as previously demonstrated. Other online tools that keep track of units automatically when making calculations include Wolfram Alpha , Google Search, and DuckDuckGo. Common conversions in mechanical engineering include those in Table 2.2

2.2 Table for Common Conversions in Mechanics

Base Unit	Converted Unit
1 m	3.28 ft
1 ft	0.31 m
1 MPa	145 psi
1 ksi (1,000 psi)	6.89 MPa

2.4 Learning Problems

• 2.4 Learning Problems on Physical Units and Standards