Dimensional Reasoning

1. Is either of these equations correct?

2. What is the common problem in the two

examples below?

Sign outside New Cuyama, CA 1998 Mars Polar Orbiter

2

2

d

vmF atd

2

1

DRILL

1. Is either of these equations correct?

atd2

1

F: kg*m / s2

m: kgd: m

v: m / sa: m / s2

2

2

d

vmF

kg*m / s2 = kg*m2 / s2

m2

= kg*m2

s2 m2

kg*m / s2 = kg / s2

2. What is the common problem in the two images below?

$125mil error: “Instead of passing about 150 km above the Martian atmosphere before entering orbit, the spacecraft actually passed about 60 km above the surface…This was far too close and the spacecraft burnt up due to friction with the atmosphere.” – BBC News

Pounds-force Newtons-force

UNITS!

Dimensional Reasoning

Lecture Outline:

1. Units – base and derived2. Units – quantitative considerations3. Dimensions and Dimensional Analysis

– fundamental rules and uses

Dimensional Reasoning Measurements consist of 2 properties:

1. a quality or dimension2. a quantity expressed in terms of “units”

Let’s look at #2 first:

THE INTERNATIONAL SI SYSTEM OF MEASUREMENT IS COMPRISED OF 7 FUNDAMENTAL (OR BASE) QUANTITIES.

THE ENGLISH SYSTEM, USED IN THE UNITED STATES, HAS SIMILARITIES AND THERE ARE CONVERSION FACTORS WHEN NECESSARY.

Dimensional Reasoning

2. a quantity expressed in terms of “units”:

THE INTERNATIONAL SI SYSTEM OF MEASUREMENT IS COMPRISED OF 7 FUNDAMENTAL (OR BASE) QUANTITIES.

BASE UNIT – A unit in a system of measurement that is defined, independent of other units, by means of a physical standard. Also known as fundamental unit.

DERIVED UNIT - A unit that is defined by simple combination of base units.

Units provide the scale to quantify measurements

SUMMARY OF THE 7 FUNDAMENTAL SI UNITS:

1. LENGTH - meter

2. MASS - kilogram

3. TIME - second

4. ELECTRIC CURRENT - ampere

5. THERMODYNAMIC TEMPERATURE - Kelvin

6. AMOUNT OF MATTER - mole

7. LUMINOUS INTENSITY - candela

Quality (Dimension) Quantity – Unit

LENGTH

YARDSTICK

METER STICK

Units provide the scale to quantify measurements

MASS

Units provide the scale to quantify measurements

TIME

ATOMIC CLOCK

Units provide the scale to quantify measurements

ELECTRIC CURRENT

Units provide the scale to quantify measurements

THERMODYNAMIC TEMPERATURE

Units provide the scale to quantify measurements

AMOUNT OF SUBSTANCE

Units provide the scale to quantify measurements

LUMINOUS INTENSITY

Units provide the scale to quantify measurements

Units 1. A scale is a measure that we use to characterize

some object/property of interest.Let’s characterize this plot of farmland:

x

y

The Egyptians would have used the length of their forearm (cubit) to measure the plot, and would say the plot of farmland is “x cubits wide by y cubits long.”

The cubit is the scale for the property length

Units

7 historical units of measurement as defined by Vitruvius

Written ~25 B.C.E.

Graphically depicted by Da Vinci’s Vitruvian Man

Units 2. Each measurement must carry some unit of

measurement (unless it is a dimensionless quantity).

Numbers without units are meaningless.

I am “72 tall”

72 what? Fingers, handbreadths, inches, centimeters??

Units

3. Units can be algebraically manipulated; also, conversion between units is accommodated.

Factor-Label Method

Convert 16 miles per hour to kilometers per second:

Units

4. Arithmetic manipulations between terms can take place only with identical units.

3in + 2in = 5in3m + 2m = 5m3m + 2in = ?

(use factor-label method)

Dimensions are intrinsic to the variables themselves

“2nd great unification of physics” for electromagnetism work (1st was Newton)

Der

ived

Bas

e

Characteristic DimensionSI

(MKS) English

Length L m foot

Mass M kg slug

Time T s s

Area L2 m2 ft2

Volume L3 L gal

Velocity LT-1 m/s ft/s

Acceleration LT-2 m/s2 ft/s2

Force MLT-2 N lb

Energy/Work ML2T-2 J ft-lb

Power ML2T-3 W ft-lb/s or hp

Pressure ML-1T-2 Pa psi

Viscosity ML-1T-1 Pa*s lb*slug/ft

Dimensional Analysis

Fundamental Rules:1. Dimensions can be algebraically manipulated.

Dimensional Analysis

Fundamental Rules:2. All terms in an equation must reduce to identical

primitive (base) dimensions.

221 attvdd oo

22T

T

LT

T

LLL

Dimensional Homogeneity

Homogeneous Equation

Dimensional Analysis

Opening Exercise #2:

Dimensional Non-homogeneity

Non-homogeneous Equation

Dimensional Analysis

Uses:1. Check consistency of equations:

atd2

1

221 attvdd oo

Dimensional Analysis Uses:2. Deduce expressions for physical phenomena.

Example: What is the period of oscillation for a pendulum?

We predict that the period T will be a function of m, L, and g:

(time to complete full cycle)

power-law expression

Dimensional Analysis 1.

2.

3.

4.

5.

6.

Dimensional Analysis

6.

7.

8.

9.

Dimensional Analysis Uses: 2. Deduce expressions for physical phenomena.What we’ve done is deduced an expression for period T.

1) What does it mean that there is no m in the final function?

2) How can we find the constant C?

The period of oscillation is not dependent upon mass m – does this make sense?

Further analysis of problem or experimentally

Dimensional Analysis Uses:2. Deduce expressions for physical phenomena.

Chalkboard Example:

A mercury manometer is used to measure the pressure in a vessel as shown in the figure below. Write an expression that solves for the difference in pressure between the fluid and the atmosphere.

Assignment

DUE WEDNESDAY:

1. Problem Set 2 and Dimensional Reasoning problems (#1-3)

2. Engineering Paper

3. Read H8 Pendulum Lab on www.bpi.edu