UNITS t

11-t., Ph~sics Cho.pteY l po.."'t 2.

DIMENSIONS
':JC>U.tu..loe .C.OM /c. va.s h u.p a Cra.sh¥..
MEASUREMENT OF LENGTH
Metnods
1
~
'Direct Indirect
we c.a..n u.se. ll..Y'I ·,nstrum~.'nt we ~tiY'()a..te. the d."1sio.nce.s
.f-rom oru_
e-5. .sc.o.\e. 1
exld to the.. othe.r
ve:rn ie.Y co.\\; peYs
d
u.~ ·, V\ d,; ff'e. t e Y\i. te cl,"' iy1..t.S .
e-~ • P°'-'to.l10v-. Me.tltlod. .
Lea.~t Cou.Y\t: (Le) ·'
the 5h"a.\les.t voJ u.e. Of o.. pn~.sica.l q,Wl.nt"itd
-t.l,,o.t CAn be
h'>~UYe.d. a.c.c.~Ya.te\'j wit~ o...v, in.st-:ru.vne. l1t is CA\le.d t~
\ea.st . COLLY\t o-t tV\a.t in.stYtAvY1e.\'\L.

~ @ 1~1 I , , , , \ , , , , 11 f~
' '
1 , , §
~
@ 1003
1.e.o.~t co u. r\t J' , tlea.st cou.nt ~ 0.1 c.m
St"tlf\ ~'ff\ meo..SUY-e."{Y)eY)ts, leo.st c.ou."'1: ~ mu.lf1ple•~tJ

f. ._~ . ~ ooJ. , tl
t~~z;t{{i'
l
\
~, ,.a
2.. 8t,
I ld~MI 'l := ~x
Ii? .J..eru;t c.o u.r.-l
0,101' ~ o.e cm
.l.o.m b",

L.c. of Vel"Yl 1e~ co.\lipe:ts

0

== 0.1 mn'\ = ,o-'1m

..I:- -5
L.c. 01 5c.."'C'ew 30.u.3e -::=. 0.01mm == lO h')
 INDIRECT METHODS OF M£ASlJQEMENT OF
LENGTH
REFLECTlON METHOD
~ L ~lt1t ""Zff .soundJ
ti--zraf tl"wo-.ve.s cFiT
use il1l. .aq;~ ~ I
-fX 31~1~ A 3fu- B ~ Jh:r ctft- ~ 'cl' 1-, ~ w ~ve
a-,ra- ti \ 2a.' J.i.sto:nc.e -t-ro.. vel d,~JP, I

clisto.."' c e == spd ,,. t i me_

5peed. -r... t ime.

~~ 4c'I
Ch2.~I i 1

E>CA.niple :
Sou."l\d i:.£Ake.s 8 sec.oY\d.S to ,e.o..c.\.-\ ~ou.. o..f te:r !jOu...
.5ee o... -½Jlti-l~d iY\ o. stoYYl'I . Ho w fo.:, is ilv... 5>t o YY>1?
Hint~ Jj3~t 't'eO.cN...S, tJcru. a...lr,10.CT ihstD..ll,tl-J 6vd ..soUly,d t~o.ve.l~
ol- ~ 34D 'y"()ls . ~
cFm ~ ~c:fle.c..t10Y)
&IT Ll.&e ~?
*~-- cl ➔7
•

Q2: Ti'r'1e to..ke ~ ~ soY)o.:, ro..da:t

a. 15 2-o ..secoY>ds. f";Yld
t ~ d islo..V'c.e of -lhe o\ojec...t .
Sona.-t o°'-d.a.-c me. sou-Ytd wo.ves ref lec..t ~f l

a youtu be . COM/ <:.Yo.sh LI p

1}( 03 We .sloY\e 1n o-. we. ll o..nd. -Sou.."' J.
d.:rop a..
re.o..c.V\5 u.s o...f t.-e.y 29. Ho w d.e.e.p is the w ell ?
h.il'll: SloY1e. w~l1 30 / ,ow:
-i½-fo.JI l\'lj f-,ee I~ .
cl- ji:+ f3t, fLL=oJ j
.SoUJY)d will come bo..c:R a.± .spe.d 1Sotmd. l
de.pt'-" -i.; 6l\-me lA.

~ cl= ½ow-J. )( t2.

totA.\ time lClkeY\ ( ciowY) -t- u.p ')
==- t ;- ti. ,,. 2.s
1

PARALLAX METHOD
Lrnf e UH~ ~ , rE"
AB(o.1:rved) ~ AB 5t-ro.i~ht Ii hl.

e = AB Cc.u:tved )
]) CRo.di tu.) J
d.
ft --. - - - -

rct= loo..si.s l
fr 9 lf'o.d.i o..)')5 * ~ (}2:5
-zJI

2TT ¥ad.io..Y1 == 36a° icol'>'lple.te O..Yl~le i"' o. circle]

=b 1. 1CAclio..Y\ ~ 5 t".3 ° j
=t> 1 Je~-ree_ rv l . r'-i 5 X 10-.1.. Ya.A.
- '1 ..,. ..1
1 o.:tc m·,n rv 2-ll \ X IO YUL.l
 3fm e ~ ~ ~ fo-~ 1; cW'"" ~
0
Tl'ijonoMe.h'J
" \.
Use cfi:t~t

Go..se. 1
Co..se 2
t(k"' e = I)
cl
r
A' li tl ➔9
1
to.m 8, = d.,/H
t().YI e = Pe¥pe.Y\diw.l0-.:r
fr
,,,,
bo...se e2. = d.
f::: "'"' 2. / H
d., +d 2 = cl
~t bo.s.e.. f' 3 e9,,w:d'io'ns . 3 va."! ·1().\.ole s...
~~- ~ We m_~a.SU.Ye d_., f9, ! ti2
f3 ~ ~ 1"1~ wcJa..= pe.-rpe.)1diu.J,.:,
we caJ cu.hte_ H
e Ll.\1:r qo ~ ~ ~
0
:::. ~

· Q: i F 8 1 = 30°, 82. = 60° o..nd. d...= Soo rn,, Find H

Q: iF po-.--roJ\o..c.Jic. £'.A.Y\~\e is 2-5°/ cl.i.sio..Y)c.e. cf ba..sis 1s
Soo k.W)., FiY'J iV\e ci'1.sta:nce -~f ih.a.. object.

TR1uoN0MerRY MFTHoD
eft G, B2.. 1 it½= d. meo.su.Y-e.
~ACD to..Y) 8 1= h/(d.t-x.) h
DBCD to.."'82 = h/>l. A ---~ - - - - c.
<. cl )t-(- )(_; -,
2 eiu..A..ti~s., 2 va:-ci a. \ale & .
ESTIMATION OF VE~"< sMfiLL
DlSTANCES
We c.:-re a.le. a. mono f() o Je c..c..d°'"" 1.a..0eY ~ : - .:- l,
- - - - t
pa; tr .ftYfAa--'>{iJ ~cirft ~Jr Jo.jel'
(JJe di\u.te o\e-·1c acid. ""
a..l co~o \ 5o ino.t we. co.."' tD
t
teduc.e "the. o.:r"'o u.."'1: of Oft t~ckt\ebS
peY d"fop. Dilllte. vfit' ®o'r- = JJ'o..l'Y'ete.Y of
o.. ~ le of
ctr1 d"fop -fr ri- -2-11.n oA o lei c. o.cicl.,
~ utt:Pa 11 a.,nJ 1..a.~eY
moY'lomole.e-L.Jo.Y ~ cl~aA-)

?"lf" M- 1. c. m3 ole ic. ui d dir 400 c..rr? Gfir .So h,.-\ io.,, <S:1--;JT
.J-fcl,~ ~ I tn: -3~); '3q~ c_m3 o.\cohol dIT <dUrl MI
1. c.-m; OA + .3q q cm3 oJc,ol-io I -~') ~ 00
'1v -So lu1ion

~ .2-.steps ~ Jilu.te. a,@ i'

1 3
+ ,q c.m o.lc.ohol -___,~ 2..0 .,.,,\
Crf'l3 OA oj !10 'fv OA .
_ _ _ _ _ _ _ _ _ _ _ _ __J
~
1cvv?r11½o
-d
1/JvOA
11
+- 1q c.m3 o..lco'1ol ) '/:,(_L =
/u, ~o
y100 rv Ofi .

~ e:x.pe.l"imei'l't it 0.:.-op bi! d~op -solu..iioYI ~- (~h1:i:>~1 / ,4r

fe 4ei 1 ~ u1fk=F W 1d-rop oolu.tioY\ ~ volt<.tt1e
fa;ct~I ~ I
lo Solu..tioY\ =
c.:rl\?:> 300 d-rop5 'J
4 )'Yleo_~tr,ecl i~ bl.(.Ye.tte. CoCM1ted.
~ ex.pcy·,m~Y\t ff (A\c.,ohol D /cra.shu.p
e.vo..poy~\e.. P- unm ~ c1-i1rf4:i
mono mt,lec.,u J o..Y" \~elf ~ ..Sttyfo..c.e o.."feo.. 411 Lffi yIJiJ2 ~I
volu.me of OA d.--rorped. = vo\u.me. of -lo..~e~ f O'i ~eJ
=t> 'Y'l >J"'ops x OA iY\ i d.Yop = vol . of k~e"'
.=t> n ~ (volu.Y\"\€. c,f),. (o.f>i ·•"") = vo\. oi' \o.. ~e:f
1.. d~op \J- d1op
G' Qm) ,. (-1 Yv) -- '--f£:j t [cl & t ·,~ cm~
1
=t> (\ x O
300
Ltoo TI

CoLVnteJ. vo\La} 1
J.iltAtion '1 d. is meo. .s u:r eJ. ea. le. wa..teJ.
cl"'fops .i d--rof 5of Lttici'n lLSiY\d -Sc.a.le. ituck~ss
.,

- ~ou.tu.be.com/cn,. shu.p

b~
Vik'fo.:f,t kir°'"
II T Kho..'fO.. j pu Y
@ @vikro..Y\tkifo-Y
E~RORS IN ME.ASIJREMENT
11t "' Ph~sic!> Cno.pt~:c 2 po.:it 3
UNITS 1 l)IMENSIO"-J S
~tt.tbe . com/C.1'~&hup • Cr-o.,~up.,.
The. u.nce-fto.."l nit~ in e.ve'i~ me.a.su -reme nt bj o__

tnel\Su.Y·,Y\~ ·,n.sf"t"u.Yl1eY\t is c..o.lleJ. e.XYo-r.

rneo..su..'fe.d vo.lue - t-ru.e.. va.lu.e. + e.,¥ot

Accu Yo. c..~ : mea.su-re of hol.,I) cJo.se. -the me.a. su ,eJ

vo.1 u.e ·,.s to ine. true. valu.e
PYe.c..i~ion : mea...suxe. of the -lim·,t oY Yesolu.-1ioY\ of the.
tneasu.-r1Y\j inst-Yu.YY\e..nt.

! ' I I

rne.a s.u..-red \ tvu.e. val i,e

vo..1 u..e (~ MlrcFi/f if,,: it- i# wrr d t I)
~ .scale. ~ dr 'rneasureJ va.(u...e/ it
-i:rc,,-- 7IM <ta,a"r ~ I

E~RORS
I
-i
s~~ terna.ti C.
Erro-rs

RAtJnoM cR.RoAs
ETroos wi-uc.V\ oc.c.u...- "f'o..v\tkom\~ o,:nd. o..Ye '(c,.Y\JOWl 1\,-\

~j~ C)..Y)! SiJVl .

e-3 · e.-rro't d u.e to v iht~iion of to.b)e wh.i le rriea..sur ,·.., .
8
SYSTEMATIC ERRO~S
I NSTRtJMENTA L. ERROR~ : eyroy Ju.e to ·,mpel'rec.,t
clesi~ 1ll o .,- poo"i' c.o.U-, b"i' oJ i oY\ of tlte. meo._!, u.Y 1"'d
iV\~J-<u meY\t. e~. Z:C.Yo eYYoY

TECHN \QUE R.ElATE=l> e-RR.oR : er'fo--r c\u.e. to im petfec.tioY>

·,V\ e1'pe Y ·, me Y\to. \ t..ecl,,Y\Ciu.e. .
e.~. ~ oleic. ~id a tR expei,rneY\t * rnolec..ules 4'tr
pe:rfe.ct sphe-re. fflrJJ filp:-w rnolec.u.l~ ~ cifu a/fr 431?9
Space ~ C.oV'lside?r riff 14, DI I ~ 0.14. 'l"tloleGt,vle s
peyfec..t .spheYe s ~ '5ft- if# l I

Pc-RsoNAL ERRo ~~ : crY-o-rs due to O.Y\ iVldividua.\s bia!; I

CtAYe.le.s5 ne.ss.
c.3. eno"t" du.e. to po..-yo.\l°';x '
I ,I
1 I I,,;,,]
~ ,' .
S~ste moJic e"(--(o"(S ~Ye. H(
I
I

oJ-rec.t\·OY\oJ o..~ Ca.,,() be e '('( OY d u.e. to ~Ya.\ \o..)(

'rY\·1Y\i~i zeA O-Y conected •

Lee1~t Cou.V\t E..TYOY : The e"t'Yo-r °'s..soc., o.teJ w·,ti,, ihe

resolu.t ion of the in.stYument .
.3-f~l'l vo.\u.e. 13. 2qg>3 cvn ~ i1t-r-~ ~ scoJe ~
rrea.su.:te. Ch°U) 1 a-,- va\ue 311p.1ft 13 .3 GY"\ . T"r~ ·, t .
CALCULATION OF ERRORS
AbStolu.te e't1o'C : ti,,e mo.1Y1'1tude of tl,,e difference
be.twe.eY\ ~ ',ndiv ,cl w:,,. \ me.Asu.reme 4 t il-u_ t-rue
vll. Iu.e of ti,. e. 't,UllVI t i tq'
bQ = I a. rneo.~u.yeJ - Qtn..1.e. valt.te l
jf tYu-e vo.\u.e is not known :
LYl- = I O.. M~Su~ed. - Q'tne°'YI \

* Dfl. ls a.\wa.~.s pos', tive

ffNAL ov fVleo.Y\ Absolute Er<oY

Meo.."' of ~.l I a. lo so IuJe e YY o "< ~
..1 Q.,.., "" Q.60., I + ILia. ,J + . . . . + l ~a.~I) /n
= ~ lha:,I
.,~1 n

Re..\CA..'l, Vt! EYroy = 6°'

a_

Pe-rc.e""to.1~ E"rY'oY :::: ~ x ,oo "I. _ &,o..

ComB1NATtoN of ERRoA.~
if z .::= A+ B i A tft eY-<oY l:lA ; Bit'enoy b.B J
6z: = LJA +~G ,Abso lu.te e -f'!O t" ~ o.dd ~
fi1\o.l 0-.Y\.St..ue:c bec.o'Y\'\QS c±bc; Ac= .6A-+bB
if ~-=-AB OY A/8
~2 = D'1 + DB
Z A e:,
relative e.""YOY 61r o.Jd. au,
 if c:::.A'3
:t;, 2:A-ftPi

=t> ~ == 6A_-\ DA -t bA = 3 6.A

Z. A A A A

~ e Yte --c-o..l
Ru..\ e.
if :c:= AP c, 1/ct

~ = p~A -t q,QG + "£JC

2 A B c

Sl~NIFICANT PIGURES
In plLt!> the..
a.l"~ m eo..su"<e meY'\t, t he --reJ io.lole di1·1t.s
fi-r5t u.nc,eria.iVI d[j it o.xe k.rlowY\ °'s s.iy.,,;f'ic.a..Y\t d.i~i t~
O"r ~13yi '1 ti C.O. 't'\t "ti~Ul'e ~ .
if le'f\~tlA of o... pen is rriea....s1....-,ecl
13. 2 9 8 3 CYY'\ _,, t, u..e. voJu.e.
J.~· 3 C.-VV"'\ , mea.suYed u.5j .,,j -Sc.a.le
l"dio.b( \ l . . . .
d.;~its ~t d.,9,t •s a.lw~.s LL.Y'ce--<k,n
( ~ cft- dnTf .3 cMeo..su-re Jt )
th.is is a.\so --reJoJeci wit'-" k.o.st c.ou.,vd e.Y-<oY .

u:r~·itiV\J 13.3 c.M

a...s t3 . 3ooc..'rf\ wou.\d be wYOV\~ .

-=P we YJleed Some --ru.le.s io write rnea..su Yecl 4-u.a.mt',t ie~

..tea.di~ ~oe.s ~ i~ Crfft~) 2 eYOC5

002003000
3f JJ2 -t;ro.ilinJ cc.:roes s ~~,if,·cat1t ~- rl7"~xpe. ri merJ ~ dT&
~ d.ec.irr1~I ~ cli1:. <Slrll~I ill~P I
 Lea.di1 2e-rnes.: oev-e'< .s~nif,·c:a.>1.t
Tro.iliVIJ ~e"foe!;: si~i,,jfica.nt if kcil'V>al IS rye,;e.,,t.

ADDITION WITH 61~- flG$,

aJ~Y\ -tne clec.1 ~a.ls o..v'ld t hen ~ound -off f he.. va.f ue.s u.e
oo Y'I' i le."' ow
'-i2,,.32 + 221. 2 t o.3o2.

, 1
,36.
3~2
,
?r fvrn-
~
values ~ ~ o-lfr
~
V1 irfA,
-t 2 2 ,- . 2.: ? ·1 ;:j<rE add efTit ~ =h2fl /
+ 0-3:0 ·2.
6 b 3 · c9! 2.. 2...
·~
! '-.... > ~
'::Jtr~ vou~
•, I - 1'-'f
0. 01
d,'}_~ ~df.riJ mrr
fYlOLTl PLICATION /o,v, 510N
2,etaiY\ t~e nu.m be.¥ of si~-fi~s of t he meo.~u-<evne.,)1.t
wiih l e.a.st .si~-fi s.
0
c.1 x 1.oC)l = 1
1 Si8f'd 3 Si3f13s

~ 1 is s "1'\ IIe" t1,,,..,, 3, !, o a. n s we• ~ hoWJ ha, ve 1 !. i

3f,g
wa.td,-. v',tleo fo"'< .SOived e..X£AV'f\plef;.

1( Coul'\leJ. vo.llM!s ho.ve no e..'f°'<o-r. The✓

0-.. wee. k. "1a~ .,Z. da j ~ 7
o.. Lvee k lia_ s t.o o ooo da.:J !. J i'

a fdOU.tv.be .c.om C..l"Clt;~ up

b~ Vi kro.'()t kj -ra."'
I
11 T K ~o..."("(). ~ PL,L'(
UNITS 2, J>IMfN&foNS

DIMENSIONS
11 PHY.Sil!» CHl'PTEA.. 2. PART Lf
yoc.tttAbe. C.orY' / C.YAS~ \AP

The di l'Y\eYI s io "~ of £1.. ph ~.si coJ q_,1.W-11ti \J

o:re iht. po we YS
io wkic.ll\ tV)e w.~e 1,,MY\tit{e5; mu.st toe ycJsecl. to
Ycpo-e..Se.Y\1 tho..i 4LA.o..'r' ti bU .

[Fo-<ce] = [M' L 1 T~ A° K 0 J mo1°] 0

=- [ML T- 2 J
fo--oce V\CAS 1 cl, ~e.-v'IS ioY\ \ V\ m°'-~S., 1 dime.Yl.~ io Yl iVi le. r1✓tV\
O..Y\d. -2. d.iWleY\s io'r\.~ i}'\ t iwie .

~~I~ r b~ q.,u.~\'\tihe.s ~ 9 dimeY\~io"'f» a.s.si~V\ /&<3 ~~·/

Q L,Lo,:n tit Unit Dirr1ev,S.i(J'() 6tj t'n bo I

LeY)st h - - - - - - me.te-r (vn) - - ( L)

Mo.S,S kilog-ro.'{'(\ Ck~) - [M}
Time - - - - - - - SecoY\d Cs) - [T]
Cll-rYe.Y\t A'YV\pe."¥e (A) - [AJ
Tempe-ra.tuYe. - - - - ke.\viYl(k) ~ = [K]
Lu..r,1·1V\ou~ h1t-eY1~i~ -= G.Y\JeJo. (c:J) - Le.cl]
Amou..Y\t of su.\asta."'c__e... Mole (\'\'lo\) - - LmoU
3-f;)J2 ~~·~ ~ r ·,Y)depeV\JeY\d q.,u.°'~tities ~ bo.~

<Sid-t(ri 1 en- ~Hf'< D '1me.Y)SIO'AS ~~ ~/ » mas.s[fv1],,

le.:r\3th [L] C}./f)d timelrJ dn- J1 dl S P-i'e.ss. u.Ye [P] 1 De Ylsi t!i D>]
a.:t.L'i -Fo-rc.e [-f"] ~ d ;me.nsioYl5 6icf-{12.Jf v=rr ~&iaJ <or J
To -t\YIJ d. iY'll e>'IS io w.. , WY·• °'
te. ~ f o Y rvt ul G1. aml -te. pIac.e.

tl,-u. Jj VY\ eN\ & io Y\A of 1,. u.a;nt j ti~ .

2
[ El'leY~ ~l= I'll~ h = [M}[LT-2_] DJ =- [M L2. T- J
[&-er~';f] ~ ½_YYYv'-z. = [M][LT-1]2 = [M L2 T-:]
[wo-rk] =- F. 5 = [ML T-2J[t.J = LJY)L'2..r-2J

Consto..nt.s co..n be. cLimeY\siovde.s&

e.~. \½_' iY\ ~ M v,_ Js 1A w,e vt~i o"'le'- s
C..OY).s to..vd s Co.ft'\ ~o..ve. d..ime.Y\ si <JY)
e •d. ,po."It,JioYlcJ eo 11 sta. ,.J" ' ~ '.

PRINCIPLE Of HOMOGENEITY Of l)JMEN510tJ~

We. CA.V) MJ 0~ Gu/oh-ad p~ _sIC.o1 'vu.a.Y\t"1 tie.A O Yl) ~ "if t\,,e.J
~°'-ve so..me. d.'i W'le Y\ s iOY\ s .
~<-1"1 'f{r eq.W:ttioY\ # tJ-t#] t__eym5:, ~ dime.)'\.SJOl-1.S .Sa.me
~ zJlf'tP l
x... = {.Lt ;- ½.a. t '2.
i J-\ l~
[L] ll:r~cu lJ. r~[T:]
= [L] ~ [L]

DEDUCING RELPTION AMON& PMV~ICAl G\.UANTITIES

we mu.st kY\oW ihe de.peY\cle.Y\c.ie.s

0
Ti""e. period. = f (rri,..l,~) la
~-~~H-J depe ndencie ~ ~ po wcv 6 o., b 2. c.. <$i' ~ "' , __ -- - --'"' TYl <f
muJtipl\1 ~ I J.,VVle~~io\l\\e.s~ c.o~~laY\t k 'fir (!4J11Yl--

T-P. -== k (m) o.(1.) 'b (~) c.

 W'1ite d.imeY\s1ol'\s 'tn LH£ 2- RH.S

[ft1°L0 T 1] = [ M]~[L] b[L T- 2Jc

== [Ma.. Lbt-C.T b-2CJ
LH.s 2- RH5 # d.imeY\510)15 Ga.me ~ z.11&2:5}
boJ°'"'~d [M] =t> o= Q ] 5olve
[L] =t> o = b+c. 3 . eiu.o.. +ioY1~
[rJ -=I> 1 = b- 2c. 3 vo.:'fia. lol~ s

:::t> a..= 0 ; lo= ½. j c.= -1;<

Put iY\ fo....-frlulo..

Tr.,.. ~ (m) oC.L) V2.S, ) V2. <" R -f1/4
Llft'ITATION'
- C.a...nnot fiY\d dirneY\si OYl\ess C...OY\~To..v(ts
- c.DoYec1 d.e..pe \I\J.eV1cies. mu.st lae. k..Y\OWY\

- kpevlC!ev.c..ies niu..st loe !iW\ite.d iV\ Y1u.mbe.~ . ~ LHS a. ~S

'ff r * Lr'~ c1.'1mev,_si0Yl.S oilr n1atCY1 d ~ -<1-1~ l
- Wo--rks el-- OY\I~ w~th mu/·tiplico..iiaY\/Jivi~iOYl.
we. cam"f" ot ~;Y\J.
._s,,, tLt + Yilt 2. evcVl iho<.,.~h s" f (.u., t,o..)

(HANGIN, f3AsE GUANTITIES

1..0e CM\ ~ i Yid divneY1 si OYIS of ev1ey~ ~ i vi d.eYI si ~[DJ, p 1e1.s~
[PJ 2- f OTC£. [.-FJ .
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