Forță centrală

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În mecanică, forța centrală este o forță ce se exercită asupra unui punct material , al cărei suport trece în permanență printr-un punct fix și depinde numai de distanța până la acel punct, numit centru de forță.

Exemple: forța electrostatică, forța gravitațională, forța elastică.

Forța centrală este o forță conservativă.

Expresie matematică

Se definește forța centrală în raport cu un punct $O$ ca fiind un vector invariant la grupul mișcărilor plane ce lasă fix punctul $O.$ Deci dreapta suport a forței trece prin $O$ iar modului acesteia depinde doar de distanța de la punctul ei de aplicație la punctul $O.$

\mathbf {F} (\mathbf {x} )=F(r){\frac {\mathbf {x} }{r}},\;\;\mathbf {x} ={\overrightarrow {OP}},\;\;r=|\mathbf {x} |,

(1.1)

unde $P$ este punctul material considerat.

Dacă $F(r)<0$ forța centrală se numește atractivă, iar dacă $F(r)>0$ forța centrală se numește repulsivă.

Din formula (1.1) rezultă că $rot\;\mathbf {F} =0,$ deci forțele centrale sunt forțe conservative.

Expresia în coordonate polare

Dacă $(r,\theta )$ sunt coordonatele polare ale punctului $P$ atunci vectorul viteză poate fi scris:

{\vec {v}}=({\dot {r}},r{\dot {\theta }})\;

(în raport cu reperul

({\vec {e}}_{r},{\vec {e}}_{\theta })

)

(1.1.1)

Fie ${\vec {\rho }}={\frac {\vec {r}}{r}}={\frac {\vec {r}}{|{\vec {r}}|}}$ versorul vectorului de poziție ${\vec {r}}.$ Atunci:

{\vec {F}}=F{\vec {\rho }}=F{\frac {\vec {r}}{r}}.

(1.1.2)

Exemple

Forța elastică

Articol principal: Elasticitate.

În cazul forței elastice $\mathbf {F} (\mathbf {x} )=-k\mathbf {x} ,$ unde $k=const.>0$ se numește modul de elasticitate. Acest rezultat se bazează pe experimente (legea lui Hooke).

Potențialul forței elastice are forma:

\Pi (x)=-{\frac {k}{2}}\sum _{i=1}^{3}x_{i}^{2}+{\mathcal {C}}

(2.1.1)

unde $x_{i},\;i={\overline {1,3}}$ sunt componentele carteziene ale vectorului $\mathbf {x} .$

Forța de atracție universală

Articol principal: Gravitație.

Forța pe care un corp de masă $M$ o exercită asupra unui corp de masă $m$ este dată de legea lui Newton:

\mathbf {F} (\mathbf {x} )=-K{\frac {Mm}{r^{2}}}\cdot {\frac {\mathbf {x} }{r}},

(2.2.1)

unde $K$ este constanta atracției universale, care este determinată experimental și are valoarea:

K=6,673\times 10^{-11}{\frac {m^{3}}{kg\times s^{2}}}.

(2.2.2)

Potențialul forței de atracție universale are forma:

\Pi (\mathbf {x} )=K{\frac {Mm}{r}}+{\mathcal {C}}.

(2.2.3)

Conservarea momentului cinetic

Din teorema momentului cinetic $\left({\frac {d{\vec {K}}_{0}}{dt}}={\vec {M}}_{0}({\vec {F}})={\vec {r}}\times {\vec {F}}=0\right)$ rezultă ${\frac {d}{dt}}{\vec {r}}\times m{\vec {v}}=0.$

Se obține integrala primă a ariilor:

	${\vec {r}}\times {\vec {v}}={\vec {c}}={\vec {r}}_{0}\times {\vec {v}}_{0}$	$(3.1)$
	${\vec {r}}(t_{0})={\vec {r}}_{0},\;{\vec {v}}(t_{0})={\vec {v}}_{0}.$

Viteza areolară a punctului $P$ este:

{\frac {d{\vec {A}}}{dt}}={\frac {1}{2}}({\vec {r}}\times {\vec {v}})={\frac {\vec {c}}{2}},\;\forall t\geq t_{0},

(3.2)

deci viteza areolară este constantă.

Prin urmare, mișcarea punctului $P$ sub acțiunea forței centrale ${\vec {F}}$ are loc astfel încât momentul cinetic și viteza areolară sunt constante vectoriale, $\forall t\geq t_{0}.$

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