Karfin kusurwa

Karfin kusurwa

Adadin motsin juyawa, wanda yayi daidai da taro (m) a cikin motsi na layi, shine lokacin inertia (I). Adadin motsin juyawa, wanda yayi daidai da saurin (v) a cikin motsi na layi, shine saurin kusurwa (ω). Don haka, abin da ke juyawa yana da motsin kusurwa wanda za'a iya ƙididdige shi ta amfani da lissafi:

L = I ω

L = ƙarfin kusurwa (kg m ² / s), I = lokacin inertia (kg m ² ), ω = saurin kusurwa (rad/s)

Misalan matsalolin da ke tattare da motsi na kusurwa

Misalin matsalar 1.

Kwayar da ke da nauyin gram 0.5 tana motsawa a cikin da'ira tare da saurin kusurwa mai 2 rad/s. Kayyade saurin kusurwar ƙwayar idan radius na hanyar ƙwayar ta kasance cm 10.

Magani:

Lokacin inertia na barbashi:

I = mr ² = (0.5 x 10 ^-3 kg)(1 x 10 ^-1 m) ² = (0.5 x 10 ^-3 kg)(1 x 10 ^-2 m ² ) = 0.5 x 10 ^-5 kg ​​​​m ²

Gudun kusurwar ƙwayar:

ω = 2 rad/s

Motsin kusurwa na barbashi:

L = (0.5 x 10 ^-5 kg ​​​​m ² )(2 rad/s) = 1 x 10 ^-5 kg ​​​​m ² / s

Dokar kiyaye ƙarfin kusurwa

Dokar kiyaye ƙarfin kusurwa ta bayyana cewa idan lokacin ƙarfi da ke fitowa a jikin mai tauri lokacin juyawa sifili ne, to ƙarfin kusurwa na jikin mai tauri lokacin juyawa koyaushe yana nan daram. Ana iya samo dokar kiyaye ƙarfin kusurwa ta hanyar lissafi ta hanyar gyara lissafin dokar Newton ta biyu ta ƙarfin kusurwa. Ga lissafin dokar Newton ta biyu akan ƙarfin kusurwa:

Idan lokacin ƙarfin da aka samu ya zama sifili, to lissafin da ke sama zai canza zuwa:

I _t = lokacin ƙarshe na inertia, I _o = lokacin farko na inertia, ω _t = saurin kusurwa na ƙarshe, ω _o = saurin kusurwa na farko, L _t = ƙarfin kusurwa na ƙarshe, L _o = ƙarfin kusurwa na farko.

Misalan matsalolin dokar kiyaye ƙarfin kusurwa

Misalin matsalar 1.

Faifan silinda mai kama da juna yana juyawa da farko akan axis ɗinsa tare da saurin 4 rad/s. Girma da radius na faifai sune kilogiram 1 da mita 0.5. Idan an sanya zobe a saman farantin wanda ke da nauyi da radius na 0.2 kg da mita 0.1 da kuma tsakiyar zoben,

a saman tsakiyar faifai, to faifan da zobe za su juya tare da saurin kusurwa na…..

Magani:

Lokacin rashin ƙarfin silinda mai ƙarfi: I = 1⁄2 mr ² = 1⁄2 (1 kg)(0.5 m) ² = (0.5)(0.25) = 0.125 kg ^m2

Lokacin rashin ƙarfin zobe: I = mr ² = (0.2 kg)(0.1 m) ² = (0.2)(0.01) = 0.002 kg m ²

Motsin kusurwa na farko (L ₁ ) = Motsin kusurwa na ƙarshe (L ₂ )

I ₁ ω ₁ = I ₂ ω ₂

(0.125 kg m ² )(4 rad/s) = (0.125 kg m ² + 0.002 kg m ² )(ω ₂ )

(0.5) = (0.127)(ω ₂ )

ω ₂ = 0.5: 0.127

ω ₂ = 4 rad/s

Tambayoyi da amsoshi na ra'ayi

1. Menene motsin kusurwa?
   amsa: Motsin kusurwa wani ma'auni ne na adadin juyawar da abu yake da shi, idan aka yi la'akari da nauyinsa da siffarsa. Wannan shine misalin juyawar motsi na layi.
2. Ta yaya ake tantance ƙarfin kusurwa ta hanyar lissafi?
   amsa: An bayyana motsin kusurwa (L) a matsayin $L=r\times p$, inda $r$ shine vector na matsayi daga axis na juyawa zuwa wurin amfani, kuma $p$ shine saurin layi.
3. Shin ƙarfin kusurwa (angle momentum) adadi ne na sikelin ko kuma yawan vector?
   amsa: Motsin kusurwa shine adadin vector.
4. Menene ma'aunin ƙarfin kusurwa a cikin tsarin SI?
   amsa: Naúrar ƙarfin kusurwa a cikin tsarin SI shine murabba'in kilogiram a kowace daƙiƙa (kg·m)²/s).
5. Ta yaya ake kiyaye ƙarfin kusurwa?
   amsa: A cikin tsarin rufewa ba tare da ƙarfin juyi na waje ba, jimlar ƙarfin kusurwa yana ci gaba da kasancewa akai-akai.
6. Menene ƙa'idar kiyaye ƙarfin kusurwa?
   amsa: Ka'idar ta bayyana cewa jimlar motsin kusurwa na tsarin rufewa yana nan daram sai dai idan karfin juyi na waje ya yi aiki da shi.
7. Ta yaya juyawar mai wasan tsere kan kankara ke da alaƙa da motsin kusurwa?
   amsa: Idan mai wasan tsere kan kankara ya ja hannayensa da ƙafafunsa kusa da jikinsa yayin juyawa, yana rage lokacin da yake yin atisaye, wanda hakan ke sa shi juyawa da sauri. Wannan yana nuna kiyaye saurin kusurwa.
8. Menene alaƙar da ke tsakanin ƙarfin juyi da ƙarfin kusurwa?
   amsa: Karfin juyi ($\tau$) shine saurin canjin motsin kusurwa. A lissafi, τ=dL/sec​.
9. Shin zai yiwu abu ya sami ƙarfin kusurwa ba tare da saurin kusurwa ba?
   amsa: A'a, idan abu yana da ƙarfin kusurwa, yana da saurin kusurwa. Girma da alkiblar ƙarfin kusurwa sun dogara ne akan lokacin inertia da saurin kusurwa.
10. Ta yaya lokacin inertia ke shafar motsin kusurwa?
    amsa: Motsin kusurwa (L) samfurin lokacin inertia (I) da saurin kusurwa (ω) shine. $L=I\omega$, ƙaruwa a lokacin inertia ga wani saurin kusurwa da aka bayar zai ƙara saurin kusurwa.
11. Mene ne bambanci tsakanin motsi na kusurwar juyawa da kuma motsi na kusurwar orbital a cikin makanikan quantum?
    amsa: Motsin kusurwar juyawa siffa ce ta ƙwayoyin halitta, kamar electrons, kuma ba ta tasowa daga motsi a sararin samaniya. Motsin kusurwar juyawa, a gefe guda, yana tasowa ne daga motsi na ƙwayar cuta a kusa da wani wuri na tsakiya.
12. Wace rawa ƙarfin kusurwa ke takawa wajen samar da tsarin duniyoyi?
    amsa: Yayin da gajimare na kwayoyin halitta ke rugujewa don samar da taurari da duniyoyi, kiyayewar motsin kusurwa yana sa kayan ya lanƙwasa ya zama faifai a kusa da tauraron da aka haifa, wanda daga ciki duniyoyi za su iya samuwa.
13. Ta yaya na'urar gyroscope ke kula da yanayin sararin samaniya?
    amsa: Gyroscope mai juyawa yana da ƙarfin kusurwa. Lokacin da ƙarfin juyawa na waje yayi ƙoƙarin canza yanayinsa, gyroscope yana motsawa, ko canza axis na juyawarsa, a cikin alkibla daidai da ƙarfin juyawa da aka yi amfani da shi saboda kiyaye ƙarfin kusurwa.
14. Me ke haifar da koma-baya a saman da ke juyawa?
    amsa: Ragewar karfin juyi a saman da ke juyawa yana faruwa ne sakamakon karfin juyi saboda nauyi da ke aiki a tsakiyar taro na saman, wanda aka daidaita shi daga inda yake juyawa. Wannan karfin juyi yana haifar da canji a alkiblar karfin juyi na saman, wanda ke sa shi yin aiki.
15. Me yasa taurari ke lanƙwasa a kan sandunansu?
    amsa: Yayin da taurari ke juyawa, suna fuskantar ƙarfin centrifugal da ke tura abu zuwa waje. Wannan tasirin ya fi ƙarfi a equator fiye da sandunan, wanda ke haifar da siffar oblate, ko kuma mai faɗi.
16. Ta yaya ma'aunin kusurwa yake da alaƙa da dokar Kepler ta biyu?
    amsa: Dokar Kepler ta biyu ta bayyana cewa sashin layi da ke haɗuwa da duniya da Rana yana share wurare daidai gwargwado a cikin tazara mai daidaito. Wannan sakamako ne na kiyaye ƙarfin kusurwa.
17. Me yasa kyanwa take sauka a ƙafafunta idan ta faɗi?
    amsa: Kuraye suna amfani da kiyaye motsin kusurwa. Ta hanyar karkatar da jikinsu da kuma canza lokutan rashin kuzarinsu a sararin sama, suna iya sake tsara kansu don su sauka a ƙafafunsu ba tare da karya dokokin kiyayewa ba.
18. Ta yaya yanayin maganadisu yake da alaƙa da motsin kusurwa a cikin electrons?
    amsa: Lokacin maganadisu na electron yana tasowa ne daga juyawarsa da kuma motsin kusurwar orbital. Lokacin maganadisu yana daidai da motsin kusurwar electron.
19. Menene motsin kusurwa dangane da daidaitattun daidaiton polar?
    amsa: A cikin daidaitattun daidaiton polar, ƙarfin kusurwa $L$ don ana iya bayyana taro mai yawa kamar haka $L=mrv$, inda $m$ shine taro, $r$ shine nisan radial, kuma $v$ shine saurin tangential.
20. Me ke faruwa da motsin kusurwa na tsarin rufewa idan abubuwa biyu suka yi karo?
    amsa: A cikin tsarin rufewa inda abubuwa biyu suka yi karo, jimlar ƙarfin kusurwa kafin karo daidai yake da jimlar ƙarfin kusurwa bayan karo, idan aka yi la'akari da cewa babu ƙarfin juyi na waje da ke aiki akan tsarin.

Matsaloli da Magani

1. Matsala: Matsakaicin ma'auni m = 2 kg yana motsawa tare da saurin v = 3 m/s a cikin da'irar radius r = 4 m. Lissafa motsin kusurwarsa game da tsakiyar da'irar.

Magani: $L=m\times r\times v$ = 2×4×3=24 kg.m ² / s

2. Matsala: Wani yanki mai ƙarfi mai nauyin kilogiram 3 da radius mai mita 0.5 yana juyawa a kusa da diamita tare da saurin kusurwa ω = 2 rad/s. Menene ƙarfin kusurwarsa?

Magani: Don wani yanki mai ƙarfi, I = 2/5 m r ²

$L=I\times \omega$

L=2/5×3×(0.5) ² ×2=0.6 kg.m ² /s

3. Matsala: Nauyin maki 3 kg yana tafiya da saurin mita 4/s a layi madaidaiciya. Menene saurin kusurwarsa game da maki 5 daga layin motsinsa?

Magani:

$L=m\times v\times r$

L=3×4×5=60 kg.m ² /s

4. Matsala: Faifan da ke da nauyin kilogiram 5 da radius mai mita 3 yana juyawa tare da saurin kusurwa na 2 rad/s. Lissafa ƙarfin kusurwarsa.

Magani: Ga faifan, I = 1/2 m r ²

$L=I\times \omega$

L=1/2×5×3 ² ×2=45 kg.m ² /s

5. Matsala: Sanda mai tsayin mita 4 da nauyi kilogiram 2 tana juyawa kusan ƙarshenta ɗaya tare da saurin kusurwa na 3 rad/s. Nemo ƙarfin kusurwarsa.

Magani: Ga sanda kusan ƙarshen ɗaya, I = 1/3 m l ²

$L=I\times \omega$

L=1/3×2×4 ² ×3=96 kg.m ² /s

6. Matsala: Menene canjin motsin kusurwa idan jikin da ke juyawa ya ninka saurin kusurwarsa sau biyu?

Magani: Alaƙar da ke tsakanin ƙarfin kusurwa (L) da saurin kusurwa (ω) ita ce $L=I\times \omega$ . Idan an ninka ω, ƙarfin kusurwar shi ma zai ninka.

7. Matsala: Idan radius na hanyar da'ira ta ƙwayar cuta mai motsi ya ninka yayin da yake kiyaye saurin gudu daidai, ta wace hanya ce saurin kusurwa ke canzawa?

Magani: Motsin kusurwa, $L=m\times r\times v$ . Idan r ya ninka, L ma zai ninka.

8. Matsala: Yaron da nauyinsa ya kai kilogiram 30 yana tsaye a gefen wani abin hawa mai hawa uku wanda yake hutawa. Radius na abin hawa uku yana da mita 2. Idan yaron ya fara gudu da gudun mita 2/s dangane da ƙasa a gefen, nemo kusurwar yaron dangane da tsakiyarsa.

Magani: $L=m\times r\times v$ = 30×2×2=120 kg.m ² / s

9. Matsala: Silinda mai nauyin kilogiram 6 da radius mai mita 2 tana juyawa a tsakiyar axis ɗinta tare da saurin kusurwa na 5 rad/s. Lissafa ƙarfin kusurwar sa.

Magani: Ga silinda, I = 1/2 m r ²

$L=I\times \omega$ = 1/2×6×2 ² ×5=120 kg.m ² / s

10. Matsala: Menene sabon motsin kusurwa na jiki idan lokacin inertia ɗinsa ya ragu kuma saurin kusurwarsa ya ninka sau uku?

Magani: Idan aka ba $L=I\times \omega$ , idan an raba $ni$ biyu kuma $\omega$ an ninka shi sau uku, sabon $L$ zai zama 1/2×3 = 1.5 × na asali.

11. Matsala: Jump mai nauyin kilogiram 2 da radius mai mita 1 yana juyawa tare da saurin kusurwa na 4 rad/s. Lissafa motsin kusurwarsa.

Magani: Don yin tsalle, I = m r ²

$L=I\times \omega$

L=2×1 ² ×4=8 kg.m ² / s

12. Matsala: Fanka tana da ruwan wukake masu tsawon mita 0.5 da jimillar nauyin kilogiram 1. Idan tana juyawa da saurin kusurwa na 10 rad/s, a tantance ƙarfin kusurwar ta.

Magani: Idan aka ɗauka cewa ruwan fanka suna kama da sanduna suna juyawa a kusa da ƙarshen ɗaya, I = 1/3 ​m l ²

$L=I\times \omega$ = 1/3×1×0.5 ² ×10=0.833 kg.m ² / s

13. Matsala: Lissafa motsin kusurwar duniya game da axis ɗinta saboda juyawarta. An bayar da: Nauyin ƙasa = 5.972×10 ²⁴ kg , Radius $r=6371\text{km}$ , ω = 7.27 × 1 0 ⁻⁵ rad/s.

Magani: Idan aka ɗauka cewa duniya ta zama wani yanki mai ƙarfi, I = 2/5 ​m r ²

$L=I\times \omega$ =2/5×5.972×10 ²⁴ ×(6371×10 ³ ) ² ×7.27×10 ^-5

L ≈ 7.07 × 1 0 ³³ kg.m ² / s

14. Matsala: Yaro yana motsawa zuwa tsakiyar dandamali mai juyawa. Shin saurin kusurwa yana ƙaruwa, yana raguwa ko kuma yana ci gaba da kasancewa iri ɗaya?

Magani: Ana kiyaye ƙarfin kusurwa. Yayin da yaron ke motsawa zuwa tsakiya, lokacin inertia yana raguwa. Don kiyaye ƙarfin kusurwa, saurin kusurwa yana ƙaruwa. Don haka, ƙarfin kusurwa yana nan iri ɗaya.

15. Matsala: Idan kuzarin motsi na jikin da ke juyawa ya ninka sau biyu, da wane dalili ne saurin kusurwarsa ke canzawa?

Magani: Makamashin Kinetic K = 1/2 ​I ω ² da kuma ƙarfin Angular $L=I\omega$ . Idan makamashin kinetic ya ninka kuma $I$ ya kasance mai daidaito, ω ² zai ninka. Wannan yana nufin cewa $\omega$ ya karu da factor na √ 2 ​.$Don\; haka,\; L$ kuma yana ƙaruwa da factor na √ 2 ​.

16. Matsala: Tayar keke mai radius 0.3 m tana da gefen nauyi na kilogiram 1.5. Idan tana juyawa da saurin kusurwa na 5 rad/s, nemo ƙarfin kusurwar ta.

Magani: Idan aka ɗauka gefen a matsayin ƙugiya, I = m r ²

$L=I\times \omega$ =1.5×0.32×5=6.75 kg.m ² /s

L = 1.5 × 0. 3 2 × 5 = 6.75 kg.m ² / s

17. Matsala: Mai wasan tsere kan kankara tana jan hannunta zuwa ciki yayin da take juyawa. Me zai faru da saurin kusurwarta?

Magani: Ana kiyaye ƙarfin kusurwar mai wasan tsere. Ta hanyar jawo hannunta, tana rage lokacin da take yin atisaye, don haka tana juyawa da sauri (ƙarin saurin kusurwar) don kiyaye ƙarfin kusurwarta.

18. Matsala: Keken tashi yana da lokacin inertia na 0.2 kg.m ² kg.m ² kuma yana juyawa da saurin kusurwa na 10 rad/s. Nemo motsin kusurwarsa.

Magani:

$L=I\times \omega$ = 0.2×10=2 kg.m ² /s

L = 0.2 × 10 = 2 kg.m ² / s

19. Matsala: Tsarin yana da karfin juyi na waje na sifili da ke aiki a kansa. Me za ku iya cewa game da karfin kusurwarsa?

Magani: Idan karfin karfin waje na yanar gizo sifili ne, to bisa ga kiyaye karfin kusurwa, jimlar karfin kusurwa na tsarin zai kasance iri daya.

20. Matsala: Matsakaicin maki m = 5 kg yana motsawa tare da saurin v = 3 m/s a tsaye zuwa layin da ke haɗa shi da wurin P. Idan nisan da ke tsakanin wurin P shine 4 m, nemo kusurwar motsinsa game da wurin P.

Magani:

$L=m\times r\times v$ = 5×4×3=60 kg.m ² /s

L = 5 × 4 × 3 = 60 kg.m ² / s