Dimensions of Physics

Dimensions of Physics

The essence of physics is to measure the observable world and describe the principles that underlie everything in creation.

This usually involves mathematical formulas.

The Metric SystemThe Metric System• first established in France

and followed voluntarily in other countries

• renamed in 1960 as the SI (Système International d’Unités)

• seven fundamental units

DimensionDimension• can refer to the number of

spatial coordinates required to describe an object

• can refer to a kind of measurable physical quantity

DimensionDimension• the universe consists of

three fundamental dimensions:• space• time• matter

LengthLength• the meter is the metric

unit of length• definition of a meter: the

distance light travels in a vacuum in exactly 1/299,792,458 second.

TimeTime• defined as a nonphysical

continuum that orders the sequence of events and phenomena

• SI unit is the second

MassMass• a measure of the tendency

of matter to resist a change in motion

• mass has gravitational attraction

The Seven Fundamental SI Units

The Seven Fundamental SI Units

• length• time• mass• thermodynamic

temperature

• meter• second• kilogram• kelvin

The Seven Fundamental SI Units

The Seven Fundamental SI Units

• amount of substance

• electric current• luminous

intensity

• mole

• ampere• candela

SI Derived UnitsSI Derived Units• involve combinations of SI

units• examples include:

• area and volume• force (N = kg • m/s²)• work (J = N • m)

Conversion FactorsConversion Factors• any factor equal to 1 that

consists of a ratio of two units

• You can find many conversion factors in Appendix C of your textbook.

Unit AnalysisUnit AnalysisFirst, write the value that you already know.

18 m

Unit AnalysisUnit AnalysisNext, multiply by the conversion factor, which should be written as a fraction.

Note that the old unit goes in the denominator.

18 ×100 cm 1 m

m

Unit AnalysisUnit AnalysisThen cancel your units.

Remember that this method is called unit analysis.

18 ×100 cm 1 m

m

Unit AnalysisUnit AnalysisFinally, calculate the answer by multiplying and dividing.

= 1800 cm18 ×100 cm 1 m

m

Unit Analysis BridgeUnit Analysis Bridge

Convert 13400 m to km.

13400 m ×1 km

1000 m= 13.4 km

Sample Problem #1Sample Problem #1

How many seconds are in a week?

1 wk ×7 d

1 wk

= 604,800 s

×24 h1 d

×60 min

1 h

×60 s

1 min

Sample Problem #2Sample Problem #2

Convert 35 km to mi, if 1.6 km ≈ 1 mi.

35 km ×1 mi

1.6 km ≈ 21.9 mi

Sample Problem #3Sample Problem #3

Principles of MeasurementPrinciples of

Measurement

Instruments Instruments • tools used to measure• critical to modern scientific

research• man-made

• comparing the object being measured to the graduated scale of an instrument

AccuracyAccuracy

• dependent upon:• quality of original design

and construction• how well it is maintained

• reflects the skill of its operator

AccuracyAccuracy

• the simple difference of the observed and accepted values• may be positive or

negative

ErrorError

• absolute error—the absolute value of the difference

ErrorError

Percent ErrorPercent Error

observed – accepted accepted

× 100%

• a qualitative evaluation of how exactly a measurement can be made

• describes the exactness of a number or measured data

PrecisionPrecision

• some quantities can be known exactly• definitions• countable quantities

PrecisionPrecision

• irrational numbers• can be specified to any

degree of exactness desired

• potentially unlimited precision

PrecisionPrecision

When you use a mechanical metric instrument (one with scale subdivisions based on

tenths), measurements should be estimated to the nearest 1/10 of the smallest

decimal increment.

The last digit that has any significance in a

measurement is estimated.

Truth in Measurements

and Calculations

Truth in Measurements

and Calculations

Remember: The last (right-most) significant digit is the

estimated digit when recording measured data.

Significant DigitsSignificant Digits

Rule 1: SD’s apply only to measured data.

Significant DigitsSignificant Digits

Rule 2: All nonzero digits in measured data are

significant.

Significant DigitsSignificant Digits

Rule 3: All zeros between nonzero digits in measured

data are significant.

Significant DigitsSignificant Digits

Rule 4: For measured data containing a decimal point:

Significant DigitsSignificant Digits

• All zeros to the right of the last nonzero digit (trailing zeros) are significant

Rule 4: For measured data containing a decimal point:

Significant DigitsSignificant Digits

• All zeros to the left of the first nonzero digit (leading zeros) are not significant

Rule 5: For measured data lacking a decimal point:

Significant DigitsSignificant Digits

• No trailing zeros are significant

Scientific notation shows only significant digits in the

decimal part of the expression.

Significant DigitsSignificant Digits

A decimal point following the last zero indicates that

the zero in the ones place is significant.

Significant DigitsSignificant Digits

...and be careful when using your

calculator!

...and be careful when using your

calculator!

Rule 1: All units must be the same before you can add or

subtract.

Adding and SubtractingAdding and Subtracting

Rule 2: The precision cannot be greater than that

of the least precise data given.

Adding and SubtractingAdding and Subtracting

Rule 1: A product or quotient of measured data

cannot have more SDs than the measurement with the

fewest SDs.

Multiplying and DividingMultiplying and Dividing

Rule 2: The product or quotient of measured data and a pure number should

not have more or less precision than the original

measurement.

Multiplying and DividingMultiplying and Dividing

Rule 1: If the operations are all of the same kind,

complete them before rounding to the correct

significant digits.

Compound CalculationsCompound Calculations

Rule 2: If the solution to a problem requires a

combination of both addition/subtraction and multiplication/division...

Compound CalculationsCompound Calculations

(1) For intermediate calculations, underline the

estimated digit in the result and retain at least one extra digit beyond the estimated digit. Drop any remaining digits.

Compound CalculationsCompound Calculations

(2) Round the final calculation to the correct significant digits

according to the applicable math rules, taking into account the underlined estimated digits

in the intermediate answers.

Compound CalculationsCompound Calculations

What about angles and trigonometry?What about angles and trigonometry?

The SI uses radians.A radian is the plane angle that subtends a circular arc equal in length to the radius

of the circle.

Angles in the SIAngles in the SI

2π radians = 360°

Angles in the SIAngles in the SI

Angles measured with a protractor should be

reported to the nearest 0.1 degree.

Degrees to Radians:

ConversionsConversions

Multiply the number of degrees by π/180.

Radians to Degrees:

ConversionsConversions

Multiply the number of radians by 180/π.

Report angles resulting from trigonometric

calculations to the lowest precision of any angles given in the problem.

Angles in the SIAngles in the SI

Assume that trigonometric ratios for angles given are

pure numbers; SD restrictions do not apply.

Angles in the SIAngles in the SI

Problem SolvingProblem Solving

Problem SolvingProblem Solving• Read the exercise carefully!

• What information is given?

• What information is sought?

• Make a basic sketch

Problem SolvingProblem Solving• Determine the method of

solution• Substitute and solve• Check your answer for

reasonableness

Reasonable AnswersReasonable Answers• Does it have the expected

order of magnitude?• Make a mental estimate• Be sure to simplify units• Express results to the

correct number of SDs