Forma mechaniczna – Wikipedia

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{DisplayStyle {BoldSymbol {r}} = x {boldsymbol {u}} _ {x}+y {boldsymbol {u}} _ {y}+z {boldsymbol {u}} _ {z}}}}}}}}}}}}}}}}}}}}}}

Prędkość punktu znajdująca się w R jest napisane

{DisplayStyle {BoldSymbol {v}} ({BoldSymbol {r}}) = {frac {{text {d}} {boldsymbol {r}}} {{text {d}} t}} = {frac {{text {boldsymbol {r}}} d}} x} {{text {d}} t}} {boldsymbol {u}} _ {x}+{frac {{text {d}} y} {{text {d}} t}} {boldsymbol {boldsymbol {boldsymbol {boldsymbol {boldsymbol { u}} _ {y}+{frac {{text {d}} z} {{text {d}} t}} {BoldSymbol {u}} _ {z}}

i przyspieszenie

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{DisplayStyle {BoldSymbol {A}} ({BoldSymbol {r}}) = {frac {{text {d}} {boldsymbol {v}}} {{text {d}} t}} = {frac {{text {text {text {boldsymbol {v}}} d}}^{2} {BoldSymbol {r}}} {{text {d}} t^{2}}} = {frac {{text {d}}^{2} x} {{text {d} } t^{2}}} {BoldSymbol {u}} _ {x}+{frac {{text {d}}^{2} y} {{text {d}} t^{2}}} {boldsymbol {u}} _ {y}+{frac {{text {d}}^{2} z} {{text {d}} t^{2}}} {BoldSymbol {u}} _ {z}}

{DisplayStyle {BoldSymbol {r}} = rho {boldsymbol {u}} _ {rho}+z {boldsymbol {u}} _ {z}}}}}}}}}

{displayStyle {BoldSymbol {v}} = {frac {{text {d}} {boldsymbol {r}}} {{text {d}} t}} = {frac {{text {d}} rho} {{text text {{text {{text {{d}} t}} = {frac {{text {text {d}} rho} {d}} t}} {BoldSymbol {u}} _ {rho}+rho {frac {{text {d}} varphi} {{text {d}} t}} {BoldSymbol {u}} _ {varphi}} +{frac {{text {d}} z} {{text {d}} t}} {boldsymbol {u}} _ {z}}

{displayStyle {BoldSymbol {a}} = {frac {{text {d}} {boldsymbol {v}}} {{text {d}} t}} = {frac {{text {d}}^{2} {{d}} t}} = {frac {{text {text {d}}^{2} { BoldSymbol {r}}} {{text {d}} t^{2}}} = left ({frac {{text {d}}^{2} rho} {{text {d}} t^{2} }}-rho po lewej ({frac {{text {d}} varphi} {{text {d}} t}} right)^{2} prawy) {boldsymbol {u}} _ {rho}+lewy (2 {2 {2 { frac {{text {d}} rho} {{text {d}} t}} {frac {{text {d}} varphi} {{text {d}} t}}+rho {frac {{text {d {d text {d {d. }}^{2} varphi} {{text {d}} t^{2}}} right) {boldsymbol {u}} _ {varphi}+{frac {{text {d}}^{2} z} {{text {d}} t^{2}}} {BoldSymbol {u}} _ {z}}

Formuły te opierają się na fakcie, że pochodna czasowa dwóch podstawowych wektorów jest niezerowa:

{displayStyle {frac {{text {d}} {boldsymbol {u}} _ {rho}} {{text {d}} t}} = {frac {{text {d}} varphi} {{text {d}} } t}} {Boldsymbol {u}} _ {varphi}}

{displayStyle {frac {{text {d}} {BoldSymbol {u}} _ {varphi}} {{text {d}} t}} =-{frac {{text {d}} varphi} {{text {d}}}} =-{frac {{text {d}} varphi} {{ }} t}} {BoldSymbol {u}} _ {rho}}

{DisplayStyle {BoldSymbol {r}} = r {BoldSymbol {u}} _ {r}}

{DisplayStyle {BoldSymbol {v}} = {frac {{text {d}} {boldsymbol {r}}} {{text {d}} t}} = {frac {{text {d}} r} {{text text {{text text} {d}} t}} {BoldSymbol {u}} _ {r}+r {frac {{text {d}} theta} {{text {d}} t}} {BoldSymbol {u}} _ {theta}}} +r {frac {{text {d}} varphi} {{text {d}} t}} sin theta {boldsymbol {u}} _ {varphi}}

{displayStyle {BoldSymbol {a}} = {frac {{text {d}} {boldsymbol {v}}} {{text {d}} t}} = {frac {{text {d}}^{2} {{d}} t}} = {frac {{text {text {d}}^{2} { Boldsymbol {r}}} {{text {d}} t^{2}}} = a_ {r} {boldsymbol {u}} _ {r}+a_ {theta} {boldsymbol {u}} _ {theta}}}}}}}}}}}}}}}}} +a_ {varphi} {boldsymbol {u}} _ {varphi}}

z:

{displayStyle a_ {r} = left ({frac {{text {d}}^{2} r} {{text {d}} t^{2}}}-rleft ({frac {{text {d}}}} theta} {{text {d}} t}} right)^{2}+rleft ({frac {{text {d}} varphi} {{text {d}} t}} right)^{2} sin^ {2} Theta right)}

{displayStyle a_ {theta} = left (r {frac {{text {d}}^{2} theta} {{text {d}} t^{2}}}+2 {frac {{text {d}}} r} {{text {d}} t}} {frac {{text {d}} theta} {{text {d}} t}}-rleft ({frac {{text {d}} varphi} {{text {text {text {text {text ext. {d}} t}} right)^{2} sin theta cos theta right)}

{displayStyle a_ {varphi} = left (r {frac {{text {d}}^{2} varphi} {{text {d}} t^{2}}} sin theta +2 {frac {{text {d. }} r} {{text {d}} t}} {frac {{text {d}} varphi} {{text {d}} t}} sin theta +2r {frac {{text {d}}} varphi} {{text {d}} t}} {frac {{text {d}} theta} {{text {d}} t}} cos theta)}}}

Albo punkt promienia wektora R w repozytorium

{DisplayStyle {Mathcal {r}}}

${mathcal R}$ . Albo inne repozytorium,

{DisplayStyle {Mathcal {r}}^{‘}}

${displaystyle {mathcal {R}}^{'}}$ , którego pochodzenie znajduje się w dziale wektorowym S W

{DisplayStyle {Mathcal {r}}}

${mathcal R}$ . Promień wektora punktu, określony w

{DisplayStyle {Mathcal {r}} ‘}

${mathcal R}'$ jest wtedy

{DisplayStyle {BoldSymbol {r}} ‘= {BoldSymbol {r}}-{BoldSymbol {s}}}}

Prędkości punktowe można zmierzyć w

{DisplayStyle {Mathcal {r}}}

${mathcal R}$ lub w

{DisplayStyle {Mathcal {r}} ‘}

${mathcal R}'$ . Są one odnotowane z indeksem

{DisplayStyle {Mathcal {r}}}

${mathcal R}$ Lub

{DisplayStyle {Mathcal {r}} ‘}

${mathcal R}'$ , a także przyspieszenia.

Wektor ilości ruchu:

${DisplayStyle {BoldSymbol {p}} = m {Boldsymbol {v}}}$
Podstawowa zasada dynamiki:

${displayStyle {frac {{text {d}} {boldsymbol {p}}} {{text {d}} t}} = sum {boldsymbol {f}};+{BoldSymbol {f}} _ {rm {i_ {i_ e}}}+{BoldSymbol {f}} _ {rm {i_ {c}}}}$
Zasada wzajemnych działań: dla dwóch ciał A I B W

${DisplayStyle {BoldSymbol {f}} _ {arightarrow b} =-{BoldSymbol {f}} _ {Brightarrow a}}$

Każda z tych energii jest zdefiniowana w pobliżu

Równanie różniczkowe formy:

${displayStyle {frac {{text {d}}^{2} u} {{text {d}} t^{2}}}+omega _ {0}^{2} u = 0}$
Czysta pulsacja:

${DisplayStyle Omega _ {0}^{2} = {frac {k} {m}}}$
Czysty okres:

${DisplayStyle t_ {0} = displayStyle {frac {2pi} {omega _ {0}}}}$
Rozwiązanie w formie:

${DisplayStyle u (t) = acos (omega _ {0} t)+bsin (omega _ {0} t)}$

Stałe A I B są określone przez warunki początkowe.

Równanie różniczkowe formy:

${displayStyle {frac {{text {d}}^{2} u} {{text {d}} t^{2}}}+2Lambda {frac {{text {d}} u} {{text {d}} } t}}+omega _ {0}^{2} u = 0}$
Trzy przypadki zgodnie z wartością dyskryminacji równania charakterystycznego:

${DisplayStyle delta = 4 (Lambda ^{2} -omega _ {0} ^{2})}$
W każdym przypadku stałe A I B są określone przez warunki początkowe.