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Length, time and massDimensional analysis

Lecture 02

General Physics (PHYS101)

www.cmt.ua.ac.be/golib/PHYS101

International System (SI) of UnitsBasic SI Units

• Length meter m• Time seconds s• Mass kilogram kg• These are the only units necessary to describe any

quantity.• [Volume] = m3 cubic meter• [Density] = kg/m3 kilogram per cubic meter• [Speed] = m/s meter per second• [Acceleration] = m/s2 meter per second squared

• [Force]: N (Newton) = kg m/s2

• [Frequency]: Hz (Hertz) = s-1

• [Pressure]: Pa (Pascal) = N/m2

• [Energy]: J (Joule) = N m• [Power]: W (Watt) = J/s

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Conversion of Units: Chain-link methodExample 1: Express 3 min in seconds?

1min = 60 s Conversion Factor?  

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Conversion of Units

1 km = 0.6 miles

Example 4: Express 200 km/h in m/s?

km/h --> m/s :3.6m/s --> km/h x3.6

1h = 60 min = 60 x 60 s = 3600 s

Example 3: Express 200 km/h in miles/s?

200 km/h= 200 x 0.6 miles/3600 s = 0.03 miles/s

1 km = 1000 m1 h = 3600 s

200 km/h= 200 x 1000 m/3600 s = 55.56 m/s

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Scientific notation Expanded form

1 x 100 1

1 x 101 10

1 x 102 100

1 x 103 1000

1 x 106 1 000 000

1 x 10-1 1/10 or 0.1

1 x 10-3 1/1000 or 0.001

1 x 10-6 0. 000 001

1.23 = 1.23 x 100

0.25 = 2.5 x 10-1

0.0007925 = 7.925 x 10-4

Scientific notationsThe following prefixes indicate

multiples of a unit.Multipli

erPrefix Symbol

1012 tera T109 giga G106 mega M103 kilo k10-3 milli m10-6 micro μ

10-9 nano n10-12 pico p10-15 femto f

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Speed of light: c=299 792 458 m/sc=2.99 792 458 x 108 m/s

•Underestimation: the following digits can just be dropped from the decimal place: 0, 1, 2, 3, an 4.

Rounding

Example 1. Round c to a nearest 1000th.

•Overestimation: digits 5 to 9 can be dropped from the decimal place during the rounding, however, one should be added to the digit in front of it.

c=2.998 x 108 m/s. Example 2. Round c to a nearest 10th. c=3.0 x 108 m/s. Example 3. Round 273.587 to a nearest integer. 274

Example 4. Round 273.587 to 2 significant figures. 270

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Uncertainties in measurements

• The accuracy of the measurements are determined by significant figures.

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Rules for Significant Figures

1. All nonzero figures are significant 359 87678 1245 987889

2. All zeros between nonzeros are significant 205 1003 508009 800009002

3. Zeros at the end are significant if there is a decimal point before them 4.200 1003.5600 30.003000 4. All other zeros are non-significant 30000 0.0000344

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Rules for Significant Figures

0 . 0 0 4 0 0 4 5 0 0

Not significantzero at the beginning

Not significantzero used only to locate the decimal point

Significantall zeros between nonzero numbers

Significantall nonzerosintegers

Significantzeros at the end ofa number to the rightof the decimal point

Just take care of zeros

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Operations with Significant Figures

•When adding or subtracting, round the results to the smallest number of decimal places of any term in the sum

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Operations with Significant Figures

•When multiplying or dividing, round the result to the same accuracy as the least accurate measurements (i.e. the smallest number of the significant figures)

Example: Calculate the surface area of a plate with dimensions 4.5 cm by 7.32 cm.

A=4.5 cm x 7.32 cm=32.94 cm2.A=33 cm2.

1.0:9= 0.11111…

= 0.11.0:9.0=

0.11111…= 0.11

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• Unit for mass is defined in terms of kilogram, based on a specific Pt-Ir cylinder kept at the International Bureau of Standards.

Why is it hidden under two glass domes?

• Another definition is based on the mass of carbon atom.

Definition of kilogramLength [L] mTime [T] sMass [M] kg

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• Defined in terms of the oscillation of radiation from a cesium atom (9 192 631 700 times frequency of light emitted).

• US time standard NIST-F1: accurate to 1 second in 80 million years.

Definition of second

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• Meter is the distance travelled by light in a vacuum during a given time (1/299 792 458 s).

• The speed of light is: c=299 792 458 m/s

Definition of meter

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Dimensional analysis: Equation analysis

• Complicated formulas can be checked for consistency by looking at the units (dimensions) to make sure that both sides of the equation match.

Example 1: Is the equation correct?

Example 2: Show that the expression dimensionally correct, where and represent velocities, is acceleration, and is a time interval.

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Dimensional analysis: Equation analysisExample 3: Is the equation

correct?𝑣=𝑠𝑡

[ [st𝐿𝑇 ≠ 𝐿𝑇 Equation is not correct!

Example 4: The volume of a sphere:

V

[V [RL3 Equation is not correct!

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Dimensional analysis: Predictions

Example. Use dimensional analysis to determine how the time of a falling apple t depends on the

height h.

h

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Summary• Scientific notations

• Rounding

• Order of magnitude: 10x

(x=1,2,3 ..)

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Summary

• Uncertainties in the measurements

• It is important to control the number of digits or significant figures in the measurements.

• Significant figures

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Summary • Dimensional analysis is a technique to check the

correctness of an equation

• Dimensional analysis can be useful in making predictions.

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