Dimensional Consistencyand Unit Conversions

Recall

Dimensions Units System of units Base units Multiple units Derived units

Dimensionless Quantities

Easier, in that they do not have units at all.

Some ways they are more complicated

Ratios…. Has to carry its title with it not unit

Angles….. an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Degree/radius

Numbers… specify what you counting

Example?

Dimensions and Units……Limitations

Customary units having wrong dimensions Mass-energy

Mass of electron=0.51MeV

Pressure… dimensions?? Wt about Blood pressure?? Be Aware what units are being used

Conventional choicesDepends upon convenience and custom.

Scientists speed…m/s, biologist studying snail motion???

Chemists Concentration…..mol/ litre,,, medical labs???

Considerations

Thermal quantities Electrical quantities Mechanical quantitiesMoralAlways find out what units are

appropriate for the task at hand and express your results accordingly.

Consistency of Units

Dimensions and units must be handled consistently.

Numerical values of two quantities may be added or subtracted only if the units are the same

Numerical values and their corresponding units may always be combined by multiplication or divisions.

Dimensional Consistency Equations involving physical quantities

must have the same dimensions on both sides, and the dimensions must be the correct ones for the quantity calculated. Dimensional Consistency/Homogenity

Consequently, the units on both sides should be the same, and must be at least equivalent and correct.

Dimensional Consistency…checking the Units

Powerful technique for uncovering errors in calculations.

Dimensions or units may be considered algebraic quantities

Some examplesDensityChecking dimensions for the formulase.g;

Example

A radar gun is used to obtain the speed of a car as it accelerates from a stop sign. A graph of speed (y-axis) vs time (x-axis) is a straight line, so the student computes a slope expecting to find the constant acceleration. How he can verify the dimensional consistency.

Contd…

Good practice to make units similar Good practice to show all units

througout a problem to test equation validity.

Identify..

Conversion of Units

Often necessary to change units in order to combine measurements made withdifferent instruments

Conversion of Units procedure is very simple if the units are again handled as algebraic quantities. The equivalence between two expressions of the same

quantity may be defined in terms of a ratio: Ratios of the form of Equations are known as

conversion factors.

To convert a quantity expressed in terms of one unit to its equivalent in terms of another unit, multiply the given quantity by the conversion factor (new unit/old unit).

Dimension Equation. One quantity is multiplied by a number of ratios Called Conversion factors of equivalent values of combinations of dimensions/units.

The numerical value depends on the units chosen. meters to millimeters nanoseconds to seconds square centimeters to square meters

Dimension Equation

Conversions within units SI,CGS and AES system Difference? Factors for converting from one system of units to

another may be determined by taking ratios of quantities AES difficulties

the occurrence of conversion factors (such as 1 ftl12 in), which, unlike those in the metric systems, are not multiples of 10;

the unit of force.

Basic Conversion Factors

•Express a speed of 50 kilometers per hour as meters per second

•Convert a concentration of 220 mg/dl to grams/liter

•Convert 3 weeks to milliseconds

•Calculate the weight in N of a 25 kg object

•A student making artificial sea water dissolves 13.1 gm of NaCl in 450 ml

of distilled water, and calculates the resulting concentration as 0.0291

gm/ml. A standard handbook claims that seawater has 29.54 gm/l of NaCl.

Comparing units, the student recalculates the concentration as 13.1 gm/

0.45 l = 29.1 gm/l, and notes that the units are now the same and the

magnitude is sufficiently close

Examples

• Convert 1 cm/s2 to it equivalent in km/yr2.

• Convert 921 kg/m3 to lbm/ft3

• The Gas Constant R= 8.314 m3-Pa/mol K. What is the value of

R in lit-bar/mol K and cal/mol-K.?

• A force of 355 poundals is exerted on a 25.0-slug object. At

what rate (m/s2) does the object accelerate?

Force and Weight

Force According to Newton's second law of motion, force is

proportional to the product of mass and acceleration (length/time2 ).

Natural force units are, therefore, kg'm/s2 (SI), g' cm/s2 (CGS), and Ibm 'ft/s2 (American engineering). To avoid having to carry around these complex units in all

calculations involving forces, derived force units have been defined in each system.

In the metric systems, the derived force units (the newton in SI, the dyne in the CGS system)

Are defined to equal the natural units:

System Conversion Units

Pound-force a pound-force (lbf)-is defined as the product of a unit mass (1 Ibm) and the

acceleration of gravity at sea level and 45° latitude, which is 32.174 ft/s2:

The symbol gc is sometimes used to denote the conversion factor from natural to derived force units:

Mass The weight of an object is the force exerted on the object

by gravitational attraction. The gravitational acceleration (g) varies directly with the

mass of the attracting body and inversely with the square of the distance between the centers of mass of the attracting body and the object being attracted.

The value of g at sea level and 45'" latitude is given below in each system of units:

The acceleration of gravity does not vary much with position on the earth's surface and (within moderate limits) altitude,