Dimensional analysis - umb.edu · Accuracy & Significant figures Scientific notation 3 105 000 m =...

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Dimensional analysis Dimensional analysis [ ] [ ] [ ][ ] 2 2 2 2 2 2 1 T L M v m v m mv E = = = = Dimension of a velocity or speed v is written [v]: [] [ ] [ ] T L t s v = Δ Δ = Dimensions of length and time: L and T Dimension of energy E is written [E]: Dimension of acceleration a is written [a]: [] [ ] [ ] 2 T L t v a = Δ Δ = Check a formula using dimensional analysis = 2 2 1 gh v [] [ ][ ] 2 3 2 T L h g v = = No = gh v [] [ ][ ] T L T L h g v = = = 2 2 Yes

Transcript of Dimensional analysis - umb.edu · Accuracy & Significant figures Scientific notation 3 105 000 m =...

  • Dimensional analysisDimensional analysis

    [ ] [ ] [ ][ ] 22222

    21

    T

    LMvmvmmvE ==⎥⎦⎤

    ⎢⎣⎡=⎥⎦

    ⎤⎢⎣⎡=

    Dimension of a velocity or speed v is written [v]:

    [ ] [ ][ ] TL

    tsv =

    ΔΔ

    =

    Dimensions of length and time: L and T

    Dimension of energy E is written [E]:

    Dimension of acceleration a is written [a]:

    [ ] [ ][ ] 2TL

    tva =

    ΔΔ

    =

    Check a formula using dimensional analysis

    ⇒= 221 ghv [ ] [ ][ ] 2

    32

    TLhgv == No

    ⇒= ghv [ ] [ ][ ]TL

    TLhgv === 2

    2Yes

  • UnitsUnits

    Unit for speed or velocity vK

    K

    ;;;.;min;;;;;

    yeardayhscmkmmmi

    tforunitsforunit

    =ΔΔ

    =

    Units for lengthUnits for length

    kmkm 101033 mm

    cmcm 1010--22 mm

    mmmm 1010--33 mm

    μμmm 1010--66 mm

    nmnm 1010--99 mm nmnm22μμmm22

    1010--66 mm22mmmm221010--44 mm22cmcm22101066 mm22kmkm22

    Units for areaUnits for area

    nmnm33μμmm33

    1010--99 mm33mmmm331010--66 mm33cmcm33101099 mm33kmkm33

    Units for volumeUnits for volume

    yearyearweekweek

    86400s86400sdayday3600 s3600 shh60 s60 sminmin

    Units for timeUnits for time

    Units for mass: lb; kg; g; …

    Change of units (Appendix C) 9.46x109.46x101515 mm1 light1 light--yearyear

    0.454 kg0.454 kg1 lb1 lb2.54 cm2.54 cm1 in1 in0.3048 m0.3048 m1 ft1 ft1609 m1609 m1 mi1 mi

  • Accuracy & Significant figuresAccuracy & Significant figures

    Scientific notation 3 105 000 m = 3.105 x 106 m3 000 s = 3 x 103 s

    sm

    smv 3363

    610035.110

    3105.3

    10310105.3

    ×=×=××

    = −

    Accuracy: 3.1416 is more accurate than 1.14

    Related to number of decimal digits

    Addition & subtraction: keep same accuracy as least accurate term3.1416 – 1.14 = 2.0016 = 2.00

    Significant figures: 6 370 000 three 6 370 001 seven

    Multiplication and division: keep smallest number of s.f. except for constants,counting from the left

    Earth’s circumference = 2π RE = 2 x 3.1416 x 6.37 x 106 = 40.02398 x 106= 4. 002398 x 107 = 4.00 x 107 m

  • ("I

    PI-

    w = b t + =

  • 6 1 4 2.1 bnl R 2 - z 0 . 6 2 w& 3-4 kc,

  • ADP18D.tmpDimensional analysisUnitsAccuracy & Significant figures