Ob lub cev uas muaj tib lub zog ntawm kev nrawm - Kev siv Newton txoj cai ntawm kev txav mus los cov teeb meem thiab kev daws teeb meem

1. Ob pawg loj m₁ = 2 kg thiab m₂ = 5 kg nyob rau ntawm ib lub dav hlau inclined thiab txuas ua ke los ntawm ib txoj hlua raws li pom hauv daim duab. Tus coefficient ntawm kinetic friction ntawm m₁ thiab qhov nkhaus yog 0.2 thiab cov coefficient ntawm kev sib txhuam kinetic nruab nrab ntawm m₂ thiab qhov nkhaus yog 0.1.

(a) Txheeb xyuas lawv acceleration

(b) Txheeb xyuas lub zog nruj

Paub:

Pawg 1 (m)₁) = 2 kg

Pawg 2 (m₂) = 4 kg

Coefficient ntawm kev sib txhuam kinetic ntawm m₁ thiab lub dav hlau inclined (μ_k1) = 0.2

Coefficient ntawm kev sib txhuam kinetic ntawm m₂ thiab lub dav hlau inclined (μ_k2) = 0.1

Acceleration vim lub ntiajteb txawj nqus (g) = 9.8 m/s²

a) Qhov loj thiab kev coj ntawm qhov kev nrawm

w₁ = hnyav 1 = m₁ g = (2 kg)(9.8 m/s)²) = 19.6 Newtons

w_1x = wb₁ kev txhaum 30^o = (19.6 N)(0.5) = 9.8 Newtons

w_1y = wb₁ coj 30^o = (19.6 N)(0.87) = 17 Newtons

N₁ = Tus lub zog quab yuam rau ntawm m₁ = wb_1y = 17 Newtons

F_k1 = Lub zog ntawm kev sib txhuam kinetic ntawm m₁ = μ_k1 N₁ = (0.2)(17 N) = 3.4 Newtons

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w₂ = qhov hnyav 2 = m₂ g = (4 kg)(9.8 m/s)²) = 39.2 Newtons

w_2x = wb₂ kev txhaum 60^o = (39.2 N)(0.87) = 34.1 Newtons

w_2y = wb₂ coj 60^o = (39.2 N)(0.5) = 19.6 Newtons

N₂ = Lub zog ib txwm muaj rau ntawm m₂ = wb_2y = 19.6 Newtons

F_k2 = Lub zog ntawm kev sib txhuam kinetic ntawm m₂ = μ_k2 N₂ = (0.1)(19.6 N) = 1.96 Newtons

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Qhov loj ntawm kev nrawm:

∑F_x = ma_x

w_2x > wb_1x yog li ntawd, qhov kev taw qhia ntawm qhov kev nrawm yog tib yam li qhov kev taw qhia ntawm w_2x.

Cov zog uas taw qhia raws qhov kev nrawm yog cov zoo thiab cov zog uas muaj kev taw qhia sib txawv rau qhov kev nrawm yog cov tsis zoo.

w_2x - F_k2 - T₂ +T₁ - w_1x - F_k1 = (m₁ +m₂) thiab_x

w_2x - F_k2 - w_1x - F_k1 = (m₁ +m₂ ) thiab_x

34.1 N – 1.96 N – 9.8 N – 3.4 N = (2 kg + 4 kg) a_x

18.94 N = (6 kg) ib_x

a_x = 18.94 N : 6 kg

a_x = 3.16 m/s²

Qhov loj ntawm qhov kev nrawm = 3.16 m/s² Kev taw qhia ntawm qhov kev nrawm = kev taw qhia ntawm T₁ = kev taw qhia ntawm w_2x

b) Qhov loj ntawm lub zog nruj

Siv Newton txoj cai thib ob rau ntawm yam khoom 2:

w_2x - F_k2 - T₂ = kuv₂ a_x

34.1 N – 1.96 N – T₂ = (4 kg)(3.16 m/s)²)

32.14 N – T₂ = 12.64 N

T₂ = 32.14 N – 12.64 N = 19.5 Newtons

Lub zog nruj = T = T₁ = T ua₂ = 19.5 Newtons

2. m₁ = 4 kg, m₂ = 2 kg. Txheeb xyuas (a) qhov loj thiab kev coj ntawm qhov kev nrawm (b) Qhov loj ntawm lub zog nruj uas txuas m₁ thiab m₂ (c) qhov loj ntawm lub zog nruj uas txuas lub pulley thiab lub ru tsev.

tshuaj

w₁ = kuv₁ g = (4 kg)(9.8 m/s)²) = 39.2 Newtons

w₂ = kuv₂ g = (2 kg)(9.8 m/s)²) = 19.6 Newtons

a) Qhov loj thiab kev coj ntawm kev nrawm

∑F_y = ma_y

w₁ > wb₂ yog li ntawd qhov kev taw qhia ntawm cov khoom yog tib yam li qhov kev taw qhia ntawm qhov hnyav 1 (w₁)Cov zog uas muaj tib qho kev taw qhia li kev nrawm yog qhov zoo thiab cov zog uas muaj qhov kev taw qhia sib txawv nrog kev nrawm yog qhov tsis zoo.

w₁ - T₁ +T₂ - w₂ = (m₁ +m₂) thiab_y

w₁ - w₂ = (m₁ +m₂) thiab_y

39.2 N – 19.6 N = (4 kg + 2 kg) a_y

19.6 N = (6 kg) ib_y

a_y = 19.6 N : 6 kg

a_y = 3.26 m/s²

Qhov loj ntawm kev nrawm = 3.26 m/s²Kev taw qhia ntawm kev nrawm = kev taw qhia ntawm w₁ .

b) Qhov loj ntawm lub zog nruj uas txuas m₁ thiab m₂

thov Txoj cai thib ob ntawm Newton rau ntawm m₂ :

∑F_y = ma_y

w₁ - T₁ = kuv₁ a_y

39.2 N – T₁ = (4 kg)( 3.26 m/s²)

39.2 N – T₁ = 13.04 N

T₁ = 39.2 N – 13.04 N

T₁ = 26.16 Newtons

Qhov loj ntawm lub zog nruj uas txuas cov khoom = T = T₁ = T ua₂ = 26.16 Newtons

c) Qhov loj ntawm lub zog nruj uas txuas lub pulley thiab lub ru tsev.

Pulley so:

∑F_y = ma_y —— ib_y = 0

∑F_y = 0

Cov zog sab saud yog cov zoo, cov zog sab hauv yog cov tsis zoo:

T₃ - T₁ - T₂ = 0

T₃ = T ua₁ +T₂

T₁ thiab T₂ muaj tib qhov loj, T₁ = T ua₂ = T = 26.16 N :

T₃ = 2T = 2(26.16 N) = 52.32 Newtons

3. Thaiv 1 (m₁ = 10 kg) thiab thaiv 2 (m₂ = 15 kg) txuas nrog los ntawm ib txoj hlua hla lub pulley tsis muaj kev sib txhuam. Coefficient ntawm kev sib txhuam zoo li qub ntawm lub thaiv 2 nrog incline = 0.6. Tus coefficient ntawm kev sib txhuam kinetic ntawm lub thaiv 2 nrog incline = 0.42. Txheeb xyuas (a) Qhov loj ntawm lub zog tsawg kawg nkaus F siv rau cov khoom kom cov khoom nrawm dua (b) Txheeb xyuas qhov loj ntawm lub zog nruj.

tshuaj

w₁ = Qhov hnyav ntawm lub thaiv 1 = m₁ g = (10 kg)(9.8 m/s)²) = 98 Newtons

w₂ = Qhov hnyav ntawm lub thaiv 2 = m₂ g = (15 kg)(9.8 m/s)²) = 147 Newtons

w_2y = wb₂ coj 30^o = (147 N)(0.87) = 127.89 Newtons

w_2x = wb₂ kev txhaum 30^o = (147 N)(0.5) = 73.5 Newtons

N₂ = Lub zog ib txwm muaj rau ntawm lub thaiv 2 = w_2y = 127.89 Newtons

F_k2 = Lub zog ntawm kev sib txhuam kinetic ntawm lub thaiv 2 = μ_k2 N₂ = (0.42)(127.89 N) = 53.7 Newtons

F_s2 = Lub zog ntawm qhov sib txhuam static rau ntawm lub thaiv 2 = μ_s2 N₂ = (0.6)(127.89 N) = 76.7 Newtons

a) Qhov loj ntawm lub zog tsawg kawg nkaus F siv rau cov khoom kom cov khoom nrawm dua

∑F_x = ma_x —— ib_x = 0

∑F_x = 0

Cov zog sab saud thiab cov zog sab xis yog cov zoo, cov zog hauv qab thiab cov zog sab laug yog cov tsis zoo.

F – F_k2 - w_2x - w₁ - T₂ +T₁ = 0

F – F_k2 - w_2x - w₁ = 0

F = F_k2 + w_2x + w₁

F = 53.7 N + 73.5 N + 98 N

F = 225.2 Newtons

b) Qhov loj ntawm lub zog nruj

Siv Newton txoj cai ntawm kev txav mus los rau ntawm thaiv 1:

∑F_y = ma_y —— ib_y = 0

∑F_y = 0

T₁ - w₁ = 0

T₁ = wb₁ = 98 Newtons

Siv Newton txoj cai ntawm kev txav mus los rau ntawm thaiv 2:

F – F_k2 - w_2x - T₂ = 0

T₂ = F – F_k2 - w_2x

T₂ = 225.2 N – 53.7 N – 73.5 N

T₂ = 98 Newtons

Qhov loj ntawm lub zog nruj = T₁ = T ua₂ = T = 98 Newtons

4. Thaiv 1 (m₁ = 16 kg) nyob rau ntawm ib qho chaw kab rov tav thiab lub thaiv 2 (m₂ = 12 kg) nyob rau ntawm ib lub dav hlau du, txuas nrog los ntawm ib txoj hlua uas hla dhau ib lub pulley me me, tsis muaj kev sib txhuam. Thaiv 3 (m₃ = 5 kg) nyob rau ntawm lub thaiv 2. Tus coefficient ntawm kev sib txhuam kinetic ntawm lub thaiv 2 thiab qhov chaw kab rov tav yog 0,4. Lub coefQhov sib txhuam zoo li qub ntawm lub thaiv 2 nrog lub thaiv 3 yog 0,3.

(A) Thaum lub kaw lus raug tso tawm ntawm qhov chaw so, lub thaiv 3 thiab lub thaiv 2 tseem swb ua ke?

(B) Yog tias muaj block 3, qhov kev nrawm ntawm block 1 thiab block 2 yog dab tsi?

Tshuaj:

a) Thaum lub kaw lus raug tso tawm ntawm qhov chaw so, lub thaiv 3 thiab lub thaiv 2 tseem swb ua ke?

w₁ = Tus qhov hnyav ntawm lub thaiv 1 = m₁ g = (16 kg)(9.8 m/s)²) = 156.8 Newtons

w_1x = wb₁ kev txhaum 60^o = (156.8 N)(0.87) = 136.4 Newtons

w_1y = wb₁ coj 60^o = (156.8 N)(0.5) = 78.4 Newtons

N₁ = Tus lub zog ib txwm siv rau ntawm lub thaiv 1 los ntawm lub dav hlau inclined = wb_1y = 78.4 Newtons

w₃ = Tus qhov hnyav ntawm lub thaiv 3 = m₃ g = (5 kg)(9.8 m/s)²) = 49 Newtons

N₂₃ = Tus lub zog ib txwm siv rau ntawm lub thaiv 3 los ntawm lub thaiv 2 = wb₃ = 49 Newtons

N₃₂ = Tus nlub zog ib txwm siv rau ntawm lub thaiv 2 los ntawm lub thaiv 3 = N₂₃ = wb₃ = 49 Newtons

(N₂₃ thiab N₃₂ yog cov khub ua haujlwm thiab cov tshuaj tiv thaiv)

F_s23 = Tus lub zog ntawm qhov sib txhuam static uas siv rau ntawm lub block 3 los ntawm lub block 2 = μ_s N₂₃ = (0.3)(49 N) = 14.7 Newton

F_s32 = Tus lub zog ntawm qhov sib txhuam static uas siv rau ntawm lub thaiv 2 los ntawm lub thaiv 3 =F_s23 = 14.7 Newtons

(F_s23 thiab F_s32 yog cov khub ua haujlwm thiab cov tshuaj tiv thaiv)

w₂ = Tus qhov hnyav ntawm lub block 2 = kuv₂ g = (12 kg)(9.8 m/s)²) = 117.6 Newtons

N₂ = Tus lub zog ib txwm muaj uas siv rau ntawm yam khoom 2 los ntawm qhov chaw kab rov tav = wb₂ + N₃₂ = 117.6 Newtons + 49

Newton = 166.6 Newton

F_k2 = Tus lub zog ntawm kev sib txhuam kinetic ntawm lub block 2 = μ_k N₂ = (0.4)(166.6 N) = 66.64 Newtons

Siv Newton txoj cai ntawm kev txav mus los rau ntawm lub block 3:

∑F_x = ma_x

F_s23 =m₃ a_x

—–> F_s23 = μ_s N₂₃ = μ_s w₃ = μ_s m₃ g

μ_s m₃ g = m₃ a_x

μ_s g = ib_x

a_x = (0.3)(9.8 m/s)²) = 2.94 m/s²

Qhov kev nrawm siab tshaj plaws ntawm lub thaiv 3 kom lub thaiv 3 thiab lub thaiv 2 tseem swb ua ke yog 2.94 m/s².

Tam sim no peb xam qhov loj ntawm lub kaw lus qhov kev nrawm tom qab raug tso tawm ntawm qhov so.

Qhov kev taw qhia ntawm qhov kev hloov chaw ntawm lub block = qhov kev taw qhia ntawm qhov kev nrawm ntawm lub block = qhov kev taw qhia ntawm T₂ = qhov kev taw qhia ntawm w_1x.

∑F_x = ma_x

w_1x - T₁ +T₂ - F_k2 - F_s32 +F_s23 = (m₁ +m₂ +m₃) thiab_x

w_1x - F_k2 = (m₁ +m₂ +m₃ ) thiab_x

136.4 N – 66.64 N = (16 kg + 12 kg + 5 kg) a_x

69.76 N = (33 kg) ib_x

a_x = 2.11 m/s²

a_x yog qhov zoo, txhais tau tias qhov kev taw qhia ntawm qhov kev hloov pauv ntawm lub thaiv lossis qhov kev taw qhia ntawm qhov kev nrawm yog tib yam li qhov kev taw qhia ntawm T₂ los yog kev taw qhia ntawm w_1x.

Qhov loj ntawm qhov kev nrawm yog 2.11 m / s² , lhwj chim tshaj 2.94 m / s² yog li peb tuaj yeem xaus lus tias thaiv 3 thiab thaiv 2 tseem swb ua ke tom qab raug tso tawm ntawm kev so.

b) Qhov loj ntawm qhov kev nrawm ntawm lub block 1 thiab lub block 2

∑F_x = ma_x

w_1x - F_k2 = (m₁ +m₂) thiab_x

—–> F_k2 = μ_k N₂ = μ_k w₂ = μ_k m₂ g = (0.4)(12 kg)(9.8 m/s)²) = 47.04 Newtons

136.4 N – 47.04 N = (16 kg + 12 kg) a_x

89.36 N = (28 kg) ib_x

a_x = 89.36 N : 28 kg = 3.19 m/s²

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1. Qhov hnyav thiab qhov hnyav
2. ib txwm muaj zog
3. Newton txoj cai thib ob ntawm kev txav mus los
4. Lub zog sib txhuam
5. Kev txav mus los ntawm qhov chaw kab rov tav yam tsis muaj kev sib txhuam
6. Kev txav ntawm ob lub cev nrog tib qhov kev nrawm ntawm qhov chaw ntxhib kab rov tav nrog lub zog sib txhuam
7. Kev txav mus los ntawm lub dav hlau inclined tsis muaj lub zog sib txhuam
8. Kev txav mus los ntawm lub dav hlau inclined nrog lub zog sib txhuam
9. Kev txav mus los hauv lub elevator
10. Kev txav ntawm lub cev yog txuas nrog los ntawm cov hlua thiab pulleys
11. Ob lub cev uas muaj tib lub zog ntawm kev nrawm
12. Kev sib hloov ntawm ib txoj kab tiaj tiaj - dynamics ntawm kev txav mus los ncig
13. Kev sib hloov ntawm cov kab nkhaus - dynamics ntawm kev txav mus los ncig
14. Kev txav mus los sib xws hauv lub voj voog kab rov tav
15. Lub zog centripetal hauv kev txav mus los sib npaug