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w'fmd'i' idudkH fm< 10 fY%aKsh - 11 fY%aKsh
fN!;sl úoHdj
.eg¨ úi`§fï myiq l%u
ksYdka; úfÊisxy
ish¨u ysñlï weúßKs l;Df.a wjirh ksis mßÈ fkd,nd fuys lsis`ÿ fldgila l=uk wdldrhlska fyda Wmqgd M, lsÍu imqrd ;ykï fõ'
m%ldYkh ( l;D m%ldYkhls uqøKh 2012
© ksYdka; úfÊisxy
mß.Kl wlaIr ixfhdackh
ysrx.d rdcmCI msgq ieliqu yd ljr ks¾udKh
ñ,dka m%shx. ñßyd.,a, uqøKh
.sydka ´*afiÜ m%skag¾ia n,f.d,a, úuiSï
0777 843229 ii
mgqk 1' .eg¨ úi£fï Ys,amh
01
2' wka;¾ cd;sl tall l%uh - SI tall
09
3' uQ,sl w¾: ±laùu
12
4' uQ,sl úIh lreKq yd iólrK úYaf,aIKh
18
5' iólrK weiqßka .eg¨ úi|Sfï myiq l%u
35
6' nyqjrK m%Yak úi£fï myiq l%u
56
7' rpkd m%Yak úi£fï myiq l%u 8' iólrK úYaf,aIK l%u fõo
112 130
9' iñlrK j¾. yd .eg¨ j¾.
139
10' úoHq;h .eg¨ úi£fï myiq l%u
161
iii
1 .eg¨ úi£fï Ys,amh fN!;sl úoJdj .eg¨ úi£u ksjerÈj yqre fkdùu ksid fndfyda wmyiq;djhkag m;ajk isiqka jeä msßila isák nj idudkH fm< iy Wiia fm< úNd.j, msQ¾Kh = 20 N º 2m = 40 Nm
olaIsKdj¾; >Q¾Kh Q
A 2m
R 2m
5N
10N
olaIsKdj¾; >Q¾Kh = 5N º 2m + 10N º 4m olaIsKdj¾; >Q¾Kh = 10Nm + 40 Nm olaIsKdj¾; >Q¾Kh = 50Nm fï ksid jdudj¾; n, >Q¾Kh 40 Nm nj o olaIsKdj¾; n, >Q¾Kh 50Nm nj o meyeÈ,s fõ' fï wkqj olaIsKdj¾; n, >Q¾Kh" jdudj¾; n, >Q¾Khg jvd úYd, nj fmfka' tneúka oKav olaIsKdj¾;j N%uKh ùug ;e;a lrhs' .eg¨jla ksjerÈj wjfndaO lr f.k úi£u i|yd úIh lreKq fyd¢ka wjfndaO lr f.k tu úIh lreKq uekúka ixúOdkh l< hq;= fõ' fï i|yd Tn úiska .eg¨ úi£u ;=%;djh ukaokh f,i y÷kajhs'
ffoYsl rdYs úYd,;ajhla yd ÈYdjla ;sfnk fN!;sl rdYs ffoYsl rdYs f,i y÷kajhs' WodyrK - úia:dmkh" m%fõ.h" ;ajrKh" ukaokh" n,h" n, >Q¾Kh" wdhdih" wd;;sh
wÈY rdYs úYd,;ajhla ;sfnk tfy;a ÈYdjla fkdue;s fN!;sl rdYs wÈY rdYs f,i y÷kajhs' WodyrK - ialkaOh" mSvkh" ld,h" >k;ajh" WIaK;ajh" fõ.h" ÿr" j¾.M,h" mßudj
iajdh;a; úp,Hh fjk;a rdYs u; r|d fkdmj;sk rdYs iajdh;a; úp,H f,i y÷kajhs' WodyrK - ld,h" ialkaOh
12
mrdh;a; úp,Hh fjk;a rdYs u; r|d mj;sk rdYs mrdh;a; úp,H f,i y÷kajhs' WodyrK - ÿr" úia:dmkh" fõ.h" m%fõ.h" ;ajrKh" ukaokh
.=re;ajc ;ajrKh mDÓúh u.ska we;s lrk .=re;ajdl¾IK n,h fya;= fldg f.k mDÓú mDIaGh wi, my¾IK n,h § we;s mDIaG folla w;f¾ l%shd lrk Wmßu >¾IK n,h iSudldß >¾IK n,h f,i y÷kajhs'
>¾IK ix.=Klh § we;s mDIaG folla w;r" l%shd lrk iSudldß >¾IK n,h tu mDIaG fol w;r fhfok wNs,ïN f;rmqu n,hg ork wkqmd;h >¾IK ix.=Klh f,i y÷kajhs'
ld¾hh n,hl fufyhùu hgf;a hï jia;=jla p,kh fõ kï tu n,h u.ska hï lsis ld¾hhla lrkafka hehs lshkq ,efí' isÿ lrkq ,nk ld¾hh m%udKh n,fha úYd,;ajfhys yd n,fha Wmfhda.S ,laIHh úia:dmkh jQ ÿfrys;a .=Ks;hg iudk fõ'
Yla;sh hï jia;=jlg we;s ld¾hh lsßfï yelshdj Yla;sh f,i y÷kajhs'
laIu;djh ld¾hh lsßfï Ys>%;djh laIu;djh f,i y÷kajhs'
n,hl iQ¾Kh ndysr n,hla fya;= fldg f.k" hï jia;=jla § we;s ,laIHhla jgd N%uKh ùfï fyj;a yerúfï l%shdj ukskq ,nkafka" tu n,fha iQ¾Kh kï jQ fN!;sl rdYsh u.sks' § we;s 13
wlaIhla jgd n,hl iQ¾Kh w¾: olajkq ,nkafka" n,fhys;a" tu n,fha l%shd f¾Ldjg" N%uK wlaIfha isg we;s ,ïNl ÿfrys;a .=Ks;h f,ih'
n, hq.auh úYd,;ajfhka tlsfklg iudk jQ n, folla tlsfklg m%;súreoaO Èydj,g l%shd f¾Ld folla Tiafia l%shd lrk úg tu jia;=j N%uKhùug fmk;ajh hï øjHhl tall mßudjl ialkaOh >k;ajh f,i y÷kajhs'
mSvkh tall j¾.M,hla u; l%shd lrk n,h mSvkh f,i y÷kajhs'
úNj Yla;sh jia;=jl hdka;%sl wjia:dj ksid fyda jia;=fõ msysàu ksid fyda tu jia;=j i;= jQ Yla;sh úNj Yla;sh f,i y÷kajhs'
pd,l Yla;sh p,kh jkakd jQ jia;=jlg ld¾hhh lsÍfï yelshdjla we;' fuh tu pia;=j i;= pd,l Yla;sh f,i y÷kajhs'
idfmalaI >k;ajh hï øjHhl >k;ajh c,fha >k;ajhg ork wkqmd;h idfmalaI >k;ajh f,i y÷kajhs' hï øjHhl idfmalaI >k;ajh" tu øjHfha hï mßudjl ialkaOh" iudk c, mßudjl ialkaOhg ork wkqmd;h f,i o w¾: ±laúh yel'
iïm%hqla; n,h jia;=jla u; l%shd lrk n, folla fyda jeä .Kkla fjkqjg fhosh iïm%hqla; n,h f,i y÷kajhs'
yels ;ks n,h
ir, hka;% hï ;ekl fhfok Ndrhla fjk;a ;eklg fhdok wdhdihla u.ska wNsnjd hdug WmldÍ jk Wml%uhla ir, hka;%hla f,i y÷kajhs'
bms¨u jia;=jla øjhla u; mdùu bms¨u f,i y÷kajhs'
,Sjr wp, ,laIHhla jgd N%uKh jk o~q tllska fyda follska iukaú; jk ir, hka;% ,Sjr f,i y÷kajhs'
14
Orh ,Sjrj, ,Sjr oKav wp, wdOdrlhla jgd N%uKh fõ' fuu wp, wdOdrlh Orh f,i y÷kajhs'
Ndrh p,kh lrkq ,nk jia;=fõ nr Ndrh f,i y÷kajhs'
wdhdih ir, hka;%hla u.ska ld¾hhla isÿ lr .ekSu Wfoid fhdokq ,nk ndysr n,h wdhdih f,i y÷kajhs'
mk øjHl wxY= ;=%;djh fõ.h f,i y÷kajhs' 15' my; i|yka lreKq weiqfrka fõ.h uksk SI tall ,nd .kak' ÿr fõ.h = ld,h ÿr uksk SI tallh fõ.h uksk SI tallh = ld,h uksk SI tallh m fõ.h uksk SI tallh = = m/s = ms-1 s fõ.h uksk SI tallh ;;amrhg óg¾ $ ms-1 fõ'
20
16' ;;amr 10 la ;=< § óg¾ 200l ÿrla p,kh jk jia;=jl fõ.h fldmuK o@ ÿr 200 m fõ.h = = ld,h 10 s
= 20 ms-1
17' tla;rd jia;=jla ;;amr 5 la ;=< § óg¾ 75 ; ÿrla .uka lf%;djh f,i y÷kajk ksid" úia:dmkh fjkia ùfï YS>%;djh m%fõ.h f,i y÷kajhs'
19' my; i|yka lreKq weiqfrka m%fõ.h uksk SI tall ,nd .kak' úia:dmkh m%fõ.h = ld,h m m%fõ.h uksk SI tallh = = m/s = ms-1 s m%fõ.h uksk SI tallh ;;amrhg óg¾ $ ms-1 fõ' 20' ;;amr 5 la ;=< óg¾ 60 la úia:dmkh jQ jia;=jl m%fõ.h fldmuK o@ úia:dmkh 60 m m%fõ.h = = = 12 ms-1 ld,h 5s 21' tla;rd jia;=jla ;;amr 10 la ;=< óg¾ 250 la úia:dmkh úh' fuu jia;=fõ m%fõ.h fldmuKo@ úia:dmkh 250 m m%fõ.h = = = 25 ms-1 ld,h 10 s 22' ir, f¾Çh ud¾.hla Tiafia p,kh jk fudag¾ r:hla iEu ;;amrhla ;=Q¾Kh = olaIsKdj¾; n, >Q¾Kh olaIsKdj¾; n, >Q¾Kh = 400 Nm úh hq;=h' 400 Nm = 1m × w 1m × w = 400 Nm 400 Nm w = 1m w = 400 N
42
24' ld¾hh = n,h × n,fha l%shdldÍ ,laaIHh úia:dmkh jQ ÿr .eg¨j : jia;=jla u; 50 N n,hla l%shd lsÍfuka th 8 m l ÿrla úia:dmkh fõ kï tu n,h u.ska isÿ lrk ,o ld¾hh m%udKh fldmuKo@ úi÷u : n,fha l%shdldÍ ,laIHh úia:dmkh jQ ÿr = 8 m jia;=j u; fhdok ,o n,h = 50 N ld¾hh = n,h × n,fha l%shdldÍ ,laaIHh úia:dmkh jQ ÿr
ld¾hh = 50 N × 8 m ld¾hh = 400 Nm ld¾hh = 400 J
25' ld¾hh = ialkaOh × .=re;ajc;ajrKh × úia:dmkh .eg¨j : c,h 1kg la 122 m by¾IK ix.=Klh 0'4 fõ' F 20kg
úi÷u : wNs,ïN f;rmqï n,h wNs,ïN f;rmqï n,h wNs,ïN f;rmqï n,h >¾IK ix.=Klh
= jia;=fõ nr = 20 kg × 10 ms-2 = 200 N = 0.4
iSudldÍ >¾IK n,h = >¾IK ix.=Klh × wNs,ïN f;rmqï n,h = 0.4 × 200 N = 80 N 29' wNs,ïN f;rmqï n,h = ialkaOh × .=re;ajc ;ajrKh .eg¨j : 25 kg nrla we;s RcqfldaKdi%dldr ,S l=Üáhla ;sria ;,h u; we;s lrk wNs,ïN f;rmqï n,h R fidhkak' R
mg
úi÷u : jia;=fõ ialkaOh = 25 kg .=re;ajc ;ajrKh =10 ms-2 R = mg ksid" wNs,ïN f;rmqï n,h = ialkaOh × .=re;ajc ;ajrKh = 25 kg × 10 ms-2 = 250 kgms-2 = 250 N 30'
>¾IK ix.=Klh =
iSudldÍ >¾IK n,h wNs,ïN f;rmqï n,h
.eg¨j : 10 kg nrla we;s RcqfldaKdi%dldr ,S l=Üáhla r¿ ;sriaa ;,hla u; ;nd we;' ,S l=Üáfha p,kh wdrïN lsÍu i|yd fhÈh hq;= wvqu n,h 70 N úh' ,S l=Üáh yd r¿ ;sria ;,h w;r >¾IK ix.=Klh fidhkak'
44
úi÷u : wNs,ïN f;rmqï n,h = jia;=fõ nr wNs,ïN f;rmqï n,h = 10 kg × 10 ms-2 wNs,ïN f;rmqï n,h = 100 N >¾IK ix.=Klh = iSudldÍ >¾IK n,h wNs,ïN f;rmqï n,h >¾IK ix.=Klh = 70 N 100 N >¾IK ix.=Klh = 0.7 31' hdka;% jdish = Ndrh wdhdih .eg¨j : 25 N wdhdihla fh§fuka 150N Ndrhla p,kh l< yels ir, hka;%hl hdka;% jdish fldmuKo@ úi÷u : Ndrh = 150N wdhdih = 25 N Ndrh hdka;% jdish = wdhdih hdka;% jdish =
hdka;% jdish =
150N 25 N 6
32' wdhdih p,kh jQ ÿr m%fõ. wkqmd;h = Ndrh p,kh jQ ÿr .eg¨j : ir, hka;%hl wdhdih 220 cm la p,kh l< úg Ndrh 11 cm la p,kh úh' fuu ir, hka;%fha m%fõ. wkqmd;h fldmuKo@ wdhdih p,kh jQ ÿr úi÷u : m%fõ. wkqmd;h = Ndrh p,kh jQ ÿr 220 cm m%fõ. wkqmd;h = 11 cm
m%fõ. wkqmd;h = 10
33' ,Sjrhl m%fõ. wkqmd;h =
wdhdi ndyqfõ È. Ndr ndyqfõ È.
.eg¨j : ,S fldghla fmr,Su i|yd Ndú;d lrk ,o w,jx.=jl È. 80 cm la jk w;r fldgfha È. Orh olajd jk w,jx.=fõ fldgfia È. 20 cm fõ' fuu w,jx.=fõ m%fõ. wkqmd;h fidhkak'
Ndr ndyqj
wdhdi ndyqj
45
úi÷u : wdhdi ndyqfõ È. = 60 cm Ndr ndyqfõ È. = 20 cm wdhdi ndyqfõ È. ,Sjrhl m%fõ. wkqmd;h = Ndr ndyqfõ È. ,Sjrfha m%fõ. wkqmd;h = ,Sjrfha m%fõ. wkqmd;h =
60 cm 20 cm 3
34' 2 ×π × lrljk ,o oKafâ È. bial=remamq celal=fõ m%fõ. wkqmd;h = wka;rd,h .eg¨j : bial=remamq celal=jl lrljk oKafâ È. 84cm fõ' bial=remamqfõ wka;rd,h 3 mm fõ' fuu bial=remamq celal=fõ m%fõ. wkqmd;h fldmuKo@ úi÷u : lrljk oKafâ È. = 84cm 3 wka;rd,h = 3 mm = cm 10 2 ×22 × 84 cm 7 bial=remamqfõ celal=fõ m%fõ. wkqmd;h = 3 cm 10 2 ×22 × 84×10 m%fõ. wkqmd;h = 7×3 m%fõ. wkqmd;h = 1760 35' wdk; ;,hl m%fõ. wkqmd;h = wdk; ;,fha È. wdk; ;,fha Wi .eg¨j : wdk; ;,hl È. 3 m fõ' fuu wdk; ;,fha Wi 1m fõ kï tys m%fõ. wkqmd;h fidhkak'
ℓ
h
úi÷u : Ndrh p,kh jk ÿr = wdk; ;,fha Wi = h wdhdih p,kh jk ÿr = wdk; ;,fha È. = ℓ wdk; ;,fha Wi = 1m wdk; ;,fha È. = 3m wdk; ;,fha È. wdk; ;,hl m%fõ. wkqmd;h = wdk; ;,fha Wi 3m wdk; ;,fha m%fõ. wkqmd;h = 1m
wdk; ;,fha m%fõ. wkqmd;h = 3
46
36'
il yd wlf¾ m%fõ. wkqmd;h =
iflys wrh wlfrys wrh
.eg¨j : vnrhl iflys wrh 70 cm fõ' tu vnrhg iïnkaO wlfrys wlfrys wrh 7 cm la fõ kï tys m%fõ. wkqmd;h fidhkak' úi÷u : iflys wrh = 70 cm wlfrys wrh = 7 cm il yd wlf¾ m%fõ. wkqmd;h =
iflys wrh wlfrys wrh
70 cm il yd wlf¾ m%fõ. wkqmd;h = 7 cm il yd wlf¾ m%fõ. wkqmd;h = 10 ^il" pl%h f,i o wlr" wlaI oKav f,io y÷kajhs'& 37' ;dm m%udKh = ialkaOh × úYsIag ;dm Odß;djh × WIaK;aj wka;rh Q = mcӨ .eg¨j : ialkaOh 500 g la jk ùÿre n÷kl c,h 2 kg la nyd we;' tys WIaK;ajh 30 C isg 80 C olajd jeä lsÍug wjYH ;dm m%udKh fjk fjku fiúh hq;=h' ;dm m%udKh fiùu i|yd ùÿre n÷kg iy c,hg fjk fjku Q = mcӨ iQ;%h fhÈh hq;=h' ùÿre n÷kg ,nd Èh hq;= ;dm m%udKh fiùu' 500 m = 500 g = kg = 0.5 kg C= 670 J / kg oC 1000 Ө = 80 oC – 30 oC = 50 oC Q = mcӨ Q = 0.5 kg × 670 J / kg oC× 50 oC Q = 15000 J c,hg ,nd Èh hq;= ;dm m%udKh fiùu m = 2 kg c= 4200 J / kg oC o Ө = 80 C – 30 oC = 50 oC Q = mcӨ Q = 2 kg × 4200 J / kg oC× 50 oC Q = 420000 J wjYH uq¿ ;dm m%udKh =15000 J + 420000 J = 435000 J = 435 kJ wjidk ms%;djh f,i neúka" m%fõ.h=
úia:dmkh ld,h
m%fõ.h uksk SI tallh = m%fõ.h uksk SI tallh =
úia:dmkh uksk SI tallh ld,h uksk SI tallh m s
m%fõ.h uksk SI tallh ms-1 ^ ;;amrhg óg¾& fï wkqj ksjerÈ msk;ajh w¾: olajkq ,nkafka" tall mßudjl ialkaOh f,i neúka" >k;ajh =
ialkaOh mßudj
mßudj hkq È.j,a ;=kl .=Ks;hls' fï ksid" mßudj = È. º È. º È. mßudj uksk SI tallh =
È. uksk È. uksk È. uksk º º SI tallh SI tallh SI tallh
mßudj uksk SI tallh = m º m º m mßudj uksk SI tallh = m3 ^>k ógr& ialkaOh uksk SI tallh kg jk ksid ialkaOh uksk SI tallh mßudj uksk SI tallh kg = 3 m
>k;ajh uksk SI tallh =
= kg m-3 ^>k ógrhg lsf,da.%Eï&
fï wkqj ksjerÈ ms¾IK n,h 500 N fõ' hï mDIaGhla u; jia;=jla ;nd ndysr n,hla fh¥ úg tu jia;=j ksYap,j mj;S kï" ndysr n,h >¾IK n,hg iudk fõ' ^41& ñáhlska weKhla .,jk rEm igykla my; olajd we;' fuys wdhdih Orh yd Ndrh wkqms¾IKh wmg jeo.;a .=Khla fõ' >¾IKh jeä lr .ekSu i|yd fhdod .kakd Wml%u (i& w;af,a iy há m;=f,a iu r¿ ù ;sîu ^ii& ghrhl lÜg ;sîu ^iii& .ia keÕSu i|yd fldyq ,Kq j,,a,la fhdod .ekSu ^iv& fldyq fl|s tlg wUrd fldyq lU iE§u ^v& mä fmak fikaá ógrhl& ialkaOh .%Eï tlla jk njo u;l ;nd .kak' c,h ,Sg¾ tlla ;=< c,h >k fikaá ógr 1000la ;sfnk ksid c,h ,Sg¾ 1l ialkaOh 1000 g fyj;a 1 kg la fõ' fï wkqj ksjerÈ msk;ajh jkafka" (i) 25/20
(ii) 20/25
(iii) (25-20) /25
(iv) (25-20) /20
mqkÍlaIK igyk yd úi÷u hï øjHhl >k;ajh c,fha >k;ajhg ork wkqmd;h idfmalaI >k;ajh f,i y÷kajhs' øjHhl idfmalaI >k;ajh =
N+ñf;,aj, >k;ajh c,fha >k;ajh
>k;ajfha w¾: ±laùug wkqj" øjHfha ialkaOh mßudj øjHhl idfmalaI >k;ajh = c,fha ialkaOh mßudj øjHfha mßudj iy c,fha mßudj iudk jk úg mßudjg mßudj lems hhs' 86
øjHhl idfmalaI >k;ajh =
øjHfha hï mßudjl ialkaOh iudk c, mßudjl ialkaOh
fï wkqj idfmalaI >k;ajh my; i|yka f,i o w¾: ±laúh yelsh' hï øjHhl idfmalaI >k;ajh hkq tu øjHfha hï mßudjl ialkaOh" thg iudk c, mßudjl ialkaOhg ork wkqmd;h fõ' idfmalaI >k;ajh >k;aj foll wkqmd;hla jk ksid thg tallhla fkdue;s nj u;l ;nd .kak' by; ,nd .;a iólrKh weiqfrka hï øjHhl" >k;ajh = øjHfha idfmalaI >k;ajh × c,fha >k;ajh nj ,nd .; yels fõ' N+ñf;,a hï mßudjl ialkaOh iudk c, mßudjl ialkaOh 20 g N+ñf;,aj, idfmalaI >k;ajh = = 20/25 fõ' 25 g N+ñf;,aj, idfmalaI >k;ajh =
fï wkqj ksjerÈ msk;ajh1000 kgm-3 jk nj u;l ;nd .kak' ^60& ,dïmq f;,a ^N+ñf;,a& j, idfmalaI >k;ajh 0'8 ls' c,fha >k;ajh1000 kgm-3 ls' ,dïmq f;,a >k ñgrhl ialkaOh fldmuK o@ (i) 0.8 × 1000 kg
(ii) 1000/0.8 kg
(iii) 0.8 /1000 kg
(iv) 1000 kg
mqkÍlaIK igyk yd úi÷u ,dïmq f;,a >k ñgrhl ialkaOh ,dïmq f;,aj, >k;ajfha úYd,;ajhg iudk fõ' fï ksid ,dïmq f;,aj, >k;ajh fiùfuka úi÷u ,nd .; yels fõ' ,dïmq f;,aj, idfmaalaI >k;ajh yd c,fha >k;ajh È ;sfnk ksid ta weiqßka ,dïmq f;,aj, >k;ajh fiúh hq;=hs' øjHhl >k;ajh = øjHfha idfmalaI >k;ajh × c,fha >k;ajh jk neúka" ,dïmq f;,aj, >k;ajh = ,dïmq f;,aj, idfmalaI >k;ajh × c,fha >k;ajh ,dïmq f;,aj, >k;ajh = 0.8 × 1000 kgm-3 87
fï wkqj ,dïmqf;,a >k ñgrhl ialkaOh 0.8 × 1000 kgm-3 fï wkqj ksjerÈ msk;ajh weiqfrka ialkaOh ,nd .; yels fõ' thg fya;=j f,dayfha >k óg¾ tall ^1m3 & ialkaOh f,dayfha >k;ajh f,i w¾: olajk neúka § ;sfnk o;a; weiqfrka c,fha >k;ajh 1000 kgm-3jk njo ,nd.; yels fõ' fï ksid" øjHhl >k;ajh = øjHfha idfmalaI >k;ajh × c,fha >k;ajh jk neúka" f,dayfha >k;ajh = øjHfha idfmalaI >k;ajh × c,fha >k;ajh f,dayfha >k;ajh = 8 × 1000 kgm-3 f,dayfha >k;ajh f,dayfha >k ñgrhl ialkaOhg iudk jk ksid f,day lene,af,a ialkaOh 8 × 1000 kg fõ' fï wkqj ksjerÈ msk;ajhg jvd jeä >k;ajfhka hq;a jia;+ka tu ;r,fha .sf,k njh' (iii) ksi, ;r,hl .s,ajd we;s jia;=jla u; we;s jk ;r, f;rmqu úia:dmkh jQ ;r,fha nrg iudk Wvql=re f;rmqula jk njh' (iv) jia;=jla ksi, ;r,hl mdfjk úg úia:dms; ;r,fha nr jia;=fõ nrg iudk jk njh' mqkÍlaIK igyk yd úi÷u hï lsis jia;=jla w¾O jYfhka fyda mQ¾K jYfhka fyda øjHhl .s,ajd we;s úg" jia;=j u; byk;ajh 1 g jvd jeä w.hla .kakd" c,fha wødjHh >k øjHh" øj c,h ;=< .sf,a' fï wkqj ksjerÈ ms¾IK ix.=Klh µ kï" A u; l%shd lrk >¾IK n,fhys úYd,;ajh jkqfha ñka ljrlao@ (i) µ F1
(ii) F2
(iii) µ F2
(iv) F4
mqkÍlaIK igyk yd úi÷u iSudldÍ >¾IK n,h (F) >¾IK ix.=Klh µ = wNs,ïN f;rmqï n,h (R) µ = F R fuu .eg,qfõ iSudldÍ >¾IK n,h F = F4 = F1 fõ wNs,ïN f;rmqï n,h R fuys F2 g fyda F2 g iudk fõ' tkï R= F2 = F3 >¾IK ix.=Klh µ jk neúka F R
µ =
iñlrKhg 91
wdfoaY lsÍfuka" µ=
F1 F2
µ F2= F1 A u; l%shd lrk >¾IK n,h µ F2 fõ' fï wkqj ksjerÈ msQ¾Kfha w¾: ±laaùu n, >Q¾Kh =
n,fha × N%uK wlaIfha isg n,fha l%shd f¾Ldjg ;sfnk ,ïNl ÿr úYd,;ajh
O wlaIh jgd N%uKh jk jia;=jla i,lkak' O wlaIh = N%uK wlaIh d = N%uK wlaIfha isg n,fha l%shd f¾Ldjg ;sfnk ,ïnl ÿr 101
n, >Q¾Kh c kï" c = F × d AB ieye,a¨ oKav iu;=,sa;j mj;sk ksid" olaIsKdj¾; n, >Q¾Kh = jdudj¾; n, >Q¾Kh 1m × w = 2m × 100Ν 200 Nm w= 1m w = 200 N fï wkqj ksjerÈ msQ¾Kh ffoYsl rdYshls' ld¾hh uksk SI tallh Nm fyj;a J fõ' kuq;a ld¾hh wÈY rdYshla fõ' ^84& x ,laIHhka t,a,d we;s ieye,a¨ oKavg fmkajd we;s mßÈ Ndr t,a¨ úg th ;sriaj msysghs' w iy x j, w.hka úh yelafla" (i) 10 N iy 0.7 m (iii) 6 N iy 2m
(ii) 7N iy 2m (iv) 3.5 N iy 1.5 m
mqkÍlaIK igyk yd úi÷u oKaav iu;=,s;j mj;sk ksid" olaIsKdj¾; n, >Q¾Kh = jdudj¾; n, >Q¾Kh X × w = 2N × 1m + 5N× 1m X × w = 2Nm + 10Nm X × w = 12Nm olaIsKdj¾; n, >Q¾Kh12Nm jk ksid" x yd w i|yd .e,fmk w.hka hq., 6N iy 2m fõ' fï wkqj ksjerÈ ms¾IK n,fha ÈYdj" tu jia;=j p,s; ùug fm¾IK n,hg Wmßu w.hla we;' fuh iSudldÍ >¾IK n,h f,i y÷kajhs' (iv) § we;s mDIaG folla i|yd iSudldÍ >¾IK n,h mDIaG w;r wNs,ïN m%;sl%shdjg iudkqmd;sl fõ' (v) wNs,ïN m%;sl%shdj fkfjkiaj mj;akd úg" mDIaG folla w;f¾ >¾IK n,h" tu mDIaG fol iam¾Y jk fmdÿ j¾.M,h u; fyda mDIaGfha yevh u; r|d fkdmj;S' tfy;a th mDIaGj, iajNdjh u; r|d mj;S' (vi) tla mDIaGhla u; ;j;a mDIaGhla p,kh ùfï§ l%shd lrk .;sl >¾IK n,h" tu mDIaG w;r iSudldÍ >¾IK n,hg jvd wvq w.hla .kS' fï wkqj mk;ajhla we;s øjh u; bmsf,hs' (iv) jia;=jla øjhlg fyÆ úg tys nrg iudk nrla we;s øj mßudjla úia:dmkh lsÍug thg fkdyels kï th øjfha .sf,hs'
110
mqkÍlaIK igyk yd úi÷u øjhl bmsf,k jia;= fyda øjHh msk;ajhla we;s ys,shï" yhsârcka jeks jdhq mqrjd we;s ne¨k jdhq f.da,h ;=< by< k.shs' tfiau idudkH jd;hg jvd wvq >k;ajhla we;s jk f;la r;a l< jd;h wvx.= ne¨k o ta wkaoñkau by< k.S' tfia by< k.skafka ne¨kh wjg we;s jd;h u.ska ne¨kh u; fhdokq ,nk Wvql=re f;rmqu ne¨kfha nrg jvd wvq jQ úghs' fï wkqj ksjerÈ ms¾IK n,h jeä ùu' (iv) ^w& ;;amr 7 isg ;;amr 10 olajd ld,dka;rh ;=< .uka l< ÿr tu fldgfia wod, j¾.M,fhka ,efí' m%ia;drfha tu wod< fldgi muKla ie,l+ úg"
;;amr 7 isg ;;amr 10 olajd ld,dka;rh ;=< .uka l< ÿr
ABCD RcqfldaKdY%fha j¾.M,h
.uka l< ÿr = 6 « 3 m .uka l< ÿr = 18 m ^wd& fg%d,sh taldldr m%fõ.fhka p,kh ù ;sfnk ksid"
fg%d,shg rn¾ máh u.ska fhdok ,o n,h
= fg%d,sfha p,s;hg tfrysj l%shd lrk n,h
fg%d,sfha p,s;hg tfrysj l%shd lrk n,h = 10 N
(v) iïm%hqla; n,h = ialkaOh « ;ajrKh ;ajrKh fiùu i|yd mk;ajh ^wd& we¨ñkshïj, idfmalaI >k;ajh = c,fha >k;ajh 2700 kgm-3 = 1000 kgm-3 = 2.7 fmd,a f;,aj, ialkaOh (ii) ^w& fmd,a f;,aj, idfmalaI >k;ajh = iudk c, mßudjl ialkaOh 45-20 g = 50-20 g 25 = 30 ^wd& øjhlska msß >k;aj l=mamshl ialkaOh fiùfï § >k;aj l=mamsfha nf|ka fkdwe,a,sh hq;= w;r tu >k;aj l=mamsh ;=, jdhq nqnq¿ isr ù fkd;sìh hq;=h' c, lf|a Wi (iii) x øjfha idfmalaI >k;ajh = øj lf|a Wi 08 = 10 = 0.8 115
(iv) ^w&
^wd& fuys§ ,nd .; hq;= mdGdxl folla we;' tkï"
(i) y øj lf|a Wi h (ii) c, lf|a Wi H
c, lf|a Wi ^we& y øjfha idfmalaI >k;ajh = y øj lf|a Wi
=
H h
^3& f¾Çh ;sria u.la Tiafia ol=Kq foig meÈ nhsisl,hl p,s;h ms¾IKh wdOdr lr .kS'ZZ fuh myod fokak' 118
úi÷u
(i) >¾IK n,fha iSudldÍ w.h ;SrKh lrk m%Odk idOl fol" iam¾Y mDIaGj, iajNdjh yd wNs,ïN f;rmqï n,h fõ' (ii) ^w& jia;=jla u; tys nr mg isriaj myF fõ'
(iii) fïih u; .fvd,a legh ;nd kQf,ka .eg .id ÿkq ;rdÈh wuqKd p,s; ùug wdikak wjia:dfõ § ÿkq ;rdÈfha mdGdxlh ,nd f.k igyka lr .kak' .fvd,a legfha wfkla me;s o fïih u; ;nd fmr mßÈu .fvd,a legh p,kh ùug wdikak wjia:dfõ § ÿkq ;rdÈfha mdGdxl ,nd .kak' by; i|yka iEu wjia:djlu ÿkq ;rdÈfha mdGdxl iudk jk nj fmkS hkq we;' (iv) fudag¾ r:fha frdao N%uKh jk úg >¾IK n,h msgq mig l%shd lrk úg thg úYd,;ajfhka iudk jQ;a ÈYdfjka m%;súreoaO jQ;a n,hla bÈßhg we;s jk ksid jdykh p,kh fõ' p,kh jk fudag¾ r:hl p,s; Èydj fjkia lsÍug fyda ksYap,;djhg m;a lsÍug o >¾IK n,h WmldÍ fõ' 119
^5& (i) zzmSvkhZZ hkqfjka woyia jkafka l=ulao@ (ii) jdhq mSvkh u.ska we;s lrk mSvkh wdo¾Ykh lsÍu i|yd Tng ilia l< yels wegjqula úia;r lrkak' (iii) i. Ndckhl we;s øjhla ;=< hï ,laIHhl mSvkh flfrys n,mdk idOl folla i|yka lrkak' ii. tla;rd n÷kl 60 cm Wig c,h mqrjd ;sfí' tu n÷k m;=f,a hï ,laIHhlÈ mSvkh .Kkh lrkak' iii. Ndckfha m;=f,a j¾.M,h 100 cm 2 m;=, u; l%shd lrk uq¿ øj f;rmqï n,h ksõgkaj,ska fidhkak' c,fha >k;ajh = 1000 kgm-3 .=re;ajc ;ajrKh = 9.8 ms-2 o f,i .kak'
(iv) Ydl uq,aj, we;s jk hï mSvkhla ksid c,h yd tys Èh jQ LKsc ,jK lf|a by%;djh cjh f,io y÷kajhs' cjh = lrk ,o ld¾hh ta i|yd .; jQ ld,h (ii) ^w& fõ.h jeä jk úg wu;rj fhÈh hq;= cjh o l%ufhka thg iudkqmd;sl f,i jeä fõ' fuhg fya;=j lkao ke.Su i|yd wu;r Yla;shla fh§ug isÿùuhs' ^wd& fõ.h fo.=K jk úg m%;sfrdaOhg úreoaOj fhdok cjh YS>%fhka jeä fõ' fuhg fya;=j" p,s;hg tfrysj fhfok m%;sfrdaOh jeä ùuhs' 125
(iii) jeä fõ.hlska lkao ke.Su i|yd iqÿiq jkafka f¾iska mdmeÈh fõ' fuhg fya;= jkafka" (a) ieye,a¨ ùu (b) .sh¾ fhdod ;sîu (c) gh¾ jvd;a isyska ùu (d) nhsisl,alre fmrg keó f¾iska mdmeÈh mÈk neúka iqk;ajh g = .=re;ajc;ajrKh øj l|la u.ska we;s lrk mSvkh flfrys n,mdk idOl" ^¸& øj lf|a Wi ^¸¸& øjfha >k;ajh ^¸¸¸& .=re;ajc;ajrKh 133
ialkaOh m ρ = mßudj v ialkaOh ialkaOh ialkaOh >k;ajh = = = = ialkaOh mßudj tall mßudjla 1 >k;ajfha w¾: ±laùu' tall mßudjl ialkaOh >k;ajh f,i y÷kajhs' ialkaOh uksk SI tallh kg >k;ajh uksk uksk SI tallh = = 3 = kgm-3 mßudj uksk SI tallh m ^17& >k;ajh =
^18& ialkaOh = >k;ajh × mßudj
m iñlrKfha m Wla; l< úg m = ρ v v tkï" ialkaOh = >k;ajh × mßudj fõ' m m ρ = m = ρ v v= v ρ ρ =
^19& U = vρg
U= Wvql=re f;rmqï n,h v = jia;=fõ mßudj ρ = øjfha >k;ajh g = .=re;ajc;ajrKh iïmQ¾Kfhka øjhla ;=< jia;=jla .s,S mj;sk úg we;s jk Wvql=re f;rmqï n,h fiùug U = vρg iólrKh fhdod .; yel'
^20& oDIH nr = mg – U, mg = jia;=fõ nr"
U = Wvql=re f;rmqï n,h
øjhla u; jia;=j mdfõ kï jia;=fõ nr" jia;=j u; we;s jk Wvql=re f;rmqï n,h ^ U & g iudk jk ksid" oDIH nr = mg - U mg = U ksid oDIH nr = Y=kH fõ' fï ksid iïmQ¾Kfhkau øjhla ;=< .s,S mj;sk jia;=jlg muKlaa oDIH nrla we;'
iólrK j¾.SlrKh fN!;sl úoHdj hgf;a tk úúO iólrK fjka fjka jYfhka wOHhkh lsÍu myiq fõ' m%Odk iólrK j¾. 5la we;' tkï"
1' w¾: ±laùu u; mokï jQ iólrK 2' wkqmd; u; mokï jQ iólrK 3' mÍlaIK u; mokï jQ iólrK 4' jHq;amkak lsÍu u; mokï jQ iólrK 5' ffioaOdka;sl wdfmdayk u; mokï jQ iólrK 134
1' w¾: ±laùu u; mokï jQ iólrK
fN!;sl rdYskaj, w¾: ±laùu u; mokï ù f.dv k.d ;sfnk iólrK fuu j¾.hg wh;a fõ' WodyrK 01' ztall ld,hl§ .uka lrk ÿr m%udKh fõ.h fõZ hk w¾: ±laùu iólrKhla njg m;a l< úg" ÿr fõ.h = hk iólrKh ,efí' ld,h WodyrK 02' ztall lafIa;%M,hla u; l%shd lrk n,h mSvkh fõZ hk w¾: ±laùu iólrKhla njg m;a l< úg" n,h mSvkh = hk iólrKh ,efí' lafIa;%M,h WodyrK 03' ztall mßudjl ^mßud tall ztllz& ialkaOh >k;ajh fõ' hk w¾: ±laùu iólrKhla njg m;a l< úg" ialkaOh >k;ajh = hk iólrKh ,efí' mßudj WodyrK 04' ztall ld,hl§ isÿjk úia:dmkh m%fõ.h fõZ' hk w¾: ±laùu iólrKhla njg m;a l< úg" úia:dmkh m%fõ.h = hk iólrKh ,efí' ld,h ie,lsh hq;=h- fuys§ tall mßudjl ^mßud tall ztllz& hk w§ ;ekaj, fhdok zlZ fyda zgZ hkafkka fn§u woyia flf¾' WodyrK 05' zhï jia;=jla u; n,hla l%shd lrk úg n,h ksid jia;=j p,kh fõ kï tu n,h yd n,fha Wmfhda.S ,laIHh úia:dmkh jQ ÿr w;r .=Ks;h ld¾hh f,io w¾: olajk ksid"
zld¾hh = n,h « n,fha Wmfhda.S ,laIHh úia:dmkh jQ ÿrZ hk iólrKh ,efí'
2' wkqmd; u; mokï jQ iólrK
we;eï fN!;sl rdYSka wkqmd; f,i w¾: olajd we;' tjeks iólrK wkqmd; u; mokï jQ iólrK kï fõ' WodyrK 01' m%;sodk Yla;sh ld¾hlaIu;djh = « 100 % m%odk Yla;sh m%;sodk Yla;sh" m%odk Yla;shg ork wkqmd;h ld¾hlaIu;djh f,i w¾: olajhs' WodyrK 02' Ndrh hdka;% jdish = wdhdih Ndrh" wdhdihg ork wkqmd;h hdka;% jdish f,i w¾: olajhs' WodyrK 03' wdhdih p,kh jQ ÿr m%fõ. wkqmd;h = Ndrh p,kh jQ ÿr 135
wdhdih p,kh jQ ÿr" Ndrh p,kh jQ ÿrg ork wkqmd;h m%fõ. wkqmd;h f,i w¾: olajhs' WodyrK 04' hï øjHhl >k;ajh idfmalaI >k;ajh = c,fha >k;ajh hï øjHhl >k;ajh" c,fha >k;ajhg ork wkqmd;h idfmalaI >k;ajh f,i w¾: olajhs' ie,lsh hq;=h- ld¾hlaIu;dj jeks wkqmd; zNd.hlaZ f,i fyda zm%;sY;hlaZ f,i fyda olajhs' Nd.h - uq¿ m%udKh 1la fyda f,i .;a úg Bg idfmalaI hï m%udKhl w.h Nd.h f,i y÷kajhs'
WodyrKh - m%odk Yla;sh 5000 J o m%;sodk Yla;sh 2000 J o jk úg m%;sodk Yla;sh 2000 J 2 ld¾hlaIu;dj = = = = 0.4 m%odk Yla;sh 5000 J 5 fuys§ ld¾hlaIu;dj zNd.hlaZ f,i olajd we;' m%;sY;h - uq¿ m%udKh 100 la f,i .;a úg Bg idfmalaIj hï m%udKhl w.h m%ldY l< úg th m%;sY;hla f,i y÷kajhs' m%;sY;h Nd.h « 100
WodyrKh - by; hka;%fha ld¾hlaIu;djh = 0.4 «100 = 40%
3' mÍlaIK u; mokï jQ iólrK
mÍlaIK rdYshla isÿ lr ,nd .;a f;dr;=re wvx.= jk iñlrK mÍlaIK u; mokï jQ iólrK f,i y÷kajhs' fuu iñlrK u.ska lsishï fN!;sl rdYshla flfrys n,mdk fN!;sl rdYSka fudkjdo lshd fidhd .; yel' Wod' ^01& øj l|la u.ska we;s lrk mSvkh P=hρg iñlrKh weiqfrka øj l|l mSvkh flfrys n,mdk idOl fidhd.; yel' tu idOl kï" 1' øj lf|a Wi (h) 2' øjfha >k;ajh (ρ) 3' .=re;ajc ;ajrKh (g) Wod' ^02& wix;=,s; n,h F = ma iñlrKh u.ska ;ajrKh (a) flfrys n,mdk idOl fidhd .; yel' ;ajrKfhys úYd,;ajh" jia;=fõ ialkaOh yd jia;=j u; l%shd lrk wix;=,s; n,h u; r|d mj;sk nj fuu iñlrKfhka ,nd .; yel' fuys (a) ;ajrKh fyda ukaokh úh yels ksid ;ajrKfhka fyda ukaokfhka p,kh jk jia;+ka u; wix;=,s; n, l%shd lrk nj ;SrKh l< yel' Wod' ^03& Odrdjla f.k hk ikakdhlhl ;dmhg mßj¾;kh jk Yla;s m%udKh H = I2Rt iólrKh u.ska ;dmhg mßj¾;kh jk Yla;sh flfrys n,mdk idOl ,nd .; yel' tkï" 1' Odrdfõ úYd,;ajh (I) 2' ikakdhlfha m%;sfrdaOh (R) 3' Odrdj .,d .sh ld,h (t) Wod' ^04& Q = mc θ iñlrKh weiqfrka øjHhlg ;dmh ,nd §fï§ isÿ jk WIaK;ajfha jeä ùu ^θ& flfrys n,mdk idOl fidhd.; yel' tkï" 1' øjHfha ialkaOh ^m& 2' øjHfha úYsIag ;dm Odß;dj ^c& 3' øjHhg ,nd fok ;dm m%udKh ^ Q & fï wdldrhg mÍlaIK u; mokï jQ iólrKj,ska jeo.;a f;dr;=re /ila ,nd .; yel' 136
4' jHq;amkak lsÍu u; mokï jQ iólrK
fjk;a iólrK folla fyda lSmhla tlg iïnkaO lr ,nd .kakd iólrK fï kñka y÷kajhs' m Wod' 01 ρ = v iólrKh iy jia;=jl nr = mg iólrK fol tlg iïnkaO lr >k;ajh yd mßudj weiqßka jia;=jl nr fidhd .; yel' m ρ = yd m = vρ ksid jia;=jl nr =mg = vρg fõ' v jia;=jl nr = vρg v = jia;=fõ mßudj ρ = jia;=j idod ;sfnk øjHfha >k;ajh V Wod' 02 H = I2Rt iy R = hk iólrK fol tlg iïnkaO l< úg" I V H = I2 t H = VIt iñlrKh ,efí' I V Wod' 03 H = VIt iñlrKh iy I = iñlrK fol tlg iïnkaO l< úg" R V V2 H= V t, H = t iñlrKh ,efí' R R fï wdldrhg iñlrK tlsfklg iïnkaO lr iñlrK /ila ,nd .; yel' tkï úúO jHq;amkak lsÍu ;=