ahupūngao

Te nekehanga ōrite i roto i te porowhita whakapae – ngā raruraru me ngā otinga

1. He pōro 0.2-kg te taumaha, e piri ana ki te pito o tētahi taura whakapae, e hurihia ana i roto i tētahi porowhita he 1 mita te whānui, ā, ko te tere mōrahi o te pōro he 10 rpm. He aha te rahi o te whakaterenga pokapū me te rahi o te kaha kume?

Mōhiotia:

Mass (m) = 0.2 kg

Pūtoro (r) = 1 m

Te orite angular (ω) = 10 whakahuri/mene = 10 whakahuri/60 s = 0.17 whakahuri/s = (0.17)(6.28 rad)/s = 1 rad/s

tere (v) = r ω = (1 m)(1 rad/s) = 1 m/s

E hiahiatia ana: a_s dan ΣF

Rongoā:

(a) Te rahi o te whakaterenga ā-pokapū

(b) Te rahi o te kaha kume

ΣF = ma

T = mā_s

T = (0.2 kg)(1 m/s)²)

T = 0.2 kg m/s²

T = 0.2 N

2. Kei te porowhita whakapae te pōro 1-kg i te pito o te aho, he 1 m te whānui o te porowhita. Ka motu te taura ina neke atu te kukū o roto i te 100 N. He aha te tere mōrahi ka taea e te pōro te eke?

Mōhiotia: Te nekehanga ōrite i roto i te porowhita whakapae – ngā raruraru me ngā otinga 2

Taumaha (m) = 1 kg

Pūtoro (r) = 1 mita

Te kaha kume (T) = kaha centripetal (ΣF) = 100 N

Hiahia: mōrahi v

Rongoā:

Te nekehanga ōrite i roto i te porowhita whakapae – ngā raruraru me ngā otinga 3

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Pānuitia atu

Te whakaawhiwhi i tētahi kōpiko peeke – ngā whanaketanga o ngā raruraru nekehanga porowhita me ngā otinga

1. He motuka e huri ana i tētahi piko whāiti. He aha te koki mō te rori he 60 mita te whānui o te piko me te tere hoahoa o te 20 m/s? Kiia kāore he waku i waenganui i te motuka me te rori.

otinga

Te whakaawhiwhi i tētahi kōpiko whakarara – ngā āhuatanga o ngā raruraru nekehanga porowhita me ngā otinga 1 N= kaha noa

N hara θ = te wāhanga whakapae o te kaha noa

N cos θ = te wāhanga poutū o te kaha noa

w = mg = te taimaha o te waka

He mea hanga te rori kia whakapūmauhia kia kore ai e whakawhirinaki ki te waku.

Ko te kaha whakapae kupenga, te te wāhanga whakapae o te kaha noa (N hara θ), e hiahiatia ana kia neke tonu te waka i roto i te porowhita huri noa i te piko.

Ka whiriwhiria e mātou te tuaka-x hei whakapae, me te tuaka-y hei poutū, kia puta ai te whakaterenga porotītaha, a_R, kei te taha whakapae. I te taha whakapae, ko te kaha anake ko te wāhanga whakapae o te kaha noa (N hara θ), e hiahiatia ana hei whakaputa i te whakaterenga pokapū. N sin θ = kaha centripetal.

Whakamahia te ture nekehanga a Newton i te ahunga poutū:

Te whakaawhiwhi i tētahi kōpiko whakarara – ngā āhuatanga o ngā raruraru nekehanga porowhita me ngā otinga 5

Whakamahia te ture nekehanga a Newton i te ahunga whakapae:

Te whakaawhiwhi i tētahi kōpiko whakarara – ngā āhuatanga o ngā raruraru nekehanga porowhita me ngā otinga 7

Whakakapingate tāpiri i a N i te whārite 1 ki roto i a N i te whārite 2 :

Te whakaawhiwhi i tētahi kōpiko whakarara – ngā āhuatanga o ngā raruraru nekehanga porowhita me ngā otinga 1

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Pānuitia atu

Te whakaawhiwhi i tētahi piko papatahi – ngā whanaketanga o ngā rapanga nekehanga porowhita me ngā otinga

1. Ka huri tetahi motuka 2000-kg i tētahi piko i runga i tētahi rori papatahi he 150 m te whānui o te radius. Ko te tauwehenga o te waku pumau ko te 0.5. Whakatauhia te tere mōrahi kia whai te motuka i te piko, ā, kia kore ai e paheke. Whakaterenga na te kaha o te kaha = 10m/s².

Mōhiotia:

Mass (m) = 2000 kg

Pūtoro (r) = 150 mita

Te tauwehenga o te waku pateko (μ_s) = 0.5

Taumaha (w) = mg = (2000 kg)(10 m/s)²) = 20,000 kg m/s²= 20,000 N

Te kaha o te waku pao (F)_s) = μ_sN = μ_sw = (0.7)(20,000 N) = 14,000 N

E hiahiatia ana : v

Rongoā:

Te whakaawhiwhi i tētahi kōpiko papatahi – ngā āhuatanga o ngā rapanga nekehanga porowhita me ngā otinga 1

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Pānuitia atu

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton

1. E rua ngā papatipu m₁= 2 kg me te m₂ = 5 kg kei runga i te papa whakarara, ā, e honoa ana mā te aho e whakaaturia ana i te pikitia. Ko te tauwehenga o te waku nekeneke i waenga i a m₁ ā, ko te pikinga he 0.2, ā, ko te tauwehenga o te te waku nekeneke i waenganui i te m₂ ā, ko te pikinga he 0.1.

(a) Whakatauhia ā rātou whakatere

(b) Whakatauhia te kaha kume

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 1

Mōhiotia:

Mass 1 (m₁) = 2 kirokaramu

Taumaha 2 (m₂) = 4 kirokaramu

Te tauwehenga o te waku nekeneke i waenga i te m₁ a papa whakarara (μ_k1) = 0.2

Te tauwehenga o te waku nekeneke i waenga i te m₂ me te papa whakarara (μ)_k2) = 0.1

Whakaterenga na te kaha o te kaha (g) = 9.8 m/s²

a) Te rahi me te ahunga o te whakaterenga

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 2

w₁ = taimaha 1 = m₁karamu = (2 kg)(9.8 m/s²) = 19.6 Niutona

w_1x =w₁ hara 30^o = (19.6 N)(0.5) = 9.8 Nīutona

w_1y =w₁ whaimana 30^o= (19.6 N)(0.87) = 17 Nīutona

N₁= Te kaha noa i runga i te m₁ =w_1y = 17 Niutona

F_k1 = Te kaha o te waku nekeneke i runga i te m₁ = μ_k1N₁ = (0.2)(17 N) = 3.4 Nīhana

---

w₂ = taumaha 2 = m₂karamu = (4 kg)(9.8 m/s²) = 39.2 Niutona

w_2x =w₂hara 60^o = (39.2 N)(0.87) = 34.1 Nīutona

w_2y =w₂ whaimana 60^o= (39.2 N)(0.5) = 19.6 Nīutona

N₂ = Te kaha noa i runga i te m₂ =w_2y= 19.6 Niutona

F_k2= Te kaha o te waku nekeneke i runga i te m₂ = μ_k2 N₂ = (0.1)(19.6 N) = 1.96 Nīhana

---

Te rahi o te whakaterenga:

∑F_x = mā_x

w_2x> w_1x nō reira he rite tonu te ahunga o te whakaterenga ki te ahunga o w_2x.

He pai ngā kaha e tohu ana i te ahunga whakaterenga, ā, he kino ngā kaha e tohu ana i te ahunga whakahāwea ki te whakaterenga.

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

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

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

18.94 N = (6 kg) a_x

a_x = 18.94 N : 6 kg

a_x= 3.16m/s²

Te rahi o te whakaterenga = 3.16 m/s² Te ahunga o te whakaterenga = te ahunga o T₁= te ahunga o te w_2x

b) Te rahi o te kaha kume

Whakamahia te ture tuarua a Newton ki te mea 2:

w_2x- F_k2- T₂ = m₂a_x

34.1 N – 1.96 N – T₂= (4 kg)(3.16 m/s²)

32.14 N – T₂ = 12.64 N

T₂= 32.14 N – 12.64 N = 19.5 Ngā Newton

Te kaha kume = T = T₁ =T₂ = 19.5 Niutona

2. m₁= 4 kg, mita₂ = 2 kg. Whakatauhia (a) te rahi me te ahunga o te whakaterenga (b) Te rahi o te kaha kume e hono ana i a m₁ me te m₂ (c) te rahi o te kaha kume e hono ana i te pūrei me te tuanui.

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 3

otinga

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 4

w₁ = m₁karamu = (4 kg)(9.8 m/s²) = 39.2 Niutona

w₂= m₂ karamu = (2 kg)(9.8 m/s²) = 19.6 Niutona

a) Te rahi me te ahunga o te whakaterenga

∑F_y = mā_y

w₁> w₂nō reira he rite tonu te ahunga o te mea ki te ahunga o te taumaha 1 (w₁)He pai ngā kaha he rite te ahunga ki te whakaterenga, ā, he kino ngā kaha he rite te ahunga ki te whakaterenga.

w₁- T₁ +T₂ - w₂= (m₁ +m₂) a_y

w₁ - w₂ = (m₁ +m₂) a_y

39.2 N – 19.6 N = (4 kg + 2 kg) a_y

19.6 N = (6 kg) a_y

a_y= 19.6 N : 6 kg

a_y = 3.26m/s²

Te rahi o te whakaterenga = 3.26 m/s². Te ahunga o te whakaterenga = te ahunga o w₁ .

b) Te rahi o te kaha kume e hono ana i a m₁ me te m₂

Anga Te ture tuarua a Newton i runga i te m₂ :

∑F_y = mā_y

w₁ - T₁ = m₁a_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 Niutona

Te rahi o te kaha kume e hono ana i ngā mea = T = T₁ =T₂ = 26.16 Niutona

c) Te rahi o te kaha kume e hono ana i te pūrakau me te tuanui.

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 5 Kei te okioki te pūrei:

∑F_y= mā_y —— he_y= 0

∑F_y= 0

He pai ngā kaha whakarunga, he kino ngā kaha whakararo:

T₃ - T₁ - T₂ = 0

T₃=T₁ +T₂

T₁ a T₂ he rite te rahi, T₁ =T₂ = T = 26.16 N :

T₃ = 2T = 2(26.16 N) = 52.32 Ngā Niutona

3. Poraka 1 (m₁ = 10 kg) me te poraka 2 (m₂= 15 kg) i honoa mā te taura i runga i te pūrei kore-waku. Te tauwehenga o te waku pūmau i waenga i te poraka 2 me te pikinga = 0.6. Ko te tauwehenga o te waku nekeneke i waenga i te poraka 2 me te pikinga = 0.42. Whakatauhia (a) Te rahi o te kaha iti rawa F i pā ki ngā mea kia tere ake ai ngā mea ki runga (b) Whakatauhia te rahi o te kaha kume.

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 6

otinga

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 7

w₁ = Te taumaha o te poraka 1 = m₁ karamu = (10 kg)(9.8 m/s²) = 98 Niutona

w₂ = Te taumaha o te poraka 2 = m₂ karamu = (15 kg)(9.8 m/s²) = 147 Niutona

w_2y =w₂ whaimana 30^o = (147 N)(0.87) = 127.89 Nīutona

w_2x =w₂hara 30^o = (147 N)(0.5) = 73.5 Nīutona

N₂ = Te kaha noa i runga i te poraka 2 = w_2y= 127.89 Niutona

F_k2 = Te kaha o te waku nekeneke i runga i te poraka 2 = μ_k2 N₂ = (0.42)(127.89 N) = 53.7 Nīhana

F_s2 = Te kaha o te waku pumau i runga i te poraka 2 = μ_s2 N₂ = (0.6)(127.89 N) = 76.7 Nīhana

a) Te rahi o te kaha iti rawa F i pā ki ngā mea kia tere ake ai ngā mea ki runga

∑F_x = mā_x —— he_x = 0

∑F_x = 0

He pai ngā kaha whakarunga me ngā kaha whakarunga matau, he kino ngā kaha whakararo me ngā kaha whakarunga maui.

Wh – Wh_k2 - w_2x - w₁ - T₂ +T₁ = 0

Wh – Wh_k2 - w_2x- w₁ = 0

F = F_k2+ w_2x + w₁

F = 53.7 N + 73.5 N + 98 N

F = 225.2 Newton

b) Te rahi o te kaha kume

Whakamahia te ture nekehanga a Newton ki te poraka 1:

∑F_y = mā_y—— he_y= 0

∑F_y = 0

T₁ - w₁ = 0

T₁ =w₁ = 98 Niutona

Whakamahia te ture nekehanga a Newton ki te poraka 2:

Wh – Wh_k2 - w_2x - T₂ = 0

T₂= F – F_k2- w_2x

T₂= 225.2 N – 53.7 N – 73.5 N

T₂ = 98 Niutona

Te rahi o te kaha kume = T₁ =T₂ = T = 98 Niutona

4. Poraka 1 (m₁ = 16 kg) kei runga i te mata whakapae, ā, ko te poraka 2 (m₂ = 12 kg) kei runga i tētahi papa whakarara maeneene, e honoa ana e tētahi taura e whiti ana i runga i tētahi porotiti iti, kore-waku. Poraka 3 (m₃ = 5 kg) kei runga i te poraka 2. Ko te tauwehenga o te waku nekeneke i waenga i te poraka 2 me te mata whakapae he 0,4. Ko te coefKo te uara o te waku pateko i waenga i te poraka 2 me te poraka 3 he 0,3.

(a) Ina tukuna te pūnaha mai i te okiokinga, ka paheke tahi tonu te poraka 3 me te poraka 2?

(B) Mena kei reira te poraka 3, he aha te whakaterenga o te poraka 1 me te poraka 2?

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 8

Rongoā:

a) Ina tukuna te pūnaha mai i te okiokinga, ka paheke tahi tonu te poraka 3 me te poraka 2?

Ngā tinana e rua he rite te rahi o te whakaterenga – Te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton 9

w₁ = Te te taumaha o te poraka 1 = m₁ karamu = (16 kg)(9.8 m/s²) = 156.8 Niutona

w_1x =w₁ hara 60^o = (156.8 N)(0.87) = 136.4 Nīutona

w_1y =w₁ whaimana 60^o = (156.8 N)(0.5) = 78.4 Nīutona

N₁ = Te te kaha noa i pā ki te poraka 1 e te papa whakarara =w_1y = 78.4 Niutona

w₃ = Te te taumaha o te poraka 3 = m₃ karamu = (5 kg)(9.8 m/s²) = 49 Niutona

N₂₃ = Te te kaha noa i pā ki te poraka 3 e te poraka 2 =w₃= 49 Niutona

N₃₂ = Te nte kaha noa i pā ki te poraka 2 e te poraka 3 = N₂₃ =w₃ = 49 Niutona

(N₂₃a N₃₂ he takirua mahi-tauhohenga)

F_s23 = Te te kaha o te waku pao i pā ki te poraka 3 e te poraka 2 = μ_s N₂₃= (0.3)(49 N) = 14.7 Newton

F_s32= Te te kaha o te waku pumau i pā ki te poraka 2 e te poraka 3 =F_s₂₃ = 14.7 Niutona

(F_s23 a F_s32 he takirua mahi-tauhohenga)

w₂ = Te te taumaha o te poraka 2 = m₂ karamu = (12 kg)(9.8 m/s²) = 117.6 Niutona

N₂ = Te te kaha noa i pā ki te mea 2 e te mata whakapae =w₂ +N₃₂ = 117.6 Ngā Newton + 49

Niutoni = 166.6 Niutoni

F_k2= Te te kaha o te waku nekeneke i runga i te poraka 2 = μ_k N₂ = (0.4)(166.6 N) = 66.64 Nīhana

Whakamahia te ture nekehanga a Newton ki te poraka 3:

∑F_x = mā_x

F_s23=m₃a_x

—–> F_s23 = μ_sN₂₃= μ_sw₃= μ_sm₃g

μ_s m₃g = m₃a_x

μ_s g = a_x

a_x = (0.3)(9.8 m/s²) = 2.94 m/s²

Ko te whakaterenga mōrahi o te poraka 3 kia paheke tonu ai te poraka 3 me te poraka 2 ko 2.94 m/s².

Nā, ka tatauhia e tātou te rahi o te whakaterenga o te pūnaha i muri i te tukunga mai i te okiokinga.

Ko te ahunga o te nekehanga o te poraka = te ahunga o te whakaterenga o te poraka = te ahunga o T₂= te ahunga o w_1x.

∑F_x = mā_x

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

w_1x - F_k2 = (m₁ +m₂+m₃ ) a_x

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

69.76 N = (33 kg) a_x

a_x = 2.11m/s²

a_x he pai, ko te tikanga he rite te ahunga o te nekehanga poraka, te ahunga rānei o te whakaterenga ki te ahunga o T₂ te ahunga rānei o w_1x.

Ko te rahi o te whakaterenga ko 2.11 m / s² , lnui atu i 2.94 m / s²Nō reira, ka taea e tātou te whakatau ka paheke tahi tonu te poraka 3 me te poraka 2 i muri i te tukunga i te okiokinga.

b) Te rahi o te whakaterenga o te poraka 1 me te poraka 2

∑F_x= mā_x

w_1x - F_k2= (m₁ +m₂) a_x

—–> F_k2 = μ_k N₂= μ_k w₂ = μ_k m₂karamu = (0.4)(12 kg)(9.8 m/s²) = 47.04 Niutona

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

89.36 N = (28 kg) a_x

a_x= 89.36 N : 28 kg = 3.19 m/s²

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Pānuitia atu

Te taurite o ngā tinana i runga i te papa whakarara – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton

1. Kei runga i tētahi papa whāiti taratara he poraka 2-kg te taumaha, he koki 37 te rahi.^oki te whakapae. Whakatauhia te rahi o te kaha o waho i pā ki te poraka, kia kore ai te poraka e paheke ki raro i te papa. (syn 37^o = 0.6, cos 37^o = 0.8, karamu = 10 ms^-2, µ_k = 0.2)

Te taurite o ngā tinana i runga i te papa whakarara – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 1 Mōhiotia:

Mass (m) = 2 kg

Whakaterenga na te kaha o te kaha (g) = 10 m/s²

Ngā Poraka taimaha (w) = mg = (2)(10) = 20 Ngā Newton

Hara 37^o = 0.6

Kohinga 37^o = 0.8

Tauwehenga o te te waku nekeneke (µ_k) = 0.2

Ko te wāhanga-y o te taumaha (w_y) =w whaimana 37^o= (20)(0.8) = 16 Ngā Newton

Ko te wāhanga-x o te taumaha (w_x) = w hara θ = (20)(hara 37) = (20)(0.6) = 12 Newtons

te kaha noa (N) = w_y= 16 Niutona

hiahia Te kaha o waho (F)

otinga :

Te taurite o ngā tinana i runga i te papa whakarara – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 2 w_x = 12 Niutona

Te kaha o te waku nekeneke (f_k) = µ_k N = (0.1)(16) = 1.6 Ngā Niutoni

Te rahi o te kaha o waho F i pā ki te poraka :

F + f_k - w_x= 0

F = w_x- f_k

F = 12 – 1.6

F = 10.4 Newton

Nui atu te kaha o waho F i te 10.4 Newton.

2. Taumaha o tētahi poraka = 2 kg, tauwehenga o te waku pateko µ_s= 0.4 me θ = 45^oTātaihia te rahi o te kaha F kia tīmata ai te poraka ki te paheke ki runga.

Te taurite o ngā tinana i runga i te papa whakarara – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 3 Mōhiotia:

Te tauwehenga o te waku pateko (µ)_s) = 0.4

Koki (θ) = 45^o

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s²

Papatipu o te poraka (m) = 2 kirokaramu

Taumaha o te poraka (w) = mg = (2 kg)(10 m/s)²) = 20 kg m/s² = 20 Niutona

Ko te wāhanga-x o te taumaha (w_x) = w hara θ = (20)(hara 45) = (20)(0.5√2) = 10√2 Newtons

Ko te wāhanga-y o te taumaha (w_y) = w cos θ = (20)(cos 45) = (20)(0.5√2) = 10√2 Newtons

hiahia Te rahi o te kaha F

Rongoā:

Te taurite o ngā tinana i runga i te papa whakarara – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 4 Ka tīmata te poraka ki te paheke ki runga, ki te mea F ≥ w_x + f_s.

Ko te wāhanga-x o te taumaha:

w_x = 10√2 Nūtene

te wāhanga-y o te taumaha :

w_y = 10√2 Nūtene

Te kaha noa :

N = w_y = 10√2 Nūtene

Te kaha o te waku pateko :

f_s = µ_sN = (0,4)(10√2) = 4√2

Te rahi o te kaha F kia tīmata ai te poraka ki te paheke ki runga :

F ≥ w_x + f_s

F ≥ 10√2 + 4√2

F ≥ 14√2 Newton

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Pānuitia atu

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūrei – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton

1. He pouaka o papatipu Kei runga i te papa whakarara te 5 kg i te koki 30^o. E tautokona ana te pouaka e te taura. Tāutuhia te kaha kume (T) me te kaha noa (N)!

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 1

otinga

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 2 ∑F_x = 0

T – w hara 30^o = 0

T = w sin 30^o

T = (5 kg)(9.8 m/s)²) hara 30^o

T = (49)(0.5)

T = 24.5 Ngā Niutona

∑F_y = 0

N – w cos 30^o = 0

N = w cos 30^o

N = (49)(0.87)

N = 43 Niutona

2. E rua ngā mea he taumaha m₁ = m₂ = 2 kg, e honoa ana e te aho kore-papatipu i runga i te pūrei kore-waku. Kimihia te kaha kume T₁ a T₂.

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 3

otinga

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 4

(a) Kauwhata tinana-kore mō te mea 1 (b) Kauwhata tinana-kore mō te mea 2

Whakamahia te ture tuatahi a Newton ki te mea 1:

∑F_y = 0

T₁ - w₁ = 0

T₁ =w₁ = m₁karamu = (2 kg)(9.8 m/s²) = 19.6 N

Anga Te ture tuatahi a Newton ki te mea 2:

∑F_y = 0

T₂ - w₂ = 0

T₂ =w₂= m₂ karamu = (2 kg)(9.8 m/s²) = 19.6 N

T₁ =T₂= 19.6 N.

3. He mea nā taimaha w_A = 30 N me tētahi mea taumaha w_B = 40 N, e herea ana e te taura māmā e whiti ana i runga i te porotaka kore-waku o te papatipu iti noa iho. Whakatauhia te tauwehenga o te mōrahi te waku pumau i waenganui i te w_B me te mata hianga, mēnā kei te okioki te pūnaha.

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 5

otinga

Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 6

(a) Kauwhata tinana-kore mō te mea w_A (b) Kauwhata tinana-kore mō te mea w_B

Whakamahia te ture tuatahi a Newton ki te mea w_A i te ahunga poutū (y):

∑F_y = 0 (kāore he whakaterenga i te ahunga poutū)

T – w_A= 0

T = w_A = 30 Niutona

Whakamahia te ture tuatahi a Newton ki te mea w_Bi te ahunga poutū (y) :

∑F_y = 0

N – w_B whaimana 45^o= 0

N = w_Bwhaimana 45^o= (40)(0.7) = 28 Ngā Newton

Whakamahia te ture tuatahi a Newton ki te mea w_Bi te ahunga whakapae (x):

∑F_x= 0

F_k + w_B hara 45^o – T = 0

μ_s N + w_B hara 45^o – T = 0

μ_s (28) + (40)(0.7) – 30 = 0

μ_s (28) + 28 – 30 = 0

μ_s(28) = 30 – 28

μ_s(28) = 2

μ_s= 2/28

μ_s= 0.07

Ko te tauwehenga o te waku pūmau mōrahi i waenga i te w_B me te mata piko = 0.07.

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Pānuitia atu

Ngā matūriki i roto i te taurite rua-ahu – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton

1. Kimihia te kaha kume T₁, T₂, a T₃Kaua e aro ki ngā taura papatipu.

Ngā matūriki i roto i te taurite rua-ahu – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 1

otinga

Ngā matūriki i roto i te taurite rua-ahu – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 2

(a) Kauwhata tinana-kore mō te mea (b) Kauwhata tinana-kore mō te taura

Whakatakotoria te Te ture tuatahi a Newton i runga i te mea:

ΣF_y = 0

T₁ – w = 0

T₁ = w = mg

T₁ = (5 kg)(9.8 m/s²)

T₁= 49 kg m/s²

T₁ = 49 N

Whakamahia te ture tuatahi a Newton ki te taura:

∑F_x = 0

T_3x - T _2x= 0

T₃whaimana 30^o - T₂ whaimana 40^o= 0

0.87 T₃– 0.77 T₂= 0

0.87 T₃ = 0.77 T₂

T₂= 0.87 T₃/ 0.77 = 1.1 T₃———- Whārite 1

∑F_y = 0

T_3y +T_2y - T_1y = 0

T₃hara 30^o +T₂hara 40^o - T₁ = 0

0.5 T₃+ 0.64 T₂– 49 N = 0 ———- Whārite 2

Te whakakapi i a T₂i roto i te whārite 2 ki roto i te whārite 2:

0.5 T₃+ 0.64 (1.1 T₃) – 49 N = 0

0.5 T₃ + 0.70 T₃- 49 = 0

1.2 T₃ - 49 = 0

1.2 T₃= 49

T₃ = 49/1.2

T₃ = 41 N

---

T₂ = 1.1 T₃

T₂ = (1.1)(40.8 N)

T₂ = 45 N

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Pānuitia atu

Ngā matūriki i roto i te taurite kotahi-ahu – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton

^{1. Mass o tētahi mea, m = 10 kg, e tautokona ana e te taura. Kimihia te kume i roto i te taura!}karamu = 10 m/s²

Ngā matūriki i roto i te taurite kotahi-ahu – te whakamahinga o ngā raruraru me ngā otinga o te ture tuatahi a Newton 1 Mōhiotia:

Taumaha (m) = 10 kg

Whakaterenga na te kaha o te kaha (g) = 10 m/s²

E hiahiatia ana: Te kaha kume (T)

Rongoā:

ΣF_y = 0

T – w = 0

T = w

T = mg

T = (10 kg)(10 m/s)²) = 100 kg m/s²

T = 100 Ngā Niutona

2. Ko te taumaha o te mea he 10 kg. Kimihia te kume i roto i te taura….. Te whakaterenga nā te kaha ā-papatipu = 10 m/s².

otinga

Mōhiotia:

Taumaha (m) = 10 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s².

E hiahiatia ana: Te kaha kume (T)

Rongoā:

^{w = taimaha = mg = (10 kg)(10 m/s2)}) = 100 kg m/s²

T₁ = te kaha kume 1

_T1x = te wāhanga-x o te kaha kume 1 = T₁ whaimana 45^o = 0.7 T₁

_T1y = te wāhanga-y o te kaha kume 2 = T₁ hara 45^o = 0.7 T₁

T₂ = te kaha kume 2

_T2x = te wāhanga-x o te kaha kume 2 = T₂ whaimana 45^o = 0.7 T₂

_T2y = te wāhanga-y o te kaha kume 2 = T₂ hara 45^o = 0.7 T₂

Te āhua taurite ΣF = 0.

tuaka-y:

ΣF_y = 0

T_1y +T_2y – w = 0

0.7T₁ + 0.7T₂ - 100 = 0

0.7T₁ + 0.7T₂ = 100 —– whārite 1

tuaka-x:

ΣF_x = 0

T_2x - T_1x = 0

0.7T₂ – 0.7T₁ = 0

0.7T₂ = 0.7T₁

T₂ =T₁ —– whārite 2

Whakatauhia te rahi o T₁ :

0.7T₁+ 0.7T₁= 100

1.4T₁ = 100

T₁ = 100/1.4

T₁= 71.4 Niutona

T₁ =T₂ nō reira T₂ = 71.4 Niutona

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Pānuitia atu

Ngā tinana e honoa ana e te taura me te pūrei – te whakamahinga o ngā raruraru me ngā otinga o te ture nekehanga a Newton

1. E rua ngā pouaka e honoa ana e te taura e rere ana i runga i te pūreirei. Kaua e aro ki te taumaha o te taura me te pūreirei me te waku i roto i te pūreirei. Mass o te pouaka 1 = 2 kg, te taumaha o te pouaka 2 = 3 kg, te whakaterenga nā te kaha ā-papa = 10m/s². Kimihia (a) Te whakaterenga o te pūnaha (b) Te kukū o te taura!

Ngā tinana e honoa ana e te taura me te pūrei - te whakamahinga o te ture nekehanga a Newton, ngā raruraru me ngā otinga 1

otinga

Ngā tinana e honoa ana e te taura me te pūrei - te whakamahinga o te ture nekehanga a Newton, ngā raruraru me ngā otinga 2 Mōhiotia:

Taumaha o te pouaka 1 (m₁) = 2 kirokaramu

Taumaha o te pouaka 2 (m₂) = 3 kirokaramu

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s²

Taumaha o te pouaka 1 (w₁) = m₁ g = (2)(10) = 20 Newton

Taumaha o te pouaka 2 (w₂) = m₂ g = (3)(10) = 30 Newton

Rongoā:

(a) te rahi me te ahunga o te whakaterenga

w₂> w₁ no reira te Ka tere ake te pouaka 2 ki raro, ka tere ake hoki te pouaka 1 ki runga.

Ngā kaha he rite te ahunga o te whakaterenga (w₂ a T₁), he pai tōna tohu. Ngā kaha e anga whakahāwea ana ki te whakaterenga (T₂ a w₁), he kino tōna tohu.

∑F = ma

w₂ - T₂ +T₁ - w₁ = (m₁ +m₂) he ——-> T₁ =T₂ =T

w₂ – T + T – w₁ = (m₁ +m₂) a

w₂ - w₁ = (m₁ +m₂) a

30 – 20 = (2 + 3) he

10 = 5

ā = 10 / 5

a = 2 m/s²

Te rahi o te whakatere he 2 m/s².

(b) Te kaha kume

Te pouaka 2:

E rua ngā kaha e pā ana ki te pouaka 2: tuatahi, te taumaha o te pouaka 2 (w₂), e tohu ana ki raro, nō reira he pai. Tuarua, ko te kaha kume i pā ki te pouaka 2 (T₂), e tohu ana ki runga, nō reira he kino. Whakamahia Te ture tuarua a Newton o te nekehanga.

∑F = ma

w₂ - T₂ = m₂ a

30 – T₂ = (3)(2)

30 – T₂ = 6

T₂ = 30 - 6

T₂ = 24 Niutona

Pouaka 1:

E rua ngā kaha e pā ana ki te pouaka 1. tuatahi, te taumaha o te pouaka 1 (w₁), e tohu ana ki raro, nō reira he kino. tuarua, te kaha kume i pā ki te pouaka 1 (T₁) e tohu ana ki runga, nō reira he pai. Whakamahia te ture tuarua o te nekehanga a Newton:

∑F = ma

T₁ - w₁ = m₁ a

T₁ – 20 = (2)(2)

T₁ - 20 = 4

T₁ = 20 + 4

T₁ = 24 Niutona

Te rahi o te kaha kume = T₁ =T₂ = T = 24 Niutona

2. He mea kei runga i tētahi mata whakapae taratara. Te taumaha o te mea 1 = 2 kg, te taumaha o te mea 2 = 4 kg, te whakaterenga nā te kaha ā-papatipu = 10 m/s², te tauwehenga o te waku pūmau = 0.4, te tauwehenga o te waku nekeneke = 0.3. Kei te okioki te pūnaha, kei te whakateretere rānei? Mena kei te whakatereterehia te pūnaha, kimihia te rahi me te ahunga o te whakateretere o te pūnaha!

Ngā tinana e honoa ana e te taura me te pūrei - te whakamahinga o te ture nekehanga a Newton, ngā raruraru me ngā otinga 3

otinga

Ngā tinana e honoa ana e te taura me te pūrei - te whakamahinga o te ture nekehanga a Newton, ngā raruraru me ngā otinga 4 Mōhiotia:

Taumaha o te mea 1 (m₁) = 2 kirokaramu

Taumaha o te mea 2 (m₂) = 4 kirokaramu

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s²

Tauwehenga o te te waku pumau (μ_s) = 0.4

Te tauwehenga o te waku nekeneke (μ_k) = 0.3

Taumaha o te mea 1 (w₁) = m₁ g = (2)(10) = 20 Newton

Taumaha o te mea 2 (w₂) = m₂ g = (4)(10) = 40 Newton

Te kaha noa i whakapaua ki te mea 1 (N) = w₁ = 20 Niutona

Te kaha o te waku pao i pā ki te mea 1 (f_s) = μ_sN = (0.4)(20) = 8 Ngā Niutoni

Te kaha o te waku nekeneke i pā ki te mea 1 (f_k) = μ_kN = (0.3)(20) = 6 Ngā Niutoni

Hiahia: whakaterenga (a)

Rongoā:

w₂ > wh_s (40 Newton > 8 Newton) nō reira ka whakaterea te mea 2 ki raro poutū, ā, ka whakaterea te mea 1 ki matau whakapae. Ko te kaha waku e pā ana ki ngā mea 1 ko te kaha o te waku nekeneke (f_k). Whakamahia te ture tuarua o te nekehanga a Newton:

∑F = ma

w₂ - te = (m₁ +m₂) a

40 – 6 = (2 + 4) he

34 = 6

ā = 34 / 6 = 17 / 3

a = 5.7 m/s²

Te rahi o te whakaterenga = 5.7 m/s²

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Pānuitia atu

Te whakamahinga o te ture nekehanga a Newton i roto i te ararewa – ngā raruraru me ngā otinga

1. He tangata 50-kg te taumaha i roto i tētahi ararewa. Whakaterenga na te kaha o te kaha = 10m/s²Whakatauhia te kaha noa ka pāngia te mea e te ararewa, mēnā:

(a) kei te okioki te ararewa

(b) kei te neke te ararewa ki raro i te tere pumau

(d) he ararewa i whakaterea ki raro i te tere pumau 5 m/s²

(e) ararewa i roto i te taka-kore

otinga

Te whakamahinga o te ture nekehanga a Newton ki runga i ngā ararewa - ngā raruraru me ngā otinga 1 Mōhiotia:

Tangata papatipu (m) = 50 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s²

Taumaha (w) = mg = (50)(10) = 500 Ngā Newton

Hiahia: Te kaha noa (N)

Rongoā:

(a) kei te okioki te ararewa

Kei te okioki te ararewa, nō reira kāore he whakaterenga (a = 0)

Ka whiriwhiria e tātou te ahunga whakarunga i te ahunga pai, me te ahunga whakararo i te ahunga kino.

ΣF = mā

N – w = 0

N = w

N = 500 Niutona

(b) kei te neke te ararewa ki raro i te tere pumau

Tere pumau, nō reira kāore he whakaterenga (a = 0)

Ka whiriwhiria e tātou te ahunga whakarunga i te ahunga pai, me te ahunga whakararo i te ahunga kino.

ΣF = mā

N – w = 0

N = w

N = 500 Niutona

Kei runga te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei runga.

N – w = ma

N = w + ma

N = 500 + (50)(5)

N = 500 + 250

N = 750 Niutona

Ka kaha ake te rongo o te tangata i te papa e pana ana ki runga i tērā i te wā e tū ana te ararewa, e neke ana rānei me te tere pumau.

Ki te tū te tangata i runga i te tauine, ka pānuihia e te tauine te rahi o te kaha whakararo e tukuna ana e te tangata i runga i te tauine. E ai ki te ture tuatoru a Newton, he rite tēnei ki te rahi o te kaha noa whakararo e tukuna ana e te tauine ki runga i te tangata.

(d) he ararewa i whakaterea ki raro i te tere pumau 5 m/s²

Kei raro te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei raro.

w – N = ma

N = w – ma

N = 500 – (50)(5)

N = 500 – 250

N = 250 Niutona

Ko te taumaha o te tangata he 250 N, he iti iho i te taumaha tuturu w = 500 N.

(e) te ararewa i te hinganga kore utu

Ko te tikanga o te hinganga kore utu he rite tonu te tere o te ararewa ki te tere o te kaha ā-papa. Ko te rahi o te tere o te kaha ā-papa he 9,8 m/s², ko tōna ahunga kei raro ki waenganui o Papatūānuku. Ka piki haere te tere i roto i te wā mā te 9,8 m/s i ia hēkona.

Kei raro te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei raro.

w – N = ma

N = w – ma

N = 500 – (50)(10)

N = 500 – 500

N = 0

2. Tātaihia te kume i roto i te taura ararewa. Ko te papatipu o te ararewa = 2000 kg.

(a) kei te okioki te ararewa

(B) ka whakaterea te ararewa ki raro i te tere pumau 5 m/s²

(d) ararewa i te hinganga kore utu

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s²

otinga

Te whakamahinga o te ture nekehanga a Newton ki runga i ngā ararewa - ngā raruraru me ngā otinga 2 Mōhiotia:

Te taumaha o te ararewa (m) = 2000 kg

Te whakaterenga o te kaha ā-papa (g) = 10 m/s²

taumaha (w) = mg = (2000)(10) = 20,000 Newton

E hiahiatia ana: Te kaha kume (T)

Rongoā:

(a) kei te okioki te ararewa

ararewa kei te okioki, nō reira kāore he whakaterenga (a = 0)

Ka whiriwhiria e tātou te ahunga whakarunga hei ahunga pai, me te ahunga whakararo hei ahunga kino.

ΣF = mā

T – w = 0

T = w

T = 20,000 Ngā Niutona

Te kukū o te taura (T) = te taumaha o te ararewa (w) = 20,000 Newton

(b) he ararewa i whakaterea ki raro i te tere pumau 5 m/s²

Kei raro te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei raro.

w – T = ma

T = w – ma

T = 20,000 – (2000)(5)

T = 20,000 – 10,000

T = 10,000 Ngā Niutona

c) ka whakaterea ake te ararewa i te tere pumau 5 m/s²

Kei raro te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei runga.

T – w = ma

T = w + ma

T = 20,000 + (2000)(5)

T = 20,000 + 10,000

T = 30,000 Ngā Niutona

(d) ararewa i te hinganga kore utu

Kei raro te ahunga o te whakaterenga, nō reira ka whiriwhiria e tātou te ahunga pai hei raro.

w – T = ma

T = w – ma

T = 20,000 – (2000)(10)

T = 20,000 – 20,000

T = 0

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