Ngā Raru Whakaoti i roto i te Ahupūngao Taketake

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 ā-pokapū me te rahi o te kaha kume?

Mōhiotia:

Taumaha (m) = 0.2 kg

Pūtoro (r) = 1 m

Tere koki (ω) = 10 hurihanga/min = 10 hurihanga/60 s = 0.17 hurihanga/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 me Σ F

Rongoā:

(a) Te rahi o te whakaterenga ā-pokapū

(b) Te rahi o te kaha kume

Σ F = ma

T = ma _s

T = (0.2 kg)(1 m/s ² )

T = 0.2 kg m/s ²

T = 0.2 N

[irp]

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:

Taumaha (m) = 1 kg

Pūtoro (r) = 1 mita

Te tohenga (T) = te kaha potakataka ( Σ F) = 100 N

E hiahiatia ana: v te mōrahi

Rongoā:

[irp]

[wpdm_package id='499′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Ko te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu

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

N= kaha noa

N sin θ = te wāhanga whakapae o te kaha noa

N cos θ = te wāhanga poutū o te kaha noa

w = mg = te taumaha o te motuka

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

Ko te kaha whakapae kupenga, te wāhanga whakapae o te kaha noa ( N sin θ) , e hiahiatia ana kia neke porowhita tonu te motuka 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:

[irp]

Whakamahia te ture nekehanga a Newton i te ahunga whakapae:

Whakakapia a N i te whārite 1 ki te N i te whārite 2 :

[wpdm_package id='497′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu

1. Ka huri te motuka 2000-kg i tētahi piko i runga i tētahi rori papatahi he 150 m te whānui. Ko te tauwehenga o te waku pao he 0.5. Tātaihia te tere mōrahi kia whai te motuka i te piko, ā, kia kore ai e paheke. Te whakaterenga nā te kaha ā-papatipu = 10 m/ ^s2.

Mōhiotia:

Taumaha (m) = 2000 kg

Pūtoro (r) = 150 mita

Tauwehenga o te waku pateko ( μs ₎ = 0.5

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

Te kaha o te waku pateko (F _s ) = μ _s N = μ _s w = (0.7)(20,000 N) = 14,000 N

E hiahiatia ana : v

Rongoā:

[irp]

[wpdm_package id='496′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu

1. E rua ngā papatipu m ₁ = 2 kg me m ₂ = 5 kg kei runga i te papa whakarara, ā, e honoa ana mā te aho e ai ki te pikitia. Ko te tauwehenga o te waku nekeneke i waenga i te m ₁ me te whakarara he 0.2, ā, ko te tauwehenga o te waku nekeneke i waenga i te m ₂ me te whakarara he 0.1.

(a) Whakatauhia tō rātou whakaterenga

(b) Whakatauhia te kaha kume

Mōhiotia:

Taumaha 1 (m1 ₎ = 2 kg

Taumaha 2 (m2 ₎ = 4 kg

Te tauwehenga o te waku nekeneke i waenga i te _m1 me te papa whakarara (μk1 ₎ = 0.2

Te tauwehenga o te waku nekeneke i waenga i te m2 _me te papa whakarara (μk2 ₎ = 0.1

Te whakaterenga nā te kaha ā-papa (g) = 9.8 m/s ²

a) Te rahi me te ahunga o te whakaterenga

w1 = taumaha ₁ = m1 _karamu = (2 kg)(9.8 m/s2 ⁾ = 19.6 Newton

w _1x = w ₁ sin 30 ^o = (19.6 N)(0.5) = 9.8 Ngā Newton

w _1y = w ₁ cos 30 ^o = (19.6 N)(0.87) = 17 Ngā Newton

N ₁ = Te kaha noa i runga i te m ₁ = w _1y = 17 Newtons

F _k1 = Te kaha o te waku nekeneke i runga i te m ₁ = μ _k1 N ₁ = (0.2)(17 N) = 3.4 Newtons

---

w2 = taumaha 2 = m2 _{karamu =} (4 kg)(9.8 m/s2 ⁾ = 39.2 Ngā Newton

w _2x = w ₂ hara 60 ^o = (39.2 N)(0.87) = 34.1 Newtons

w _2y = w ₂ cos 60 ^o = (39.2 N)(0.5) = 19.6 Ngā Newton

N ₂ = Te kaha noa i runga i te m ₂ = w _2y = 19.6 Newtons

F _k2 = Te kaha o te waku nekeneke i runga i te m ₂ = μ _k2 N ₂ = (0.1)(19.6 N) = 1.96 Newtons

---

Te rahi o te whakaterenga:

∑ F _x = ma _x

w _2x > w _1x nō reira he rite 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 = (m ₁ + m ₂ ) 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 = 3.16 m/s ²

Te rahi o te whakaterenga = 3.16 m/s² ^. Te ahunga o te whakaterenga = te ahunga o T1 ₌ te ahunga o _w2x

b) Te rahi o te kaha kume

Whakamahia te ture tuarua a Newton ki te mea 2:

w _2x – F _k2 – T ₂ = m ₂ a _x

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

32.14 N – T ₂ = 12.64 N

T ₂ = 32.14 N – 12.64 N = 19.5 Ngā Newton

Te kaha kume = T = T ₁ = T ₂ = 19.5 Newton

[irp]

2. m ₁ = 4 kg, m ₂ = 2 kg. Whakatauhia (a) te rahi me te ahunga o te whakaterenga (b) te rahi o te kaha kume e hono ana i a m ₁ me m ₂ (c) te rahi o te kaha kume e hono ana i te pūrei me te tuanui.

otinga

w1 ₌ m1 _karamu = (4 kg)(9.8 m/s2 ⁾ = 39.2 Ngā Newton

w2 = m2 _karamu = (2 kg)(9.8 m/s2 ⁾ = 19.6 Ngā _Newton

a) Te rahi me te ahunga o te whakaterenga

∑ F _y = ma _y

Ko te w ₁ > w _2, nō reira he rite te ahunga o te mea ki te ahunga o te taumaha 1 ( w ₁ ) . He pai ngā kaha he rite te ahunga ki te whakaterenga, ā, he kino ngā kaha he rerekē te ahunga o 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) ia _tau

a _y = 19.6 N : 6 kg

a _y = 3.26 m/s ²

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 _m2

Whakamahia te ture tuarua a Newton ki te m ₂ :

∑ F _y = ma _y

w1 – _T1 = _m1 a _y​

39.2 N – T ₁ = (4 kg)( 3.26 m/s ² )

39.2 N – T ₁ = 13.04 N

T ₁ = 39.2 N – 13.04 N

T ₁ = 26.16 Newton

Te rahi o te kaha kume e hono ana i ngā mea = T = T ₁ = T ₂ = 26.16 Newton

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

Kei te okioki te pūrei:

∑ F _y = ma _y —— a _y = 0

∑ F _y = 0

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

T ₃ – T ₁ – T ₂ = 0

T3 = _T1 + _T2

He rite te rahi o _T1 me T2 _, T1 ₌ T2 ₌ T = 26.16 N :

T ₃ = 2T = 2(26.16 N) = 52.32 Ngā Newton

[irp]

3. Ko te Poraka 1 ( _m1 = 10 kg) me te poraka 2 (m2 ₌ 15 kg) i honoa mā te taura i runga i te pūrei kore-waku. Ko te tauwehenga o te waku pumau 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.

otinga

w ₁ = Te taumaha o te poraka 1 = m ₁ karamu = (10 kg)(9.8 m/s ² ) = 98 Newton

w ₂ = Te taumaha o te poraka 2 = m ₂ karamu = (15 kg)(9.8 m/s ² ) = 147 Newton

w _2y = w ₂ cos 30 ^o = (147 N)(0.87) = 127.89 Ngā Newton

w _2x = w ₂ hara 30 ^o = (147 N)(0.5) = 73.5 Newtons

N ₂ = Te kaha noa i runga i te poraka 2 = w _2y = 127.89 Newtons

F _k2 = Te kaha o te waku nekeneke i runga i te poraka 2 = μ _k2 N ₂ = (0.42)(127.89 N) = 53.7 Newtons

F _s2 = Te kaha o te waku pumau i runga i te poraka 2 = μ _s2 N ₂ = (0.6)(127.89 N) = 76.7 Newtons

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

∑ F _x = ma _x —— a _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.

F – F _k2 – w _2x – w ₁ – T ₂ + T ₁ = 0

F – F _k2 – w _2x – w ₁ = 0

F = F _k2 + w _2x + w ₁

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 = ma _y —— a _y = 0

∑ F _y = 0

T ₁ – w ₁ = 0

T ₁ = w ₁ = 98 Newton

Whakamahia te ture nekehanga a Newton ki te poraka 2:

F – F _k2 – w _2x – T ₂ = 0

T ₂ = F – F _k2 – w _2x

T ₂ = 225.2 N – 53.7 N – 73.5 N

T ₂ = 98 Newton

Te rahi o te kaha kume = T ₁ = T ₂ = T = 98 Newton

[irp]

4. Kei runga i te mata whakapae te poraka 1 (m _{1 = 16 kg), ā} , kei runga i te papa whakarara maeneene te poraka 2 (m _{2 = 12 kg), e honoa ana e te taura e whiti ana i runga i tētahi pūreri iti, kore-waku. Kei runga i te poraka 2 te poraka 3 (m 3} = 5 kg). Ko te tauwehenga o te waku nekeneke i waenga i te poraka 2 me te mata whakapae ko te 0,4. Ko te tauwehenga o te waku pūmau i waenga i te poraka 2 me te poraka 3 ko te 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?

Rongoā:

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

w ₁ = Te taumaha o te poraka 1 = m ₁ karamu = (16 kg)(9.8 m/s ² ) = 156.8 Newton

w _1x = w ₁ sin 60 ^o = (156.8 N)(0.87) = 136.4 Ngā Newton

w _1y = w ₁ cos 60 ^o = (156.8 N)(0.5) = 78.4 Ngā Newton

N ₁ = Te kaha noa i tukuna ki te poraka 1 e te papa whakarara = w _1y = 78.4 Newtons

w ₃ = Te taumaha o te poraka 3 = m ₃ karamu = (5 kg)(9.8 m/s ² ) = 49 Newton

N ₂₃ = Te kaha noa i tukuna ki te poraka 3 e te poraka 2 = w ₃ = 49 Newtons

N ₃₂ = Te kaha noa i tukuna ki te poraka 2 e te poraka 3 = N ₂₃ = w ₃ = 49 Newton

(Ko N ₂₃ me N ₃₂ he takirua mahi-tauhohenga )

F _s23 = Te kaha o te waku pūmau i pā ki te poraka 3 e te poraka 2 = μ _s N ₂₃ = (0.3)(49 N) = 14.7 Newtons

F _s32 = Te kaha o te waku pūmau i pā ki te poraka 2 e te poraka 3 = F _{s 23} = 14.7 Newton

(Ko te F _s23 me te F _s32 he takirua mahi-tauhohenga )

w ₂ = Te taumaha o te poraka 2 = m ₂ karamu = (12 kg)(9.8 m/s ² ) = 117.6 Newton

N ₂ = Te kaha noa i tukuna ki te mea 2 e te mata whakapae = w ₂ + N ₃₂ = 117.6 Newtons + 49

Niutoni = 166.6 Niutoni

F _k2 = Te kaha o te waku nekeneke i runga i te poraka 2 = μ _k N ₂ = (0.4)(166.6 N) = 66.64 Newtons

Whakamahia te ture nekehanga a Newton ki te poraka 3:

∑ F _x = ma _x

F _s23 = m ₃ a _x

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

μ _s m ₃ g = m ₃ a _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 tahi 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 T ₂ = te ahunga o w _1x.

∑ F _x = ma _x

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

w _1x – F _k2 = (m ₁ + m ₂ + m ₃ ) 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 = 2.11 m/s ²

He pai te _x , arā, ko te ahunga o te nekehanga poraka, te ahunga rānei o te whakaterenga, he rite tonu ki te ahunga o T ₂ , ki te ahunga rānei o w _1x.

Ko te rahi o te whakaterenga he 2.11 m/s² ^, he iti iho i te 2.94 m/s² ^, nō reira ka taea e tātou te whakatau kei te 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 = ma _x

w _1x – F _k2 = (m ₁ + m ₂ ) a _x

—–> F_k2 = μ_k N₂ = μ_k w₂ = μ_k m₂ karamu = (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/s ²

[wpdm_package id='493′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu

1. Kei runga i tētahi papa whakarara he poraka 2-kg te taumaha, he koki 37 ^° ki te whakapae. Tātaihia te kaha o waho i pā ki te poraka, kia kore ai te poraka e paheke ki raro i te papa. (sin 37 ^° = 0.6, cos 37 ^° = 0.8, g = 10 ms ^-2 , µ _k = 0.2)

Mōhiotia:

Taumaha (m) = 2 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

Taumaha o te poraka (w) = mg = (2)(10) = 20 Newton

Sin 37 ^o = 0.6

Cos 37 ^o = 0.8

Tauwehenga o te waku nekeneke (µk ₎ = 0.2

Ko te wāhanga-y o te taumaha (w _y ) = w cos 37 ^o = (20)(0.8) = 16 Newton

Ko te wāhanga-x o te taumaha (w _x ) = w sin θ = (20)(sin 37) = (20)(0.6) = 12 Ngā Newton

te kaha noa (N) = w _y = 16 Newton

E hiahiatia ana : Te kaha o waho (F)

Otinga :

w_x = 12 Niutona

Te kaha o te waku nekeneke (f _k ) = µ _k N = (0.1)(16) = 1.6 Newtons

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.

[irp]

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

Mōhiotia:

Ko te tauwehenga o te waku pumau (µs ₎ = 0.4

Koki (θ) = 45 ^o

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

Papatipu o te poraka (m) = 2 kirokaramu

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

Ko te wāhanga-x o te taumaha (w _x ) = w sin θ = (20)(sin 45) = (20)(0.5√2) = 10√2 Ngā Newton

Ko te wāhanga-y o te taumaha (w _y ) = w cos θ = (20)(cos 45) = (20)(0.5√2) = 10√2 Ngā Newton

E hiahiatia ana : Te rahi o te kaha F

Rongoā:

Ka tīmata te poraka ki te paheke ki runga, ki te mea F ≥ w_x + f_s.

Ko te wāhanga-x o te taumaha:

wx = 10√2 _Newton

te wāhanga-y o te taumaha :

w _y = 10√2 Newton

Te kaha noa :

N = w _y = 10√2 Newton

Te kaha o te waku pateko :

f _s = µ _s N = (0,4)(10√2) = 4√2

Ko 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

[wpdm_package id='492′]

1. Ngā matūriki i roto i te taurite kotahi-ahu
2. Ngā matūriki i roto i te taurite rua-ahu
3. Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero
4. Te taurite o ngā tinana i runga i te papa whakarara

Pānuitia atu

1. Kei runga i te papa piko tētahi pouaka he 5 kg te taumaha i te koki 30 ^° . E tautokona ana te pouaka e tētahi taura. Tātaihia te kaha kume (T) me te kaha noa (N)!

otinga

∑F_x = 0

T – w sin 30 ^o = 0

T = w sin 30 ^o

T = (5 kg)(9.8 m/s ² ) sin 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

[irp]

2. E rua ngā mea he taumaha m ₁ = m ₂ = 2 kg, e honoa ana e te aho kore taumaha i runga i te pūrei kore waku. Kimihia ngā kaha kume T ₁ me T ₂.

otinga

(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

T ₁ – w ₁ = 0

T ₁ = w ₁ = m ₁ karamu = (2 kg)(9.8 m/s ² ) = 19.6 N

Whakamahia te ture tuatahi a Newton ki te mea 2:

∑ F _y = 0

T ₂ – w ₂ = 0

T ₂ = w ₂ = m ₂ karamu = (2 kg)(9.8 m/s ² ) = 19.6 N

T ₁ = T ₂ = 19.6 N.

[irp]

3. He mea taumaha w _A = 30 N me te mea taumaha w _B = 40 N, e herea ana e te taura māmā e whiti ana i runga i te pūrei kore-waku he iti noa te papatipu. Whakatauhia te tauwehenga o te waku pūmau mōrahi i waenga i te w _B me te mata piko, mena kei te okioki te pūnaha.

otinga

(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 Newton

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

∑ F _y = 0

N – w _B cos 45 ^o = 0

N = w _B cos 45 ^o = (40)(0.7) = 28 Ngā Newton

Whakamahia te ture tuatahi a Newton ki te mea w _B i te ahunga whakapae (x):

∑ F _x = 0

F _k + w _B sin 45 ^o – T = 0

μ _s N + w _B hara 45 ^o – T = 0

μ _s (28) + (40)(0.7) – 30 = 0

μs (28) + 28 _– 30 = 0

μs (28) = 30 – ₂₈

μs (28) = ₂

μs ₌ 2 / 28

μs ₌ 0.07

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

[wpdm_package id='490′]

1. Ngā matūriki i roto i te taurite kotahi-ahu
2. Ngā matūriki i roto i te taurite rua-ahu
3. Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero
4. Te taurite o ngā tinana i runga i te papa whakarara

Pānuitia atu

1. Kimihia ngā kaha kume T1 _, T2 _, me T3 _. Kaua e aro ki te taumaha o te taura.

otinga

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

Whakamahia te ture tuatahi a Newton ki te mea:

ΣF _y = 0

T ₁ – w = 0

T ₁ = w = mg

T ₁ = (5 kg)(9.8 m/s ² )

T ₁ = 49 kg m/s ²

T ₁ = 49 N

Whakamahia te ture tuatahi a Newton ki te taura:

∑ F _x = 0

T _3x – T _2x = 0

T ₃ cos 30 ^o – T ₂ cos 40 ^o = 0

0.87 T ₃ – 0.77 T ₂ = 0

0.87 T ₃ = 0.77 T ₂

T ₂ = 0.87 T ₃ / 0.77 = 1.1 T ₃ ———- Whārite 1

[irp]

-

∑ F _y = 0

T _3y + T _2y – T _1y = 0

T ₃ hara 30 ^o + T ₂ hara 40 ^o – T ₁ = 0

0.5 T ₃ + 0.64 T ₂ – 49 N = 0 ———- Whārite 2

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

0.5 T ₃ + 0.64 (1.1 T ₃ ) – 49 N = 0

0.5 T ₃ + 0.70 T ₃ – 49 = 0

1.2 T ₃ – 49 = 0

1.2 T ₃ = 49

T ₃ = 49 / 1.2

T ₃ = 41 N

---

T ₂ = 1.1 T ₃

T ₂ = (1.1)(40.8 N)

T ₂ = 45 N

[wpdm_package id='488′]

1. Ngā matūriki i roto i te taurite kotahi-ahu
2. Ngā matūriki i roto i te taurite rua-ahu
3. Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero
4. Te taurite o ngā tinana i runga i te papa whakarara

Pānuitia atu

^{1. Te taumaha o tētahi mea, m = 10 kg, e tautokona ana e te taura. Kimihia te kume i roto i te taura!} g = 10 m/s ²

Mōhiotia:

Taumaha (m) = 10 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

E hiahiatia ana: Te kaha kume (T)

Rongoā:

ΣF _y = 0

T – w = 0

T = w

T = mg

T = (10 kg)(10 m/s² ⁾ = 100 kg m/ ^s²

T = 100 Ngā Niutona

[irp]

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

T ₁ = te kaha kume 1

_{T 1x} = te wāhanga-x o te kaha kume 1 = T ₁ cos 45 ^o = 0.7 T ₁

_{T 1y} = te wāhanga-y o te kaha kume 2 = T ₁ sin 45 ^o = 0.7 T ₁

T ₂ = te kaha kume 2

_{T 2x} = te wāhanga-x o te kaha kume 2 = T ₂ cos 45 ^o = 0.7 T ₂

_{T 2y} = te wāhanga-y o te kaha kume 2 = T ₂ sin 45 ^o = 0.7 T ₂

Ko te āhua taurite ΣF = 0.

tuaka-y:

ΣF _y = 0

T _1y + T _2y – w = 0

0.7T ₁ + 0.7T ₂ – 100 = 0

0.7T ₁ + 0.7T ₂ = 100 —– whārite 1

tuaka-x:

ΣF _x = 0

T _2x – T _1x = 0

0.7T ₂ – 0.7T ₁ = 0

0.7T ₂ = 0.7T ₁

T ₂ = T ₁ —– whārite 2

Whakatauhia te rahi o T ₁ :

0.7T ₁ + 0.7T ₁ = 100

1.4T ₁ = 100

T ₁ = 100 / 1.4

T ₁ = 71.4 Newton

T ₁ = T ₂ nō reira ko T ₂ = 71.4 Ngā Newton

[wpdm_package id='486′]

1. Ngā matūriki i roto i te taurite kotahi-ahu
2. Ngā matūriki i roto i te taurite rua-ahu
3. Te taurite o ngā tinana e honoa ana e ngā taura me ngā pūwero
4. Te taurite o ngā tinana i runga i te papa whakarara

Pānuitia atu

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. Te taumaha o te pouaka 1 = 2 kg, te taumaha o te pouaka 2 = 3 kg, te whakaterenga nā te kaha ā-papatipu = 10 m/s2 ^. Kimihia (a) Te whakaterenga o te pūnaha (b) Te kumenga i roto i te taura!

otinga

Mōhiotia:

Taumaha o te pouaka 1 (m1 ₎ = 2 kg

Taumaha o te pouaka 2 (m2 ₎ = 3 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

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

Taumaha o te pouaka 2 (w ₂ ) = m ₂ karamu = (3)(10) = 30 Newtons

Rongoā:

(a) te rahi me te ahunga o te whakaterenga

w ₂ > w _1, nō reira ka tere ake te pouaka 2 ki raro, ka tere ake hoki te pouaka 1 ki runga.

Ko ngā kaha he rite te ahunga ki te whakaterenga (w2 _me te w1 ₎ , he pai tō rātou tohu. Ko ngā kaha he rerekē te ahunga ki te whakaterenga (T2 _me te w1 ₎ , he kino tō rātou tohu.

∑ F = ma

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

w ₂ – T + T – w ₁ = (m ₁ + m ₂ ) a

w ₂ – w ₁ = (m ₁ + m ₂ ) a

30 – 20 = (2 + 3) he

10 = 5

ā = 10 / 5

a = 2 m/s ²

Ko te rahi o te whakaterenga he 2 m/ ^s².

(b) Te kaha kume

Te pouaka 2:

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

∑ F = ma

w ₂ – T ₂ = m ₂ a

30 – T ₂ = (3)(2)

30 – T ₂ = 6

T ₂ = 30 – 6

T ₂ = 24 Newton

Pouaka 1:

E rua ngā kaha e pā ana ki te pouaka 1. Tuatahi , ko te taumaha o te pouaka 1 (w ₁ ), e anga ana ki raro, nō reira he kino. Tuarua , ko te kaha kume i pā ki te pouaka 1 (T ₁ ) e anga ana ki runga, nō reira he pai. Whakamahia te ture nekehanga tuarua a Newton:

∑ F = ma

T ₁ – w ₁ = m ₁ a

T ₁ – 20 = (2)(2)

T ₁ – 20 = 4

T ₁ = 20 + 4

T ₁ = 24 Newton

Te rahi o te kaha kume = T ₁ = T ₂ = T = 24 Newton

[irp]

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

otinga

Mōhiotia:

Taumaha o te mea 1 (m1 ₎ = 2 kg

Taumaha o te mea 2 (m2 ₎ = 4 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

Tauwehenga o te waku pateko ( μs ₎ = 0.4

Ko te tauwehenga o te waku nekeneke ( μk ₎ = 0.3

Taumaha o te mea 1 (w ₁ ) = m ₁ karamu = (2)(10) = 20 Ngā Newton

Taumaha o te mea 2 (w ₂ ) = m ₂ karamu = (4)(10) = 40 Ngā Newton

Te kaha noa i pā ki te mea 1 (N) = w ₁ = 20 Newtons

Te kaha o te waku pūmau i pā ki te mea 1 (f _s ) = μ _s N = (0.4)(20) = 8 Newton

Te kaha o te waku nekeneke i pā ki te mea 1 (f _k ) = μ _k N = (0.3)(20) = 6 Newtons

E hiahiatia ana: te whakaterenga (a)

Rongoā:

w ₂ > f _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 nekehanga tuarua a Newton:

∑ F = ma

w ₂ – te = (m ₁ + m ₂ ) a

40 – 6 = (2 + 4) he

34 = 6

ā = 34 / 6 = 17 / 3

a = 5.7 m/s ²

Te rahi o te whakaterenga = 5.7 m/s ²

[wpdm_package id='484′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu

1. He tangata 50-kg te taumaha i roto i tētahi ararewa. Te whakaterenga nā te kaha ā-papatipu = 10 m/s ^2. Tātaihia te kaha noa i pā ki te mea e te ararewa, mēnā:

(a) kei te okioki te ararewa

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

(c) i tere ake te ararewa ki runga i te tere pumau 5 /s ²

(d) i whakaterea te ararewa ki raro i te tere pumau 5 m/s ²

(e) te ararewa i te hinganga kore utu

otinga

Mōhiotia:

Papatipu o te tangata (m) = 50 kg

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

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

E hiahiatia ana: 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.

Σ F = ma

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.

Σ F = ma

N – w = 0

N = w

N = 500 Niutona

(c) i whakaterea ake te ararewa i te tere pumau 5 m/s ²

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) i whakaterea te ararewa ki raro i te tere pumau 5 m/s ²

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 he heke ki raro ki waenganui o te Ao. 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

[irp]

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) i whakaterea te ararewa ki raro i te tere pumau 5 m/s ²

(c) i whakaterea ake te ararewa i te tere pumau 5 m/s ²

(d) ararewa i te hinganga kore utu

Te whakaterenga nā te kaha ā-papa (g) = 10 m/s ²

otinga

Mōhiotia:

Te taumaha o te ararewa (m) = 2000 kg

Te whakaterenga o te kaha ā-papa (g) = 10 m/s ²

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

E hiahiatia ana: Te kaha kume (T)

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 hei ahunga pai, me te ahunga whakararo hei ahunga kino.

Σ F = ma

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) i whakaterea te ararewa ki raro i te tere pumau 5 m/s ²

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) i tere ake te ararewa ki runga i te tere pumau 5 m/s ²

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

[wpdm_package id='482′]

1. Papatipu me te taumaha
2. Te kaha noa
3. Te ture tuarua o te nekehanga a Newton
4. Te kaha waku
5. Te nekehanga i runga i te mata whakapae me te kore he kaha waku
6. Te nekehanga o ngā tinana e rua me te tere tere ōrite i runga i te mata whakapae taratara me te kaha waku
7. Te nekehanga i runga i te papa whakarara me te kore he kaha waku
8. Te nekehanga i runga i te papa whakarara taratara me te kaha waku
9. Te nekehanga i roto i te ararewa
10. Ka honoa te nekehanga o ngā tinana e ngā taura me ngā pūrere
11. E rua ngā tinana he rite te rahi o te whakaterenga
12. Te whakaawhiwhi i tētahi piko papatahi – ngā nekehanga porowhita
13. Te whakaawhiwhi i tētahi kōpiko peeke – ngā hihiri o te nekehanga porowhita
14. Te nekehanga ōrite i roto i te porowhita whakapae
15. Te kaha pokapū i roto i te nekehanga porowhita ōrite

Pānuitia atu