Tātai Pūmau o te Puna
Ko ngā puna tētahi o ngā huānga matua o te miihini e whakamahia ana hei penapena me te tuku pūngao. I roto i ngā tono hangarau maha, ka whakamahia ngā puna hei whakaiti i ngā wiri, hei penapena pūngao, me te whakarato i te kaha whakahoki i roto i ngā pūnaha miihini maha. Ko te pūmau puna, e mōhiotia ana ko te tauwehenga puna, ko te pūmau kaha puna rānei, he tawhā nui e whakatau ana i te pakari o te puna. Ka matapakihia e tēnei tuhinga te pūmau puna, ōna tātai e pā ana, me ōna tono i roto i ngā mara maha.
Te Mārama ki te Pūmau o te Puna
Ko te pūmau puna (\(k\)) he ine i te pakari o te puna. E whakaahua ana tēnei pūmau i te whanaungatanga i waenga i te kaha e pā ana ki te puna me te rerekētanga e puta mai ana (te roa, te whakapoto rānei). Mā te Ture a Hooke, ka whakaatuhia tēnei whanaungatanga:
\[
F = k \cdot x
\]
kāore i te mana:
– Ko te kaha e pā ana ki te puna (Newton, N) ko \(F\),
– Ko te pūmau puna (Newton ia mita, N/m) te \(k\),
– Ko te \(x\) te toronga, te whakapoto rānei o te puna mai i tōna tūranga taurite (mita, m).
Ka pā te ture a Hooke ki ngā puna kei roto i ō rātou rohe rapa, ka hoki te puna ki tōna āhua taketake i muri i te tangohanga o te kaha i whakamahia.
Te Tātai i te Pūmau Puna
Hei tatau i te pūmau puna, me mōhio tātou ki te kaha e pā ana me te roa, te poto rānei e puta mai ana. I runga i te Ture a Hooke, ka taea te tatau i te pūmau puna mā te whakamahi i te tātai:
\[
k = \frac{F}{x}
\]
Hei tauira, ki te toro atu tētahi puna mā te 0,1 mita ina pāngia e te kaha o te 10 Newton, ko tōna pūmau puna ko:
\[
k = \frac{10 \, \text{N}}{0.1 \, \text{m}} = 100 \, \text{N/m}
\]
Pūngao Pūmanawa o te Koanga
Ka rongoa hoki ngā puna i te pūngao pūmanawa ina pāngia e te kaha. Ko te pūngao pūmanawa o te puna (\(U\)) e rongoatia ana i roto i te puna kua whakapōturitia ka hoatu e te tātai:
\[
U = \frac{1}{2} kx^2
\]
E whakaatu ana tēnei tātai he rite tonu te kaha pūmanawa o te puna ki te tapawhā o tōna āhua rerekētanga. Ka tukuna tēnei kaha ina hoki te puna ki tōna tūranga taurite.
Taupānga Pūmau o te Puna
He maha ngā whakamahinga o te pūmau puna i roto i ngā mara maha. Ko ētahi o ēnei ko:
1. Motuka: I roto i te pūnaha whakatārewatanga o te waka, ka whakamahia ngā puna hei whakaiti i ngā wiri me te whakarato i te whakamarie mō te haere. Ko te pūmau puna i roto i te whakatārewatanga e whakatau ana i te pakari, te ngawari rānei o te whakatārewatanga.
2. Ngā Āhuahira: Ka whakamahia ngā puna i roto i ngā momo tikanga mīhini, pērā i ngā taputapu ine kaha (ngā dynamometer), ngā pūnaha taurite, me ngā ārai wiri. Mā te pūmau puna ka tika te mahi a ēnei pūnaha, ā, ka tutuki hoki ngā whakaritenga hoahoa.
3. Hikohiko: I roto i ētahi pūoko me ngā kaiwhakaoho, ka whakamahia ngā puna hei whakarato i te kaha whakahoki e tika ana mō te mahi tika. Hei tauira, i roto i ngā pana, i ngā pātene rānei, ka tukuna e ngā puna he urupare pā ki te kaiwhakamahi.
4. Ngā Hākinakina me ngā Whakangahau: Ka whakamahia ngā puna i roto i ngā taputapu hākinakina pēnei i ngā trampoline, ngā kopere, me ngā taputapu whakapakari tinana. Ka pā te pūmau puna ki te mahi me te haumaru o ngā taputapu.
5. Hauora: I roto i ngā taputapu hauora pēnei i ngā prostheses me ngā āwhina nekenekehanga, ka whakamahia ngā puna hei whakatauira i ngā nekehanga ā-taiao o te tinana. Mā ngā pūmau puna tika ka tino pai te whakamarie me te mahi a te kaiwhakamahi.
Te Tātaitanga o te Pūmau Puna i te Ara iahiko Puna
Ka taea te hono i ngā puna i roto i ngā whirihoranga rerekē, pērā i te raupapa me te whakarara. Ka rerekē te tatau i te pūmau puna i roto i ēnei ara iahiko i runga i te whirihoranga.
Ngā Puna i roto i te Raupapa
Ina honoa ngā puna e rua, neke atu rānei i roto i te raupapa, ka homai te pūmau puna katoa (\(k_{\text{total}}\)) e:
\[
\frac{1}{k_{\text{tapeke}}} = \frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}
\]
I roto i te whirihoranga raupapa, ka pāngia ia puna e te kaha kotahi, engari ko te huringa katoa ko te tapeke o ngā huringa o ia puna.
Ngā Puna i te Whakarara
Ina honoa ngā puna e rua, neke atu rānei, i te taha whakarara, ka homai te pūmau puna katoa (\(k_{\text{total}}\)) e:
\[
k_{\kuputuhi{katoa}} = k_1 + k_2 + \cdots + k_n
\]
I roto i te whirihoranga whakarara, ka pāngia ia puna e te rerekētanga ōrite, engari ko te kaha katoa ko te tapeke o ngā kaha i pā ki ia puna.
Tauira o te Tatau i te Pūmau Puna i roto i te Porowhita
Me kī he puna e rua ā tātou me ngā pūmau puna \(k_1 = 200 \, \text{N/m}\) me \(k_2 = 300 \, \text{N/m}\). E hiahia ana tātou ki te tatau i te pūmau puna katoa mō ngā whirihoranga raupapa me ngā whirihoranga whakarara.
Whirihoranga Raupapa
\[
\frac{1}{k_{\text{tapeke}}} = \frac{1}{200 \, \text{N/m}} + \frac{1}{300 \, \text{N/m}}
\]
\[
\frac{1}{k_{\text{tapeke}}} = 0.005 + 0.00333 = 0.00833
\]
\[
k_{\kuputuhi{katoa}} = \frac{1}{0.00833} \approx 120 \, \kuputuhi{N/m}
\]
Whirihoranga Whakarara
\[
k_{\kākau{katoa}} = 200 \, \kākau{N/m} + 300 \, \kākau{N/m} = 500 \, \kākau{N/m}
\]
Pūmau Puna i roto i ngā Pūnaha Hihiri
I roto i te tātari pūnaha hihiri, ka whakamahia hoki te pūmau puna i roto i ngā whārite nekehanga mō te pūnaha papatipu-puna. Ko te whārite rerekētanga e whakaahua ana i te nekehanga orooro māmā ko:
\[
m \frac{d^2x}{dt^2} + kx = 0
\]
kāore i te mana:
– Ko te taumaha o te mea (kg) ko te \(m\),
– Ko te whakaterenga (m/s²) te \(\frac{d^2x}{dt^2}\)
– Ko te pūmau puna (N/m) te \(k\),
– Ko te \(x\) te nekehanga mai i te tūranga taurite (m).
Ko te otinga mō tēnei whārite e whakaatu ana ka oioi te pūnaha me te auautanga maori (\(\omega\)) e homai ana e:
\[
\omega = \sqrt{\frac{k}{m}}
\]
Ine Pūmau o te Puna
Ka taea te ine i te pūmau puna mā te whakamahi i ngā taputapu whakamātautau māmā. Ko tētahi tikanga noa ko te tāpiri i tētahi taumaha ki te puna, me te ine i te roanga e puta mai ana. Mā te mōhio ki te kaha (te taumaha o te taumaha) me te roanga, ka taea te tatau i te pūmau puna mā te whakamahi i te Ture a Hooke.
Whakamutunga
He tawhā nui te pūmau puna e whakatau ana i te pakari o te puna, ā, he mea nui hoki tana mahi i roto i ngā tono hangarau me ngā tono hangarau. Mā te mārama ki ngā tātai e pā ana ki te pūmau puna, ka taea e tātou te tātari me te hoahoa i ngā pūnaha e whakamahi pai ana i ngā puna. Mai i ngā whakatārewatanga waka ki ngā taputapu hauora, ka whakarite te pūmau puna i te mahi tino pai me te mahi e tumanakohia ana. Mā te māramatanga hōhonu ki te pūmau puna ka taea te whakawhanake i ngā hangarau matatau ake me ngā tono whānui ake i roto i te oranga o ia rā.
