Units of Measure

Samuel Clemens obtained his pseudonym, Mark Twain, from a Mississippi River phrase. A weighted rope would be lowered to measure the depth of the river. The rope was marked in fathoms, and the marks would be called out. Mark Twain meant you had passed the second mark, or were two fathoms deep.

How deep was the river at “Mark Twain?”

Fundamental Units•Length•Mass•Time•Electric Current•Temperature•Amount of substance•Luminous intensity

Supplemental Units•Plane Angle•Solid angle

Derived Units

•Velocity•Acceleration•Time•Frequency•Pressure•Force•Moment (torque)•Energy•Mass Density

Units: LengthWhat is the length of this line?

Units we use are arbitrary. Key is that we agree on a unit so that we all know what we are talking about.

• Based on things that made sense to people• previously known as English (or British)• 1 inch = 3 dry, round, barleycorns end-to-end• foot = length of King Edward I’s foot• mile = 1000 double paces of Roman soldier • 12 in/ft; 4 in/hand; 3 ft/yd; 5280 ft/mile

US Customary System (USC)

Systeme Internationale (SI)

• Commonly called metric system, although different• attempted to be less arbitrary• 1 meter

• original: one ten-millionth of the distance from the equator to either pole

• current: based on wavelength of light

Conversion between systems:Exact: 25.4 mm / 1 inchApproximate: 1 mile / 1.6 kilometers

Prefixes

10-1 Deci- 101 Deca-

10-2 Centi- 102 Hecto-

10-3 Milli- 103 Kilo-

10-6 Micro- 106 Mega-

10-9 Nano- 109 Giga-

10-12 Pico- 1012 Tera-

Only prefixes with powers of three are officially part of SI system. We will use centimeter, as it is the same order of magnitude as an inch.

What is 101 cards? How about a 10-6 scope? Or a 10-2 pede?

Example: Length conversion

How many yards are there in 10 km?

How many fathoms in a furlong?

Units: Area

• SI• square meter (m2)• hectare (ha) = 104 m2

• USC• square inch, square foot• acre: land a team of oxen could plow in a day• acre is 40 x 4 rods, or 43560 ft2

engage

Area often described as a length-squared, but applies to any shaped area. What is the area of the teal letters in the engage logo?

Example: Area conversion

What is the area (in in2) of a 1m x 2m area?

How many 1 inch x 1 inch squares fit into a 1 m x 2 m area (get area in terms of in2)?

Units: Volume

• SI• Liter; 1 L = (10 cm)3 = 1000 cm3

• USC• Gallon; 1 gallon = 231 in3

Volume often described as a length-cubed, but applies to any shaped volume. We are really talking about the number of unit cubes that fit into something. What is the volume of the potato?

Example: Volume conversion

What is the volume of a 2m x 2m x 1m box in ft3?

What is the volume of a 0.2m x 0.2m x 0.1m box in ft3?

Units: Angles

Most common unit: degree Why are there 360 degrees in a circle?

– convenience– divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18

Degrees are sub-divided into 60 minutes– minute subdivided into 60 seconds.– Notation 43°1837 = 43 degrees, 18 minutes, 37 seconds

Units: Angles – Engineering

sr

θr

s

The unit of radians often used to measure angles– draw angle at the center of a circle– angle defines an arc on the circle’s circumference– ratio of arc length to circle radius is angle in radians

» dimensionless

How many radians in 30°?

Measurements

Measure width of block using three different instruments.

Instrument Width

Child’s ruler

Metal ruler

Calipers

How do we communicate the difference between the different measurements?

SIGNIFICANT DIGITSWhich measurement is better?

Child’s ruler

Metal ruler

Caliper

Accuracy and Precision

Accuracy – measure of the nearness of a value to the correct value

Precision – repeatability of the measurement

EstimationWhat is the volume of Estabrook 111 in cubic meters?•30 second – use it to get the right order of magnitude•5 minute – use quick measurements•30 minute – use more accurate measurements, may need a more detailed problem definition

•Use units that are readily available•Don’t worry about details – this is an estimate•Include an estimate of the accuracy of your estimate

Height Estimation•Use ratios•Arm-length / ruler•Shadows

Significant Digits

Significant digits: communicate level of uncertainty.

If we say the length of a line is 12.7”, what does that mean?

Length between 12.65” and 12.75”

How is it different if we say the length of a line is 12.70”?

Length between 12.695” and 12.705”

The numbers you put down communicate the level of precision, or how sure you are of the number.

• Most engineering data are assumed to have 3 significant digits

• Unless told otherwise, give answers to 3 significant digits

• Carry 4 to 5 significant digits in calculations• Be realistic about the number of significant digits you

report– How precisely do you know the reported value?– Full credit on exams requires appropriate use of

significant figures

Significant Digits: Summary

Reasonableness

• Think about what your answer means• Take five seconds to think about every

answer• Use estimation and simplification of the

problem to get an order-of-magnitude estimate

EF 101 Final Example

A passenger jet is taking off from Hartsfield International airport in Atlanta. Assuming it starts from rest, and accelerates at a constant rate of 2.6 m/s2, how long does it take to cover the 3.8 km runway?

A) 1.5 s B) 38 s C) 54 sD) 99 s E) 140 s F) 1460 s

Which two answers are unreasonable?

Problem Solving Define the problem

– Identify the critical data of the problem. Do not be misled by data that is extraneous, erroneous, or insignificant.

Diagram– A diagram or schematic of the system being analyzed is often very

helpful, and may be required. Governing equations

– Determine what type of problem is being solved. Recognize when certain equations apply and when they do not apply. The governing equations should be written out in symbolic form before substituting in numerical quantities.

Calculations– Carry out your calculations only after you have completed the first

three steps. Check to make sure units are consistent. Solution check

– Make sure you solved the problem that was posed. If possible, use an independent method or equation to check your result. Check to see that your solution is physically reasonable. Make sure both the magnitude and sign of the answer makes sense.