Ko te pokapū o te taumaha, te pokapū papatipu rānei, he ariā taketake i roto i te ahupūngao me te miihini e whakamahia ana hei whakatau i te taurite me te pumau o tētahi mea. Ko te pokapū o te taumaha te pūwāhi e whakaarohia ana he kukū te taumaha o tētahi mea, ā, e whakaarohia ana te kaha o te taumaha e mahi ana. He mea nui te mārama ki tēnei ariā i roto i ngā momo tono, mai i te hoahoa hanganga hanga ki te tātari i te nekehanga o te mea. Ka matapakihia e tēnei tuhinga te whakamāramatanga o te pokapū o te taumaha, me pēhea te tatau i te pokapū o te taumaha mō ngā āhua o ngā mea, me ētahi tauira raruraru hei whakamārama i tēnei ariā.
Te Whakamāramatanga o te Pokapū o te Taumaha
Ko te pokapū o te taumaha (pokapū o te papatipu) te pūwāhi i roto i tētahi mea e taea ai te whakaaro he mea kukū te papatipu katoa o te mea hei tatau i ngā kaha me ngā au. I roto i tētahi pūnaha taunga Cartesian, ka taea te tatau i te pokapū o te taumaha o tētahi mea he papatipu tohatoha mā te whakamahi i te tātai e whai ake nei:
\[
x_{\kuputuhi{cm}} = \frac{\tapeke (x_i \cdot m_i)}{\tapeke m_i}
\]
\[
y_{\text{cm}} = \frac{\sum (y_i \cdot m_i)}{\sum m_i}
\]
\[
z_{\text{cm}} = \frac{\sum (z_i \cdot m_i)}{\sum m_i}
\]
Ko \( (x_i, y_i, z_i) \) ngā taunga o te huānga papatipu \( m_i \).
Te Pokapū o te Āhua Taumaha mō ngā Āhua Rerekē o ngā Mea
1. Te Pokapū Taumaha mō ngā Mea Taurite
Mō ngā mea ōrite (he rite te matotoru), ka taea te whakatau i te pokapū o te taumaha mā te huarahi māmā ake. Hei tauira:
– Tokotoko Angiangi: Kei waenganui o te tokotoko te pokapū taumaha o tētahi tokotoko angiangi, ōrite te roa \( L \), arā, kei \( x = \frac{L}{2} \).
– Papa Tapawhā: Ko te pokapū o te taumaha o tētahi papa tapawhā ōrite me te roa \( L \) me te whānui \( W \) kei te whakawhiti o ngā whakarara, arā, kei \( x = \frac{L}{2} \) me \( y = \frac{W}{2} \).
– Pereti Tapatoru: Kei runga i te hautoru o ia waenga o te tapatoru te pokapū o te taumaha o tētahi pereti tapatoru ōrite. Mō tētahi tapatoru me ngā taunga tihi \( A(x_1, y_1) \), \( B(x_2, y_2) \), me \( C(x_3, y_3) \):
\[
x_{\kuputuhi{cm}} = \frac{x_1 + x_2 + x_3}{3}
\]
\[
y_{\text{cm}} = \frac{y_1 + y_2 + y_3}{3}
\]
2. Te Pokapū Taumaha mō ngā Mea Kāore i te ōrite
Mō ngā mea kāore i te ōrite (me te kiato kore-ōrite), me tatau te pokapū o te taumaha mā te wehewehe i te mea kia iti te papatipu, me te tatau i tō rātou pokapū o te taumaha mā te whakamahi i te tātai whakauru. Hei tauira, mō tētahi mea he rerekē te kiato \( \rho(x, y, z) \):
\[
x_{\kuputuhi{cm}} = \frac{\int x \cdot \rho(x, y, z) \, dV}{\int \rho(x, y, z) \, dV}
\]
\[
y_{\kuputuhi{cm}} = \frac{\int y \cdot \rho(x, y, z) \, dV}{\int \rho(x, y, z) \, dV}
\]
\[
z_{\kuputuhi{cm}} = \frac{\int z \cdot \rho(x, y, z) \, dV}{\int \rho(x, y, z) \, dV}
\]
Ngā Tauira Pātai mō te Pokapū o te Āhuatanga Taumaha
Tauira Pātai 1: Te Pokapū o te Āhuatanga Taumaha o tētahi Toko Angiangi
Pātai:
Tātaihia te pokapū o te kaha taumaha o tētahi tokotoko angiangi, ōrite, he 10 mita te roa.
Otinga:
Nā te mea he ōrite te tokotoko, kei waenganui o te tokotoko te pokapū o te taumaha:
\[
x_{\text{cm}} = \frac{L}{2} = \frac{10 \, \text{m}}{2} = 5 \, \text{m}
\]
Nō reira, e 5 mita te pokapū o te kaha taumaha o te tokotoko angiangi mai i tetahi pito o te tokotoko.
Tauira Pātai 2: Te Pokapū o te Taumaha o tētahi Pereti Tapawhā
Pātai:
Tātaihia te pokapū o te taumaha o tētahi pereti tapawhā rite, he 8 mita te roa, he 4 mita te whānui.
Otinga:
Kei te wāhi e honoa ai ngā whakarara te pokapū o te taumaha o tētahi pereti tapawhā rite, arā:
\[
x_{\text{cm}} = \frac{L}{2} = \frac{8 \, \text{m}}{2} = 4 \, \text{m}
\]
\[
y_{\text{cm}} = \frac{W}{2} = \frac{4 \, \text{m}}{2} = 2 \, \text{m}
\]
Nō reira, ko te pokapū taumaha o te pereti tapawhā rite ko (4 m, 2 m).
Tauira Pātai 3: Te Pokapū o te Āhuatanga Taumaha o tētahi Pereti Tapatoru
Pātai:
Tātaihia te pokapū o te taumaha o tētahi pereti tapatoru ōrite me ngā pito kei ngā taunga \( A(0, 0) \), \( B(6, 0) \), me \( C(3, 6) \).
Otinga:
Ka taea te tatau i te pokapū o te taumaha o tētahi pereti tapatoru ōrite mā te whakamahi i te tātai:
\[
x_{\kuputuhi{cm}} = \frac{x_1 + x_2 + x_3}{3} = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \, \kuputuhi{m}
\]
\[
y_{\text{cm}} = \frac{y_1 + y_2 + y_3}{3} = \frac{0 + 0 + 6}{3} = \frac{6}{3} = 2 \, \text{m}
\]
Nō reira, ko te pokapū taumaha o te pereti tapatoru ko (3 m, 2 m).
Tauira Pātai 4: Te Pokapū o te Taumaha o tētahi Pūnaha Matūriki
Pātai:
Kei roto i tētahi pūnaha e toru ngā matūriki he rite te taumaha o ia matūriki, he 2 kg te taumaha, kei ngā taunga \( (1, 2) \), \( (3, 4) \), me \( (5, 6) \). Tātaihia te pokapū taumaha o te pūnaha matūriki.
Otinga:
I te mea he ōrite ngā taumaha o ngā matūriki, ka taea e tātou te whakamahi i tētahi tātai māmā hei tatau i te pokapū o te kaha taumaha:
\[
x_{\kuputuhi{cm}} = \frac{\tapeke (x_i \cdot m_i)}{\tapeke m_i} = \frac{(1 + 3 + 5) \cdot 2}{3 \cdot 2} = \frac{9}{3} = 3 \, \kuputuhi{m}
\]
\[
y_{\text{cm}} = \frac{\sum (y_i \cdot m_i)}{\sum m_i} = \frac{(2 + 4 + 6) \cdot 2}{3 \cdot 2} = \frac{12}{3} = 4 \, \text{m}
\]
Nō reira, ko te pokapū o te taumaha o te pūnaha matūriki ko (3 m, 4 m).
Whakamutunga
He ariā matua nui te pokapū o te taumaha i roto i te ahupūngao me te hangarau. He mea nui te mārama ki te tatau i te pokapū o te taumaha mō ngā āhua rerekē o ngā mea me ngā pūnaha matūriki hei tātari i te taurite me te pumau. Kua matapakihia e tēnei tuhinga te whakamāramatanga o te pokapū o te taumaha, me pēhea te tatau i te pokapū o te taumaha mō ngā mea ōrite me ngā mea kāore i te ōrite, ā, kua whakaratohia he tauira raruraru hei āwhina i te whakamārama i tēnei ariā.
I roto i te oranga o ia rā, he tino whai hua te mārama ki te pokapū o te taumaha i roto i ngā momo mahi, mai i te hoahoa whare ki te whakawhanaketanga hangarau. Mā te mārama me te whakamahi i te ariā o te pokapū o te taumaha, ka taea e tātou te hoahoa i ngā hanganga pumau ake, haumaru ake hoki, ā, ka pai ake te mārama ki ngā nekehanga o te mea.
