Фізика, збірник задач

Fhf_gl Lg_jpL€ lLeZ \L^ghkgh aZ^Zgh€ hkL

n

J = ∑ mi ri2 ,

i=1

^_ ri – \L^klZgv _e_f_glZ fZkb mi \L^ hkL h[_jlZggy

Fhf_gl Lg_jpL€ lhgdh]h h[jmqZ \L^ghkgh hkL ydZ i_ji_g^b- dmeyjgZ ^h iehsbgb h[jmqZ • ijhoh^blv q_j_a ch]h p_glj

J = mR2.

^_ R – jZ^•mk h[jmqZ

Fhf_gl Lg_jpL€ ^bkdZ \L^ghkgh hkL ydZ i_ji_g^bdmeyjgZ ^h iehsbgb ^bkdZ • ijhoh^blv q_j_a ch]h p_glj

J= 21 mR 2 .

Fhf_gl Lg_jpL€ kljb`gy \L^ghkgh hkL ydZ ijhoh^blv q_j_a k_j_^bgm kljb`gy L i_ji_g^bdmeyjgZ ^h gvh]h

J = 121 m 2 ,

^_ • – ^h\`bgZ kljb`gy

Fhf_gl Lg_jpL€ dmeL \L^ghkgh hkL sh ijhoh^blv q_j_a €€ p_glj

J= 52 mR 2 .

Fhf_gl •fimevkm l•eZ \•^ghkgh hk• z

Lz = Jω,

^_ ω – dmlh\Z r\b^d•klv h[_jlZggy l•eZ

J•\gyggy ^bgZf•db h[_jlZevgh]h jmom l•eZ gZ\dheh g_jmohfh€ hk•

G / = 0 -ε = 0 GW

^_ εz – ijh_dp•y \_dlhjZ dmlh\h]h ijbkdhj_ggy ε l•eZ gZ \•kv z.

AZdhg a[_j_`_ggy fhf_glm Lfimevkm aZfdg_gh€ kbkl_fb

^_ Li (Lzi) – fhf_gl Lfimevkm i-]h lLeZ \L^ghkgh aZ^Zgh]h € p_gljZ hkL

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Jh[hlZ klZeh]h fhf_glm kbeb F sh ^L} gZ lLeh yd_ h[_jlZ}lvky

A = Mϕ,

^_ ϕ – dml ih\hjhlm lLeZ

Fbll}\Z ihlm`gLklv sh jha\b\Z}lvky i•^ qZk h[_jlZggy lLeZ

N= Mω.

DLg_lbqgZ _g_j]Ly lLeZ yd_ h[_jlZ}lvky

•f H[qbkeblb fhf_gl •g_jp•€ Jk kljb`gy \•^ghkgh hk• sh

ijhoh^blv q_j_a k_j_^bgm kljb`gy i_ji_g^bdmeyjgh ^h gvh]h • fhf_gl •g_jp•€ J \•^ghkgh hk• sh ijhoh^blv q_j_a d•g_pv kljb`gy

d] f2 d] f2)

h[_jlZ}lvky a qZklhlhx Q h[ k Dheb \bfdgmeb ijb\•^ fZoh\bd ajh[b\rb N = 200 h[_jl•\ i•^ ^•}x l_jly amibgb\ky

<bagZqblb fhf_gl kbeb l_jly FL sh ^•y\ gZ fZoh\bd G f

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sh ijhoh^blv q_j_a p_glj ^bkdZ Fhf_gl kbeb l_jly sh ^•} gZ ^bkd ijyfh ijhihjp•cgbc ^h dmlh\h€ r\b^dhkl• FL <&, ^_

< G f k jZ^ AgZclb dmlh\m r\b^d•klv &2 ^bkdZ q_j_a qZk t = = k i•key ijbibg_ggy ^•€ ah\g•rgvh]h fhf_glm kbe Kd•evdb h[_j-

l•\ ajh[blv ^bkd mijh^h\` pvh]h qZkm" jZ^ k h[

5.7.Ih ]hjbahglZevghfm klhe• fh`_ dhlblbky [_a dh\aZggy kmp•evgbc pbe•g^j fZkhx m d] gZ ydbc gZfhlZgZ gbldZ >h \•evgh]h

d•gpy gbldb ydbc i_j_dbgmlbc q_j_a e_]dbc [ehd i•^\•r_gbc \ZglZ` lZdh€ kZfh€ fZkb m <bagZqblb ijbkdhj_ggy \ZglZ`m .1 • kbem l_jly FL f•` pbe•g^jhf • klhehf f k2 G

5.8. <Ze m \b]ey^• kmp•evgh]h pbe•g^jZ fZkhx m1 d] gZkZ^`_gbc gZ ]hjbahglZevgm \•kv GZ pbe•g^j gZfhlZgbc rgmj ^h \•evgh]h d•gpy ydh]h i•^\•r_gbc \ZglZ` fZkhx m2 d] A ydbf ijbkdh- j_ggyf . [m^_ himkdZlbky \ZglZ`" f k2)

5.9.GZ fZoh\bd ^•Zf_ljhf ' f gZfhlZgbc g_\Z]hfbc rgmj ^h

\•evgh]h d•gpy ydh]h ijb\¶yaZgbc \ZglZ` fZkhx P d] H[_j- lZxqbkv j•\ghijbkdhj_gh i•^ ^•}x kbeb ly`•ggy \ZglZ`m fZoh\bd

5.10. >\Z \ZglZ`• fZkZfb m1 = 5,2 d] • m2 d] a¶}^gZg• g_\Z]hfhx gbl- dhx ydZ i_j_dbgmlZ q_j_a g_jmohfbc [ehd m \b]ey^• h^ghj•^gh]h km-

p•evgh]h ^bkdZ fZkhx P d] G_olmxqb l_jlyf \ hk• [ehdZ \bagZ-

qblb ijbkdhj_ggy . a ydbf [m^mlv jmoZlbky \ZglZ`• • kbeb gZly]m L1 • L2 gbldb ih h[b^\Z [hdb [ehdZ f k2 G G

5.11. GZ [ZjZ[Zg jZ^•mkhf 5 f fhf_gl •g_jp•€ ydh]h J d] f2, gZfhlZgbc rgmj ^h d•gpy ydh]h ijb\¶yaZgbc \ZglZ` fZkhx P d]. I_j_^ ihqZldhf h[_jlZggy [ZjZ[ZgZ \bkhlZ \ZglZ`m gZ^ i•^eh]hx

Kf. <bagZqblb d•g_lbqgm _g_j]•x ?d \ZglZ`m \ fhf_gl m^Zjm

h[ i•^eh]m qZk t aZ ydbc \ZglZ` himklblvky ^h i•^eh]b • kbem gZly]m L rgmjZ L_jlyf ag_olm\Zlb >` k G

5.12. H[jmq lZ kmp•evgbc ^bkd h^gZdh\h€ fZkb dhlylvky [_a ijhdh\am\Zggy a h^gZdh\hx r\b^d•klx D•g_lbqgZ _g_j]•y ^bkdZ ?d^ >`. AgZclb d•g_lbqgm _g_j]•x ?dh[ h[jmqZ >`

23

5.13.H[jmq kmp•evgbc ^bkd • dmey kdhqmxlvky [_a ijhdh\am\Zggy a ihobeh€ iehsbgb a dmlhf gZobem . = 300 IhqZldh\• r\b^dhkl• l•e

^hj•\gxxlv gmex <bagZqblb e•g•cg• ijbkdhj_ggy p_glj•\ fZk pbo l•e .h[ f k2 .^ f k2 .d f k2)

5.14. H^ghj•^gbc lhgdbc kljb`_gv ^h\`bghx • f ydbc aZdj•ie_-

gbc lZd sh \•g fh`_ h[_jlZlbkv gZ\dheh ]hjbahglZevgh€ hk• ydZ

ijhoh^blv i_ji_g^bdmeyjgh ^h kljb`gy q_j_a h^bg a ch]h d•gp•\ \•^\h^ylv \•^ \_jlbdZevgh]h iheh`_ggy gZ dml . = 600 • ihl•f

\•^imkdZxlv <bagZqblb r\b^d•klv v gb`gvh]h d•gpy kljb`gy \ fhf_gl ijhoh^`_ggy gbf iheh`_ggy j•\gh\Z]b f k

5.15. He•\_pv ^h\`bghx • f ydbc klhy\ \_jlbdZevgh iZ^Z} gZ kl•e Ydm dmlh\m & lZ e•g•cgm v r\b^dhkl• [m^mlv fZlb \ d•gp• iZ^•ggy k_j_^bgZ lZ \_jog•c d•g_pv he•\py" (&1= &2 jZ^ k v1 f k; v2 f k

h[_jlZ}lvky a qZklhlhx Q h[ k

gZ\dheh hk• sh ijhoh^blv q_j_a €€ p_glj Ydm jh[hlm : g_h[o•^gh

\bdhgZlb sh[ a[•evrblb dmlh\m r\b^d•klv dme• \^\•q•" =mklbgZ aZe•aZ ! d] f3. >`

5.17.GZ ieZlnhjf• m \b]ey^• ^bkdZ kb^blv ex^bgZ • ljbfZ} m \bly]gmlbo jmdZo ]bj• fZkhx ih P d] dh`gZ <•^klZgv \•^ dh`gh€ ]bj• ^h hk•

h[_jlZggy ieZlnhjfb •1 f IeZlnhjfZ h[_jlZ}lvky a qZklhlhx n1 h[ k \•^ghkgh hk• sh ijhoh^blv q_j_a p_glj fZk ex^bgb •

ieZlnhjfb KmfZjgbc fhf_gl •g_jp•€ ex^bgb • ieZlnhjfb \•^ghkgh hk• h[_jlZggy J0 d] f2 Ex^bgZ a[eb`m} jmdb lZd sh \•^klZgv \•^

ieZlnhjfb Ydhx [m^_ e•g•cgZ r\b^d•klv ex^bgb v \•^ghkgh i•^eh]b" f k

24

5.19. >\Z ]mfh\• ^bkdb a `hjkldbfb ih\_jogyfb h[_jlZxlvky gZ\dheh hk_c sh e_`Zlv gZ h^g•c \_jlbdZe• ijbqhfm iehsbgb ^bkd•\

<_jog•c ^bkd iZ^Z} gZ gb`g•c • a¶}^gm}lvky a gbf <bagZqblb dmlh\m r\b^d•klv & ^bkd•\ • af•gm €o d•g_lbqgh€ _g_j]•€ û?d.

jZ^ k >`

F?O:G,QG, DHEB<:GGY

Hkgh\gL nhjfmeb

>bn_j_gp•Zevg_ jL\gyggy ]ZjfhgLqgbo dheb\Zgv L ch]h jha\’yahd

x + ω02 x = 0

x = Asin(ω0t +ϕ0 )

Z[h [ = $cos(ω0W + ϕ0 ),

(ω = ω02 −δ 2 ).

Eh]ZjbnfLqgbc ^_dj_f_gl a]ZkZggy

>bn_j_gp•Zevg_ jL\gyggy \bfmr_gbo dheb\Zgv L ch]h jha\’yahd

x + 2δx +ω02 x = Fmo cosωt, x = Acos(ωt +ϕ),

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I_jLh^ dheb\Zgv ijm`bggh]h fZylgbdZ

T = 2π mk ,

^_ m – fZkZ lLeZ k – `hjkldLklv ijm`bgb

I_jLh^ dheb\Zgv fZl_fZlbqgh]h fZylgbdZ

T = 2π g ,

^_ l – ^h\`bgZ fZylgbdZ g – ijbkdhj_ggy \Levgh]h iZ^LggyI_j•h^ dheb\Zgv n•abqgh]h fZylgbdZ

T = 2π mgJ

^_ J – fhf_gl •g_jp•€ fZylgbdZ \•^ghkgh lhqdb €€ i•^\•km • – \•^klZgv f•` lhqdhx i•^\•km • p_gljhf fZk fZylgbdZ

>h\`bgZ o\be•

λ =vT = v

J•\gyggy iehkdh€ o\be• ydZ ihrbjx}lvky \a^h\` hk• x

ξ(x,t) = Acos (ωt −kx +ϕo ),

^_ k – o\bevh\_ qbkeh k = 2λπ ).

=mklbgZ ihlhdm _g_j]•€ sh i_j_ghkblvky o\be_x \_dlhj Mfh\Z

→→

j = wv ,

^_ w – h[¶}fgZ ]mklbgZ _g_j]•€ o\be•

6.1.FZl_j•ZevgZ lhqdZ \•^ghkgh iheh`_ggy j•\gh\Z]b \bdhgm}

]Zjfhg•qg• dheb\Zggy \a^h\` ^_ydh€ ijyfh€ a i_j•h^hf L k • Zfie•lm^hx : f <bagZqblb k_j_^gx r\b^d•klv <v> lhqdb

aZ qZk mijh^h\` ydh]h \hgZ ijhoh^blv reyo sh ^hj•\gx} i_jr•c iheh\bg• Zfie•lm^b ^jm]•c iheh\bg• • mk•c Zfie•lm^• f k

f k f k

26

6.2.FZl_j•ZevgZ lhqdZ a^•ckgx} ]Zjfhg•qg• dheb\Zggy \a^h\` ]hjb-

6.4.GZ l•eh fZkhx P d] ^•} kbeZ ydZ af•gx}lvky aZ aZdhghf

G, & k-1 M fhf_gl qZkm t = 0 af•s_ggy l•eZ \•^ iheh`_ggy j•\gh\Z]b x = 0 • r\b^d•klv v = 0.

>h\_klb sh lZdbc jmo } dheb\Zevgbf <bagZqblb i_j•h^ dheb\Zgv L fZdkbfZevg_ agZq_ggy af•s_ggy xmax • fZdkbfZevg_ agZq_ggy

r\b^dhkl• vmax. k f f k

6.5.FZl_j•ZevgZ lhqdZ h^ghqZkgh [_j_ mqZklv m ^\ho dheb\Zggyo

pbde•qgm qZklhlm & • ihqZldh\m nZam dheb\Zgv 3 Qhfm ^hj•\gx}

af•s_ggy x0 r\b^d•klv v0 • ijbkdhj_ggy .0 \ ihqZldh\bc fhf_gl qZkm t = 0? f k-1 Œ f f k f k2)

6.7.E_]dbc kljb`_gv fh`_ \•evgh dheb\Zlbky gZ\dheh ]hjbahglZevgh€

hk• GZ \•^klZg• 6 kf \•^ hk• gZ gvhfm aZdj•ie_gh g_\_ebdm dmevdm • ^Ze• gZ \•^klZgyo G kf h^gZ \•^ h^gh€ – s_ ^\• lZd• kZf• dmevdb AgZclb i_j•h^ dheb\Zgv p•}€ kbkl_fb k

6.8.GZ ]eZ^_gvdhfm ]hjbahglZevghfm klhe• e_`blv l•eh fZkhx

m1 d] yd_ ijbdj•ie_gh ^h kl•gdb ijm`bghx `hjkld•klx

27

qblb i_j•h^ L dheb\Zgv l•eZ lZ Zfie•lm^m :. k f

6.9.FZl_j•ZevgZ lhqdZ fZkhx P d] \bdhgm} ]Zjfhg•qg• dheb-

\Zggy aZ aZdhghf [ VLQ W Œ kf H[qbkeblb fZdkbfZevgm kbem Fmax sh ^•} gZ lhqdm • ih\gm _g_j]•x ? lhqdb sh dheb-

\Z}lvky fG fd>`

6.10.:fie•lm^Z ]Zjfhg•qgbo dheb\Zgv fZl_j•Zevgh€ lhqdb : f,

ih\gZ _g_j]•y dheb\Zgv ? fd>` Ijb af•s_gg• o \•^ iheh-

kh€^Zevgbo dheb\Zggyo h^gh]h gZijyfm a h^gZdh\bfb Zfie•lm^Zfb : f • h^gZdh\bfb i_j•h^Zfb L k J•agbpy nZa f•` pbfb dheb\Zggyfb 32 – 31 Œ IhqZldh\Z nZaZ h^gh]h a pbo dheb- \Zgv ^hj•\gx} gmex GZibkZlb j•\gyggy j_amevlmxqh]h jmom o

= FRV ŒW Œ

6.12.FZl_j•ZevgZ lhqdZ [_j_ mqZklv h^ghqZkgh m ^\ho \aZ}fgh i_ji_g-

^bdmeyjgbo dheb\Zggyo sh hibkmxlvky j•\gyggyfb [ VLQ &W •

\ VLQ &W GZibkZlb j•\gyggy ljZ}dlhj•€ lhqdb lZ gZjbkm\Zlb ljZ}dlhj•x (16x4 – 16x2 + y2 = 0)

• d•evd•klv dheb\Zgv N yd• \bdhgZ\ fZylgbd k

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6.16.L•eh fZkhx P d] yd_ i•^\•r_g_ ^h ijm`bgb `hjkld•klx N G f \bdhgm} \ ^_ydhfm k_j_^h\bs• ijm`g• dheb\Zggy Eh]Zjbn- f•qgbc ^_dj_f_gl a]ZkZggy dheb\Zgv æ = 0,01 Q_j_a ydbc ijhf•`hd qZkm ût _g_j]•y dheb\Zgv l•eZ af_grblvky m n = 7,4 jZaZ k

6.17.FZl_j•ZevgZ lhqdZ dheb\Z}lvky m \Zdmmf• a pbde•qghx qZklhlhx

6.18.<ZglZ` fZkhx P d] ydbc i•^\•r_gbc gZ ijm`bg• `hjkld•klx N G f dheb\Z}lvky m \¶yadhfm k_j_^h\bs• a dh_n•p•}glhf

hihjm U d] k GZ \_jog•c d•g_pv ijm`bgb ^•} \bfmrmxqZ kbeZ sh af•gx}lvky aZ aZdhghf F = F0 FRV &W ^_ F0 G <bagZqblb

^ey p•}€ dheb\gh€ kbkl_fb dh_n•p•}gl a]ZkZggy / • j_ahgZgkgm Zfie•lm^m :j_a. k-1 f

6.19. :fie•lm^b \bfmr_gbo ]Zjfhg•qgbo dheb\Zgv ^hj•\gxxlv h^gZ h^g•c ydsh pbde•qg• qZklhlb &1 k-1 • &2 k-1 <bagZqblb qZklhlm &j aZ ydh€ Zfie•lm^Z pbo \bfmr_gbo dheb\Zgv } fZdkb- fZevghx k-1)

6.20.L•eh fZkhx m d] a^•ckgx} a]ZkZxq• dheb\Zggy a ihqZldh\hx

Zfie•lm^hx :0 f ihqZldh\hx nZahx 30 = 0 • dh_n•p•}glhf a]ZkZggy / k-1 GZ p_ l•eh ihqZeZ ^•ylb ah\g•rgy i_j•h^bqgZ kbeZ

F i•^ ^•}x ydh€ \klZgh\bebky \bfmr_g• dheb\Zggy J•\gyggy \bfmr_gbo dheb\Zgv fZ} \b]ey^ o ,10 cos(10ŒW – 3Œ f. KdeZklb j•\gyggy a]ZkZxqbo dheb\Zgv l•eZ • j•\gyggy ah\g•rgvh€ i_j•h^bqgh€ kbeb (x = 0,12 e-2,0tcos 10,6π t

6.21.Klj•edZ qmleb\h]h ijbeZ^m dheb\Z}lvky [•ey iheh`_ggy j•\gh\Z]b 2€ ihke•^h\g• djZcg• iheh`_ggy lZd• n1 = 26,4; n2 = 10,7, n3 = 20,5. AgZclb ih^•edm ydZ \•^ih\•^Z} j•\gh\Z`ghfm iheh`_ggx klj•edb ydsh €€ ^_dj_f_gl a]ZkZggy } klZebf m qZk• (16,7)

29

=,>JH>BG:F,D: Hkgh\gL nhjfmeb

1.JL\gyggy g_i_j_j\ghklL kljmf_gy

Sv = const,

^_ S – iehsZ ihi_j_qgh]h i_j_jLam ljm[db l_qL€ υ – r\b^d•klv l_qL€JL\gyggy ;_jgmeeL ^ey L^_Zevgh€ g_klbkdm\Zevgh€ jL^bgb

p + ρv2 2 + ρgh = const,

^_ p – klZlbqgbc lbkd ρ – ]mklbgZ j•^bgb v – r\b^dhklL jL^bgb \ pbo

i_j_jLaZo ρv 2

2

ghfm i_j_j•a• ljm[db l_q•€

3. R\b^dLklv \bl•dZggy jL^bgb a fZeh]h hl\hjm m \L^djblLc rbjh- dLc ihkm^bgL

v = 2gh,

^_ h – ]eb[bgZ gZ ydLc jhaf•s_gh hl\Lj \L^ghkgh jL\gy jL^bgb \ ihkm^bgL

7.1. J•agbpy j•\g•\ \h^b \ ljm[p• IjZg^ley

djm]ebc hl\•j ^•Zf_ljhf G f AgZclb r\b^d•klv v ihgb- `_ggy j•\gy \h^b \ ihkm^bg• \\Z`Zxqb \h^m g_\¶yadhx j•^bghx

ff k

7.3.Pbe•g^jbqgbc [Zd \bkhlhx H f aZih\g_gbc ^h djZ€\ \h^hx

M ^g• [ZdZ ml\hjb\ky hl\•j iehsZ S2 ydh]h m N = 300 jZa•\ f_grZ \•^ iehs• ch]h ihi_j_qgh]h i_j_j•am S1 AZ ydbc qZk t \ky \h^Z \bl_q_ q_j_a hl\•j" k

30