Lub sijhawm ntawm inertia rau cov khoom me me - teeb meem thiab kev daws teeb meem

Lub sijhawm ntawm inertia rau cov khoom me me - teeb meem thiab kev daws teeb meem

1. Ob lub pob txuas nrog ib tug pas nrig, raws li pom hauv daim duab hauv qab no. Tsis quav ntsej txog qhov hnyav ntawm tus pas nrig . Qhov hnyav ntawm lub pob P yog 600 grams thiab qhov hnyav ntawm lub pob Q yog 400 grams. Lub sijhawm inertia ntawm lub kaw lus txog AB yog dab tsi?

Paub:

Lub axis ntawm kev sib hloov yog AB.

m _p = 600 grams = 0.6 kg, m _q = 400 grams = 0.4 kg

r _p = 20 cm = 0.2 m, r _q = 50 cm = 0.5 m

Xav tau: Lub sijhawm ntawm inertia ntawm lub system

Tshuaj:

_Kuv = _mprp2 + ^mqrq2​_​​_​^​

Kuv = (0.6 kg)(0.2 m) ² + (0.4 kg)(0.5 m) ²

Kuv = (0.6 kg)(0.04 m2 ⁾ + (0.4 kg)(0.25 ^m2 )

Kuv = 0.024 kg ^m2 + 0.1 kg ^m2

Kuv = 0.124 kg ^m2

2. Tus pas nrig AB uas hnyav 2-kg tig ib ncig ntawm qhov chaw A, lub sijhawm inertia ntawm tus pas nrig yog 8 kg m2 ^. Yog tig ib ncig ntawm qhov chaw O (AO = OB), lub sijhawm inertia ntawm tus pas nrig yog dab tsi.

Paub:

Qhov hnyav ntawm tus pas AB (m) = 2 kg

Yog tias tig ib ncig ntawm qhov chaw A kom lub vojvoog ntawm kev tig (r) = qhov ntev ntawm AB = r ces lub sijhawm ntawm inertia (I) = 8 kg m ²

Xav tau: Yog tias tig ib ncig ntawm qhov chaw O kom lub vojvoog ntawm kev tig (r) = qhov ntev ntawm AO = qhov ntev ntawm OB = 1/2 r ces lub sijhawm ntawm inertia ntawm tus pas nrig yog dab tsi.

Tshuaj:

Kuv = Mr. ²

8 kg ^m2 = (2 kg) ^r2

8 m2 ⁼ (2) ^r2

r2 = 8 ^m2 / ²

r2 = ^{4 m2}

r = 2 meters

Yog tias tig ib ncig ntawm qhov chaw O yog li ½ r = 1 'meter', ces lub sijhawm ntawm inertia:

Kuv = mr ² = (2 kg)(1 m) ² = (2 kg)(1 m ² ) = 2 kg m ²

3. Ob lub pob uas txuas nrog ib tug pas nrig raws li qhia hauv daim duab hauv qab no. Tsis txhob xav txog qhov hnyav ntawm tus pas nrig. Lub sijhawm inertia ntawm lub cev yog dab tsi.

Paub:

Qhov hnyav ntawm lub pob A (m_A) = 200 grams = 0.2 kg

Qhov hnyav ntawm lub pob B (mB ₎ = 400 grams = 0.4 kg

Qhov nrug ntawm lub pob A thiab lub axis ntawm kev sib hloov (r _A ) = 0

Qhov nrug ntawm lub pob B thiab lub axis ntawm kev sib hloov (r _B ) = 25 cm = 0.25 m

Xav tau: Lub sijhawm ntawm inertia ntawm lub system

Tshuaj:

Lub sijhawm ntawm inertia ntawm pob A:

Kuv _A = (m _A )(r ^A2 ₎ = (0.2)(0) ² = 0

Lub sijhawm ntawm inertia ntawm pob B:

IB = (m _B )(rB2 _{) = (0.4)(} ^0.25 ) ² = (0.4)(0.0625) = 0.025 _kg ^m2

Lub sijhawm ntawm inertia ntawm lub system:

Kuv = Kuv _A + Kuv _B = 0 + 0.025 = 0.025 kg m2 ⁼ 25 x 10 ^-3 kg ^m2

4. Plaub lub khoom me me uas muaj qhov hnyav sib txawv, raws li qhia hauv daim duab hauv qab no. Txheeb xyuas lub sijhawm ntawm inertia ntawm lub kaw lus ncig txoj kab kab rov tav P.

tshuaj

Lub axis ntawm kev tig yog kab kab rov tav P.

Paub:

Qhov hnyav ntawm cov khoom me me A (m _A ) = m

Qhov hnyav ntawm cov khoom me me B (mB ₎ = 2m

Qhov hnyav ntawm cov khoom me me C (mC ₎ = 3m

Kev hla ntawm cov khoom me me D (m _D ) = 4m

Qhov nrug ntawm cov khoom me me A thiab lub axis ntawm kev tig (r _A ) = b

Qhov nrug ntawm cov khoom me me B thiab lub axis ntawm kev tig (r _B ) = b

Qhov nrug ntawm cov khoom me me C thiab lub axis ntawm kev tig (r _C ) = 2b

Qhov nrug ntawm cov khoom me me D thiab lub axis ntawm kev tig (r _D ) = 2b

Xav tau: Lub sijhawm ntawm inertia ntawm lub kaw lus ncig kab rov tav P

Tshuaj:

Kuv = m _A r _A ² + m _B r _B ² + m _C r _C ² + m _D r _D ²

I = (m)(b) ² + (2m)(b) ² + (3m)(2b) ² + (4m)(2b) ²

I = mb ² + 2 mb ² + (3m)(4b ² ) + (4m)(4b ² )

I = mb ² + 2 mb ² + 12 mb ² + 16 mb ²

Kuv = 31 ^mb2

5. Plaub lub khoom me me uas txuas nrog ib tug pas nrig. Tsis txhob xav txog qhov hnyav ntawm tus pas nrig. Txheeb xyuas lub sijhawm ntawm inertia ncig lub axis ntawm kev tig los ntawm cov khoom me me m ₁ thiab m ₂ , raws li qhia hauv daim duab hauv qab no.

Paub Tias

Qhov hnyav ntawm cov khoom me me 1 (m₁) = 1/4 kg

Qhov hnyav ntawm cov khoom me me 2 (m2 ₎ = 1/2 kg

Qhov hnyav ntawm cov khoom me me 3 (m3 ₎ = 1/4 kg

Qhov hnyav ntawm cov khoom me me 4 (m4 ₎ = 1/4 kg

Qhov nrug ntawm cov khoom me me 1 thiab lub axis ntawm kev tig (r ₁ ) = 0

Qhov nrug ntawm cov khoom me me 2 thiab lub axis ntawm kev tig (r ₂ ) = 0

Qhov nrug ntawm cov khoom me me 3 thiab lub axis ntawm kev tig (r ₃ ) = 10 cm = 10/100 m = 1/10 m

Qhov nrug ntawm cov khoom me me 4 thiab lub axis ntawm kev tig (r ₄ ) = 10 cm = 10/100 m = 1/10 m

Xav tau: Lub sijhawm ntawm inertia

Tshuaj:

Kuv = m ₁ r ₁ ² + m ₂ r ₂ ² + m ₃ r ₃ ² + m ₄ r ₄ ²

Kuv = (1/4)(0) ² + (1/2)(0) ² + (1/4)(1/10) ² + (1/4)(1/10) ²

Kuv = 0 + 0 + (1/4)(1/100) + (1/4)(1/100)

Kuv = 1/400 + 1/400

Kuv = 2/400

Kuv = 1/200 ^kg.m2

1. Lub sijhawm ntawm inertia rau ib qho khoom me me yog dab tsi?
   • teb: Rau ib qho khoom me me ntawm lub cev $m$ ntawm qhov deb $r$ los ntawm ib qho axis ntawm kev sib hloov, nws lub sijhawm ntawm inertia $I$ yog muab los ntawm I=m×r².
2. Vim li cas lub sijhawm ntawm inertia feem ntau hu ua "kev sib piv ntawm kev sib hloov" ntawm pawg?
   • tebIb yam li qhov hnyav yog qhov ntsuas ntawm ib yam khoom txoj kev tiv thaiv rau kev hloov pauv hauv nws txoj kev txav mus los (vim yog Newton txoj cai thib ob), lub sijhawm ntawm inertia yog qhov ntsuas ntawm ib yam khoom txoj kev tiv thaiv rau kev hloov pauv hauv nws txoj kev txav mus los.
3. Kev hloov qhov deb ntawm ib qho khoom me me ntawm nws txoj kab sib hloov cuam tshuam li cas rau nws lub sijhawm inertia?
   • tebLub sijhawm ntawm inertia yog proportional rau lub square ntawm qhov deb ntawm lub axis ntawm kev sib hloov. Yog tias koj ob npaug ntawm qhov deb, lub sijhawm ntawm inertia yuav nce los ntawm ib qho ntawm plaub.
4. Vim li cas qhov kev ncua deb yog lub xwmfab (r)²) siv rau hauv cov mis rau lub sijhawm ntawm inertia es tsis yog qhov deb xwb?
   • teb: Lub square ntawm qhov deb yog siv vim yog txoj kev uas lub zog kinetic hauv kev sib hloov ua haujlwm. Hauv kev txav mus los, txhua lub khoom me me ntawm ib yam khoom pab txhawb rau lub zog kinetic sib hloov raws li nws qhov hnyav thiab nws qhov deb ntawm lub axis squared.
5. Lub sijhawm ntawm inertia ntawm ib qho khoom me me hloov pauv li cas yog tias nws qhov hnyav yog tripled thaum khaws qhov deb ntawm lub axis tas li?
   • tebYog tias qhov hnyav raug peb npaug thiab qhov deb raug khaws cia tas li, lub sijhawm ntawm inertia kuj yuav peb npaug vim nws yog proportional ncaj qha rau qhov hnyav.
6. Puas muaj lub sijhawm inertia ntawm xoom uas ib lub khoom me me puas muaj? Yog tias muaj, nyob rau hauv qhov xwm txheej twg?
   • tebYog, ib qho khoom me me yuav muaj lub sijhawm inertia ntawm xoom yog tias nws nyob ncaj qha rau ntawm lub axis ntawm kev sib hloov, ua rau nws qhov deb $r$ los ntawm lub axis sib npaug rau xoom.
7. Vim li cas cov khoom sib txawv uas muaj tib lub cev hnyav thiab qhov loj me muaj lub sijhawm sib txawv ntawm inertia thaum tig mus rau ntau lub axes?
   • tebQhov kev faib tawm ntawm qhov hnyav nyob ib ncig ntawm txoj kab ntawm kev tig txiav txim siab lub sijhawm ntawm inertia. Txawm tias ob yam khoom muaj tib lub cev hnyav thiab qhov loj me, lawv cov kev faib tawm ntawm qhov hnyav piv rau txoj kab ntawm kev tig yuav txawv, ua rau muaj ntau lub sijhawm ntawm inertia.
8. Lub sijhawm ntawm inertia puas yog qhov ntau ntawm scalar lossis vector?
   • teb: Lub sijhawm ntawm inertia yog qhov ntau scalar. Txawm li cas los xij, rau cov lub cev khov kho nrog cov duab nyuaj thiab ntau lub axis ntawm kev sib hloov, nws yog sawv cev los ntawm tensor.
9. Yog tias ob lub particles, txhua lub mass $m$, nyob ntawm qhov deb r_1​ thiab r₂​ los ntawm lub axis ntawm kev sib hloov, lub sijhawm sib xyaw ua ke ntawm inertia yog dab tsi?
   • tebLub sijhawm ntawm inertia yog qhov ntxiv rau cov khoom sib cais. Yog li, lub sijhawm ua ke ntawm inertia I=m×r₁^{2 ​}+ m×r₂^2​.

10. Lub sijhawm ntawm inertia cuam tshuam li cas rau kev txuag ntawm angular momentum?

    • teb: Lub zog ntawm lub kaum sab xis $L$ yog cov khoom ntawm lub sijhawm ntawm inertia $I$ thiab qhov ceev ntawm lub kaum sab xis $\omega$, sawv cev los ntawm qhov sib npaug $L=I\times \omega$Yog tias tsis muaj lub zog sab nraud ua haujlwm rau ib lub cev, lub zog ntawm lub kaum sab xis yuav nyob ruaj khov. Qhov no txhais tau tias yog tias lub sijhawm ntawm inertia hloov pauv (zoo li hauv tus neeg caij skater rub hauv lawv txhais tes), lub zog ntawm lub kaum sab xis yuav tsum kho kom cov khoom ruaj khov.